Pre-calculus Workbook for Dummies

Pre-calculus Workbook for Dummies
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Workbook
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by Michelle Rose Gilman,
Christopher Burger,
Karina Neal
Pre-Calculus Workbook For Dummies®
Published by
Wiley Publishing, Inc.
111 River St.
Hoboken, NJ 07030-5774
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Copyright © 2009 by Wiley Publishing, Inc., Indianapolis, Indiana
Published by Wiley Publishing, Inc., Indianapolis, Indiana
Published simultaneously in Canada
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About the Authors
Michelle Rose Gilman is proud to be known as Noah’s mom (Hi, Noah!). A
graduate of the University of South Florida, Michelle found her niche early —
at 19 she was already working with emotionally disturbed and learningdisabled students in hospital settings. At 21, she made the trek to California.
There she discovered her passion for helping teenage students become more
successful in school and life. What started as a small tutoring business in the
garage of her California home quickly expanded and grew to the point where
traffic control was necessary on her residential street.
Today, Michelle is the founder and CEO of the Fusion Learning Center/Fusion
Academy, a private school and tutoring/test-prep facility in Solana Beach,
California, serving more than 2,000 students per year. She has taught tens of
thousands of students since 1988. In her spare time, Michelle created the
Mentoring Approach to Learning and authored The ACT For Dummies,
Pre-Calculus For Dummies, AP Biology For Dummies, AP Chemistry For Dummies,
Chemistry Workbook For Dummies, and The GRE For Dummies. She currently
specializes in motivating the unmotivated adolescent, comforting shellshocked parents, and assisting her staff of 27 teachers.
Michelle lives by the following motto:
“There are people content with longing; I am not one of them.”
Christopher Burger graduated with a Bachelor of Arts degree in mathematics
from Coker College in Hartsville, South Carolina, with minors in art and theater. He has taught math for more than 10 years and has tutored subjects ranging from basic math to calculus for 20 years. He is currently the Director of
Independent Studies for Fusion Learning Center and Fusion Academy in
Solana Beach, California, where he not only teaches students one-on-one but
also writes curriculum, oversees a staff of 27 teachers, and maintains a high
level of academic rigor within the school. When not at school, Christopher
can be found in local theaters directing, acting, stage managing, or doing
pretty much any job that they’ll let him do. Chris is also one of the authors
of Pre-Calculus For Dummies.
Karina Neal graduated with a Bachelor of Science degree in combined sciences with an emphasis in psychology from Santa Clara University.
Additionally, she received her certificates in educational therapy and in college counseling from the University of California, San Diego. From an early age,
teaching and tutoring have been her passions — from starting her own tutoring business in high school to helping found the Fusion Learning Center and
Fusion Academy in Solana Beach, California. As that institution’s Director of
Tutoring and Mentoring, Karina teaches all levels of mathematics and science,
provides special education and college counseling consultation, and oversees
a staff of 27 tutors and teachers. Karina has over 18 years of experience in the
education field and continues to tutor and teach students in a wide range of
subjects, from remedial writing to calculus. Besides being a closet math and
science geek, Karina is dedicated to the success of her students and believes
that all students can learn.
Dedication
We would like to dedicate this book to every student we’ve ever taught —
each one of you taught us something in return. Also, to our families and
friends who supported us during the writing.
Authors’ Acknowledgments
We would like to acknowledge Bill Gladstone, our wonderful agent from
Waterside, for the opportunity to write this book; Nicholas Angelo for being
the scanning king; Natalie Harris, project editor extraordinaire; copy editor
Todd Lothery; technical editor David Herzog; acquisitions editors Tracy
Boggier and Erin Mooney, who, for unknown reasons, continue to want to
work with us; and to everyone who has lent a helping hand or eye or brain,
we couldn’t do it without you: TL, VS, BN, NG, and Clyde.
Publisher’s Acknowledgments
We’re proud of this book; please send us your comments through our Dummies online registration form located
at http://dummies.custhelp.com. For other comments, please contact our Customer Care Department
within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002.
Some of the people who helped bring this book to market include the following:
Acquisitions, Editorial, and Media Development
Composition Services
Project Editor: Natalie Faye Harris
Project Coordinator: Patrick Redmond
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Debbie Stailey, Director of Composition Services
Contents at a Glance
Introduction .................................................................................1
Part I: Foundation (And We Don’t Mean Makeup!) ........................5
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus................................................................7
Chapter 2: Get Real!: Wrestling with Real Numbers................................................................................25
Chapter 3: Understanding the Function of Functions ............................................................................41
Chapter 4: Go Back to Your Roots to Get Your Degree...........................................................................73
Chapter 5: Exponential and Logarithmic Functions ...............................................................................91
Part II: Trig Is the Key: Basic Review,
the Unit Circle, and Graphs .......................................................105
Chapter 6: Basic Trigonometry and the Unit Circle .............................................................................107
Chapter 7: Graphing and Transforming Trig Functions .......................................................................127
Part III: Advanced Trig: Identities, Theorems, and Applications .....143
Chapter 8: Basic Trig Identities ...............................................................................................................145
Chapter 9: Advanced Identities ...............................................................................................................161
Chapter 10: Solving Oblique Triangles ...................................................................................................177
Part IV: And the Rest . . ...........................................................193
Chapter 11: Complex Numbers and Polar Coordinates .......................................................................195
Chapter 12: Conquering Conic Sections.................................................................................................211
Chapter 13: Finding Solutions for Systems of Equations .....................................................................243
Chapter 14: Sequences, Series, and Binomials — Oh My!....................................................................275
Chapter 15: The Next Step Is Calculus ...................................................................................................287
Part V: The Part of Tens ............................................................299
Chapter 16: Ten Uses for Parent Graphs ................................................................................................301
Chapter 17: Ten Pitfalls to Pass Up in Pre-Calc .....................................................................................309
Index .......................................................................................313
Table of Contents
Introduction ..................................................................................1
About This Book.........................................................................................................................1
Conventions Used in This Book ...............................................................................................2
Foolish Assumptions .................................................................................................................2
How This Book Is Organized.....................................................................................................2
Part I: Foundation (And We Don’t Mean Makeup!).......................................................2
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs ............................2
Part III: Advanced Trig: Identities, Theorems, and Applications ...............................3
Part IV: And the Rest . . ...................................................................................................3
Part V: The Part of Tens...................................................................................................3
Icons Used in This Book............................................................................................................3
Where to Go from Here..............................................................................................................4
Part I: Foundation (And We Don’t Mean Makeup!) .........................5
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus ....................................7
Reviewing Order of Operations: The Fun in Fundamentals..................................................7
Keeping Your Balance While Solving Equalities...................................................................10
A Picture Is Worth a Thousand Words: Graphing Equalities and Inequalities .................12
Graphing using the plug and chug method.................................................................13
Graphing using the slope-intercept form ....................................................................13
Using Graphs to Find Information (Distance, Midpoint, Slope).........................................15
Finding the distance.......................................................................................................15
Calculating the midpoint ...............................................................................................16
Discovering the slope ....................................................................................................16
Answers to Problems on Fundamentals................................................................................20
Chapter 2: Get Real!: Wrestling with Real Numbers .....................................................25
Solving Inequalities..................................................................................................................25
Expressing Inequality Solutions in Interval Notations ........................................................28
Don’t Get Drastic with Radicals and Exponents — Just Simplify Them! ..........................30
Getting Out of a Sticky Situation or Rationalizing................................................................33
Answers to Problems on Real Numbers................................................................................36
Chapter 3: Understanding the Function of Functions.....................................................41
Battling Out Even versus Odd ................................................................................................41
Leaving the Nest: Transforming Parent Graphs...................................................................43
Quadratic functions .......................................................................................................43
Square root functions ....................................................................................................44
Absolute value functions...............................................................................................44
Cubic functions...............................................................................................................44
Cube root functions .......................................................................................................45
Table of Contents
Vertical transformations................................................................................................45
Horizontal transformations...........................................................................................46
Translations ....................................................................................................................46
Reflections.......................................................................................................................47
Combinations of transformations ................................................................................47
Lucid Thinking? Graphing Rational Functions .....................................................................50
Picking Up the Pieces: Graphing Piece-Wise Functions ......................................................53
Operating on Functions: No Scalpel Necessary ...................................................................54
Evaluating Composition of Functions....................................................................................56
Working Together: Domain and Range ..................................................................................58
Finding the Inverse of a Function (Who Knew It Was Lost?) ..............................................60
Answers to Questions on Functions......................................................................................62
Chapter 4: Go Back to Your Roots to Get Your Degree...................................................73
Reason Through It: Factoring a Factorable Polynomial......................................................73
Get Your Roots Done while Solving a Quadratic Polynomial .............................................76
Completing the square ..................................................................................................76
Quadratic formula ..........................................................................................................76
Climb the Mountains by Solving High Order Polynomials .................................................78
Determining positive and negative roots: Descartes’ Rule of Signs ........................78
Counting on imaginary roots ........................................................................................78
Getting the rational roots ..............................................................................................79
Synthetic division finds some roots.............................................................................79
Strike That! Reverse It! Using Roots to Find an Equation ...................................................81
Graphing Polynomials ...................................................................................................82
Answers to Questions on Finding Roots ...............................................................................86
Chapter 5: Exponential and Logarithmic Functions .......................................................91
Things Get Bigger (Or Smaller) All the Time – Solving Exponential Functions ...............91
The Only Logs You Won’t Cut: Solving Logarithms .............................................................93
Putting Them Together: Solving Equations Using Exponents and Logs ...........................96
Solving Exponential Word Problems . . . Argh! .....................................................................98
Answers to Problems on Exponential and Logarithmic Functions .................................100
Part II: Trig Is the Key: Basic Review,
the Unit Circle, and Graphs........................................................105
Chapter 6: Basic Trigonometry and the Unit Circle......................................................107
It’s All Right-Triangle Trig — Finding the Six Trigonometric Ratios................................107
Solving Word Problems with Right Triangles .....................................................................110
Unit Circle and the Coordinate Plane: Finding Points and Angles...................................112
Isn’t That Special? Finding Right Triangle Ratios on the Unit Circle...............................115
Solving Trig Equations...........................................................................................................117
Making and Measuring Areas ...............................................................................................119
Answers to Problems.............................................................................................................121
Chapter 7: Graphing and Transforming Trig Functions ................................................127
Getting a Grip on Period Graphs..........................................................................................127
Sine and Cosine: Parent Graphs and Transformations .....................................................128
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Pre-Calculus Workbook For Dummies
Tangent and Cotangent: Mom, Pops, and Children ...........................................................131
Secant and Cosecant: Generations ......................................................................................134
Answers to Problems on Graphing and Transforming Trig Functions............................137
Part III: Advanced Trig: Identities, Theorems, and Applications ....143
Chapter 8: Basic Trig Identities........................................................................................145
Using Reciprocal Identities to Simplify Trig Expressions .................................................145
Simplifying with Pythagorean Identities .............................................................................147
Discovering Even-Odd Identities..........................................................................................148
Solving with Co-Function Identities .....................................................................................149
Moving with Periodicity Identities.......................................................................................150
Tackling Trig Proofs ...............................................................................................................151
Answers to Problems on Basic Trig Identities ...................................................................153
Chapter 9: Advanced Identities........................................................................................161
Simplifying with Sum and Difference Identities..................................................................161
Using Double Angle Identities ..............................................................................................163
Reducing with Half-Angle Identities.....................................................................................165
Changing Products to Sums..................................................................................................166
Expressing Sums as Products...............................................................................................168
Powering Down: Power-Reducing Formulas.......................................................................169
Answers to Problems on Advanced Identities ...................................................................170
Chapter 10: Solving Oblique Triangles ...........................................................................177
Solving a Triangle with the Law of Sines: ASA and AAS ....................................................177
Tackling Triangles in the Ambiguous Case: SSA ................................................................179
Conquering a Triangle with the Law of Cosines: SAS and SSS .........................................180
Using Oblique Triangles to Solve Word Problems .............................................................182
Figuring Flatness (Area) ........................................................................................................184
Answers to Problems on Solving Triangles ........................................................................186
Part IV: And the Rest . . ............................................................193
Chapter 11: Complex Numbers and Polar Coordinates ...............................................195
Performing Operations with and Graphing Complex Numbers .......................................195
Round a Pole: Graphing Polar Coordinates ........................................................................199
Changing to and from Polar ..................................................................................................202
Graphing Polar Equations .....................................................................................................204
Archimedean spiral ......................................................................................................204
Cardioid .........................................................................................................................204
Rose................................................................................................................................204
Circle ..............................................................................................................................204
Lemniscate ....................................................................................................................204
Limaçon .........................................................................................................................205
Answers to Problems on Complex Numbers and Polar Coordinates..............................207
Table of Contents
Chapter 12: Conquering Conic Sections.........................................................................211
A Quick Conic Review............................................................................................................211
Going Round and Round with Circles..................................................................................212
Graphing Parabolas: The Ups and Downs ..........................................................................213
Standing tall: Vertical parabolas.................................................................................214
Lying down on the job: Horizontal parabolas ..........................................................216
Graphing Ellipses: The Fat and the Skinny .........................................................................218
Short and fat: The horizontal ellipse .........................................................................219
Tall and skinny: The vertical ellipse ..........................................................................220
Graphing Hyperbolas: No Caffeine Required......................................................................222
Hip horizontal hyperbolas (alliteration!) ..................................................................223
Vexing vertical vyperbolas (er, hyperbolas).............................................................225
Identifying Conic Sections ....................................................................................................227
Converting from Parametric Form to Polar Coordinates and Back.................................230
Parametric form for conic sections ...........................................................................230
Changing from parametric form to rectangular form ..............................................232
Conic sections on the polar coordinate plane..........................................................233
Answers to Problems on Conic Sections ............................................................................235
Chapter 13: Finding Solutions for Systems of Equations.............................................243
A Quick-and-Dirty Technique Overview..............................................................................243
Solving Two Linear Equations with Two Variables............................................................244
The substitution method.............................................................................................244
The elimination method ..............................................................................................245
Not-So-Straight: Solving Nonlinear Systems .......................................................................247
One linear equation and one not................................................................................247
Two equations that are nonlinear ..............................................................................248
Systems of equations disguised as rational equations ...........................................248
Solving More than Two (Equations and/or Variables) ......................................................249
Graphing Systems of Inequalities.........................................................................................251
We’re Partial to . . . Decomposing Partial Fractions! .........................................................253
There Is No Spoon: Working with a Matrix .........................................................................255
Getting It in the Right Form: Simplifying Matrices.............................................................257
Solving Systems of Equations Using Matrices....................................................................259
Gaussian elimination....................................................................................................260
Inverse matrices ...........................................................................................................261
Cramer’s Rule................................................................................................................262
Answers to Problems on Systems of Equations.................................................................264
Chapter 14: Sequences, Series, and Binomials — Oh My! ........................................275
Major General Sequences and Series: Calculating Terms.................................................275
Working Out the Common Difference: Arithmetic Sequences and Series ......................277
Be Fruitful and Multiply: Simplifying Geometric Sequences and Series .........................279
Expanding Polynomials Using the Binomial Theorem ......................................................281
Answers to Problems on Sequences, Series, and Binomials ............................................283
Chapter 15: The Next Step Is Calculus ...........................................................................287
Finding Limits: Graphically, Analytically, and Algebraically ............................................287
Graphically ....................................................................................................................288
Analytically....................................................................................................................289
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Pre-Calculus Workbook For Dummies
Algebraically .................................................................................................................290
Knowing Your Limits..............................................................................................................292
Determining Continuity .........................................................................................................293
Answers to Problems on Calculus .......................................................................................296
Part V: The Part of Tens .............................................................299
Chapter 16: Ten Uses for Parent Graphs .........................................................................301
Squaring Up with Quadratics ...............................................................................................301
Cueing Up for Cubics .............................................................................................................302
Rooting for Square Roots and Cube Roots .........................................................................302
Graphing Absolutely Fabulous Absolute Value Functions................................................303
Flipping over Rational Functions .........................................................................................304
Exploring Exponential Graphs and Logarithmic Graphs ..................................................304
Seeing the Sine and Cosine ...................................................................................................305
Covering Cosecant and Secant.............................................................................................306
Tripping over Tangent and Cotangent ................................................................................306
Chapter 17: Ten Pitfalls to Pass Up in Pre-Calc ............................................................309
Going Out of Order (Of Operations) ....................................................................................309
FOILed Again! FOILing Binomials Incorrectly.....................................................................309
Gettin’ Down with Denominators.........................................................................................310
Combining Terms That Can’t Be Combined .......................................................................310
Forgetting to Flip the Fraction..............................................................................................310
Losing the Negative (Sign) ....................................................................................................311
Oversimplifying Roots ...........................................................................................................311
Avoiding Exponent Errors.....................................................................................................311
Canceling Too Quickly...........................................................................................................312
Dealing with Distribution ......................................................................................................312
Index ........................................................................................313
Introduction
Y
ou’ve just picked up the best workbook ever to help you with pre-calculus, if we do say
so ourselves. If you’ve gotten this far in your math career, congratulations! Many students choose to stop their math education after they complete Algebra II, but not you!
If you’ve picked up this book (obviously you have; you’re reading this sentence, duh!),
maybe some of the concepts in pre-calc are giving you a hard time, or perhaps you just want
more practice. Maybe you’re deciding whether you even want to take pre-calc at all. This
book fits the bill for all those reasons. And we’re here to cheerlead you on during your precalc adventure. Look, if you’ve gotten this far in math, you’re no “Dummie,” so don’t let the
title throw you!
We know that you’ll find this workbook chock-full of valuable practice problems and explanations. In instances where you feel you may need a more thorough explanation, please refer to
Pre-Calculus For Dummies (we wrote that one too — yes, we are math geeks). In some areas of
the book, we even refer you to Pre-Calculus For Dummies ourselves. We set up this workbook
to directly coincide with the format of Pre-Calculus For Dummies in an effort to make it really
easy for you to use the two together, if you wish. This book, however, is a great stand-alone
workbook if you need extra practice, need just a brushup in certain areas, or just can’t stand
our jokes in the other book.
About This Book
Don’t let pre-calc scare you. When you realize that you already know a whole bunch from
Algebra I and Algebra II, you’ll see that pre-calculus is really just using that old information in
a new way. And even if you’re scared, we’re here with you, so no need to panic. Before you
get ready to start this new adventure, you need to know a few things about this book.
This book isn’t a novel. It’s not meant to be read in order from beginning to end. You can read
any topic at any time, but we’ve structured it in such a way that it follows the “normal” curriculum. This is hard to do because most states don’t have state standards for what makes
pre-calc pre-calc. We looked at a bunch of curriculums, though, and came up with what we
think is a good representation of a Pre-Calc course. Sometimes, we may include a reference
to material in another chapter, and we may send you there for more information.
Instead of placing this book on a shelf and never looking at it again, or using it as a doorstop
(thanks for the advertisement, in either case), we suggest you follow one of two alternatives:
⻬ Look up what you need to know when you need to know it. The index, table of contents, and even the contents at a glance section will all direct you where to look.
⻬ Start at the beginning and read through. This way, you may be reminded of an old topic
that you had forgotten (anything to get those math wheels churning inside your head).
Besides, practice makes perfect, and the problems in this book are a great representation of the problems found in pre-calc textbooks.
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Pre-Calculus Workbook For Dummies
Conventions Used in This Book
For consistency and ease of navigation, this book uses the following conventions:
⻬ Math terms are italicized when they’re introduced or defined in the text.
⻬ Variables are italicized to set them apart from letters.
⻬ The symbol for imaginary numbers is a lowercase i.
Foolish Assumptions
We don’t assume that you love math the way we do as professional math geeks. We do
assume, though, that you picked this book up for a reason of your own. Maybe you
want a preview of the course before you take it, or perhaps you need a refresher on the
topics in the course, or maybe your kid is taking the course and you’re trying to help
him be successful.
Whatever your reason, we assume that you’ve encountered most of the topics in this
book before, because for the most part, the topics are reviews of ones you’ve seen in
algebra or geometry.
How This Book Is Organized
This book is divided into five parts dealing with the most commonly taught topics of
pre-calc.
Part I: Foundation (And We
Don’t Mean Makeup!)
The chapters in Part I begin at the beginning. First we review basic material from
Algebra II. We then cover real numbers and what you’ll be asked to do with them. Next
up are functions of all kinds (polynomials, rational, exponential, and logarithmic):
graphing them and performing operations with them.
Part II: Trig Is the Key: Basic Review,
the Unit Circle, and Graphs
The chapters in Part II review trig ratios and word problems for trig. Then we show
you how to build the unit circle, how to solve trig equations, and how to graph trig
functions. Some of these topics may be review for you as well; that really depends on
how much trig was covered in your Algebra II course.
Introduction
Part III: Advanced Trig: Identities,
Theorems, and Applications
The chapters in Part III cover basic and advanced identities. We cover the tricky
trig proofs in this part. If you’re asked to do trig proofs in your Pre-Calc course,
you definitely want to check out our tips on how to handle them like a pro. We also
cover some trig applications that can be solved using the Law of Sines or the Law
of Cosines.
Part IV: And the Rest . . .
The chapters in Part IV cover the topics from the remainder of the Pre-Calc
course. We introduce complex numbers and how to work with them, and we
explain conic sections and how to graph them. Because systems of equations tend
to get harder in pre-calc, we begin with a review and build up to the tougher
topics. Your Pre-Calc course may only focus on a couple of these topics, so be sure
to pay attention to the table of contents here. Next, we move into sequences and
series and introduce the binomial theorem, which helps you raise binomials to
high powers. Last, we introduce the first topics of a calc course. Sometimes, these
are the last topics you’ll see in pre-calc, so we want to be sure to go over them.
Part V: The Part of Tens
This book has two handy lists at the end. The first list includes ten parent graphs:
how to recognize them, how to graph them, and how to transform them. The
second list covers common mistakes we often see that we’d like to help you avoid.
Icons Used in This Book
Throughout this book you’ll see icons in the margins to draw your attention to
something important that you need to know.
Pre-calc rules are exactly what they say they are — the rules of pre-calculus.
Theorems, laws, and properties all make Pre-Calc an ironclad course — they must
be followed at all times.
Tips are great, especially if you wait tables for a living! These tips are designed to
make your life easier, which are the best tips of all!
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Pre-Calculus Workbook For Dummies
The Remember icon is used one way: It asks you to remember old material from a previous math course.
Warnings are big red flags that draw your attention to common mistakes you may get
tripped up on.
Where to Go from Here
Pick a starting point in the book and go practice the problems there. If you’d like to
review the basics first, start at Chapter 1. If you feel comfy enough with your algebra
skills, you may want to skip that chapter and head over to Chapter 2. Most of the
topics there are reviews of Algebra II material, but don’t skip over something because
you think you’ve got it under control. You’ll also find in pre-calc that the level of difficulty in some of these topics gets turned up a notch or two. Go ahead — dive in and
enjoy the world of pre-calc!
Part I
Foundation (And We Don’t
Mean Makeup!)
P
In this part . . .
re-calculus is really just another stop on the road to
calculus. You started with the village of Algebra I,
moved on to the small town of Geometry, made your way
to Algebra II city, and now find yourself in the megametropolis known as Pre-Calculus. The skills, for the most
part, are the same. This part takes those skills and reviews
them (and, in some cases, expands on them).
The chapters here begin with a review of the basics: using
the order of operations, solving and graphing equations
and inequalities, and using the distance and midpoint formulas. Some new material pops up in the form of interval
notation, so be sure and check that out. Then we move on
to real numbers, including radicals. Everything you ever
wanted to know about functions is covered in one of the
chapters: graphing and transforming parent graphs, rational functions, and piece-wise functions. We also go over
performing operations on functions and how to find the
inverse. We then move on to solving higher degree polynomials using techniques like factoring, completing the
square, and the quadratic formula. You also learn how to
graph these complicated polynomials. Lastly, you discover
exponential and logarithmic functions and what you’re
expected to know about them.
Chapter 1
Beginning at the Very Beginning:
Pre-Pre-Calculus
In This Chapter
䊳 Brushing up on order of operations
䊳 Solving equalities
䊳 Graphing equalities and inequalities
䊳 Finding distance, midpoint, and slope
P
re-calculus is the stepping stone for Calculus. It’s the final hurdle after all those years of
math: Pre-algebra, Algebra, Geometry, and Algebra II. Now all you need is Pre-calculus
to get to that ultimate goal — Calculus. And as you may recall from your Algebra II class, you
were subjected to much of the same material you saw in Algebra and even Pre-algebra (just
a couple steps up in terms of complexity — but really the same stuff). As the stepping stone,
pre-calculus begins with certain concepts that you’re expected to solidly understand.
Therefore, we’re starting here, at the very beginning, reviewing those concepts. If you feel
you’re already an expert at everything algebra, feel free to skip past this chapter and get the
full swing of pre-calc going. If, however, you need to review, then read on.
If you don’t remember some of the concepts we discuss in this chapter, or even in this book,
you can pick up another For Dummies math book for review. The fundamentals are important. That’s why they’re called fundamentals. Take the time now to review — it will save you
countless hours of frustration in the future!
Reviewing Order of Operations:
The Fun in Fundamentals
You can’t put on your sock after you put on your shoe, can you? The same concept applies
to mathematical operations. There’s a specific order to which operation you perform first,
second, third, and so on. At this point, it should be second nature. However, because the
concept is so important as we continue into more complex calculations, we review it here.
8
Part I: Foundation (And We Don’t Mean Makeup!)
Please excuse who? Oh, yeah, you remember this one — my dear Aunt Sally! The old
mnemonic still stands, even as you get into more complicated problems. Please
Excuse My Dear Aunt Sally is a mnemonic for the acronym PEMDAS, which stands for:
⻬ Parentheses (including absolute value, brackets, and radicals)
⻬ Exponents
⻬ Multiplication and Division (from left to right)
⻬ Addition and Subtraction (from left to right)
The order in which you solve algebraic problems is very important. Always work
what’s in the parentheses first, then move on to the exponents, followed by the multiplication and division (from left to right), and finally, the addition and subtraction
(from left to right). Because we’re reviewing fundamentals, now is also a good time to
do a quick review of properties of equality.
When simplifying expressions, it’s helpful to recall the properties of numbers:
⻬ Reflexive property: a = a. For example, 4 = 4.
⻬ Symmetric property: If a = b, then b = a. For example, if 2 + 8 = 10, then 10 = 2 + 8.
⻬ Transitive property: If a = b and b = c, then a = c. For example, if 2 + 8 = 10 and
10 = 5 · 2, then 2 + 8 = 5 · 2.
⻬ Commutative property of addition (and of multiplication): a + b = b + a. For
example, 3 + 4 = 4 + 3.
⻬ Commutative property of multiplication: a · b = b · a. For example, 3 · 4 = 4 · 3.
⻬ Associative property of addition (and of multiplication): a + (b + c) = (a + b) + c.
For example, 3 + (4 + 5) = (3 + 4) + 5.
⻬ Associative property of multiplication: a · (b · c) = (a · b) · c. For example,
3 · (4 · 5) = (3 · 4) · 5.
⻬ Additive identity: a + 0 = a. For example, 4 + 0 = 4.
⻬ Multiplicative identity: a · 1 = a. For example, –18 · 1 = –18.
⻬ Additive inverse property: a + (–a) = 0. For example, 5 + –5 = 0.
⻬ Multiplicative inverse property:
. For example, –2 · (–1⁄2) = 1.
⻬ Distributive property: a(b + c) = a · b + a · c. For example, 5(3 + 4) = 5 · 3 + 5 · 4.
⻬ Multiplicative property of zero: a · 0 = 0. For example. 4 · 0 = 0.
⻬ Zero product property: If a · b = 0, then a = 0 or b = 0. For example, if x(2x – 3) = 0,
then x = 0 or 2x – 3 = 0.
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
Q.
Simplify:
A.
The answer is 5. Following our rules of
order of operations, simplify everything in
parentheses first.
.
Radicals and absolute value marks
act like parentheses. Therefore, if
any of the operations are under radicals or within absolute value marks,
do those first before simplifying the
radicals or taking the absolute value.
Q.
Simplify:
A.
The answer is 3. Using the associative
property of addition, rewrite the expression to make the fractions easier to add:
.
. Add the fractions with
Simplify the parentheses by taking the
square root of 25 and the absolute value
common denominators,
of –4:
reduce the resulting fraction:
=
=
. Now that the parentheses are
, and
. Next,
find a common denominator for the fractions in the numerator and denominator:
simplified, you can deal with the exponents.
. Add these:
Square the 6 and the –2: =
Although they’re not written, parentheses are implied around the
terms above and below a fraction
bar. In other words, the expression
this expression is a division problem,
, multiply by the inverse and
simplify:
can also be written as
. Therefore, you must
simplify the numerator and denominator before dividing the terms following the order of operations:
=
=
. Recognizing that
.
=
= 5.
=
=
=
= 3.
9
10
Part I: Foundation (And We Don’t Mean Makeup!)
1.
Simplify:
2.
Simplify:
.
Solve It
Solve It
3.
.
Simplify: (23 – 32)4(–5).
4.
Simplify:
.
Solve It
Solve It
Keeping Your Balance While
Solving Equalities
Just as simplifying expressions is the basics of pre-algebra, solving for variables is the
basics of algebra. Both are essential to more complex concepts in pre-calculus. Solving
basic algebraic equations should be easy for you; however, it’s so fundamental to precalculus, we give you a brief review here.
Solving linear equations with the general format of ax + b = c, where a, b, and c are constants, is relatively easy using properties of numbers. The goal, of course, is to isolate
the variable, x.
One type of equation you can’t forget is absolute value equations. The absolute value is
defined as the distance from 0. In other words,
. As such, an absolute
value has two possible solutions: one where the quantity inside the absolute value
bars is positive and another where it’s negative. To solve these equations, it’s important to isolate the absolute value term and then set the quantity to the positive and
negative values.
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
Q.
Solve for x: 3(2x – 4) = x – 2(–2x + 3).
Q.
Solve for x:
A.
x = 6. First, using the distributive property,
distribute the 3 and the –2: 6x – 12 = x + 4x
– 6. Combine like terms and solve using
algebra: 6x – 12 = 5x – 6; x – 12 = –6; x = 6.
A.
x = 7, –1. First, isolate the absolute value:
Solve: 3 – 6[2 – 4x(x + 3)] = 3x(8x + 12) + 27.
6.
5.
Solve It
7.
Solve
Solve It
.
. Next, set the quantity inside the
absolute value bars to the positive solution: x – 3 = 4. Then, set the quantity inside
the absolute value bars to the negative
solution: –(x – 3) = 4. Solve both equations
to find two possible solutions: x – 3 = 4,
x = 7; and –(x – 3) = 4, x – 3 = –4, x = –1.
Solve
.
Solve It
.
8.
Solve 3 – 4(2 – 3x) = 2(6x + 2).
Solve It
11
12
Part I: Foundation (And We Don’t Mean Makeup!)
9.
Solve
Solve It
.
10.
Solve 3(2x + 5) + 10 = 2(x + 10) + 4x + 5.
Solve It
A Picture Is Worth a Thousand Words:
Graphing Equalities and Inequalities
Graphs are visual representations of mathematical equations. In pre-calculus, you’ll be
introduced to many new mathematical equations and then be expected to graph them.
We give you lots of practice graphing these equations when we cover the more complex equations. In the meantime, it’s important to practice the basics: graphing linear
equalities and inequalities.
These graphs are graphed on the Cartesian coordinate system. This system is made
up of two axes: the horizontal, or x-axis, and the vertical, or y-axis. Each point on the
coordinate plane is called a Cartesian coordinate pair (x, y). A set of these ordered
pairs that can be graphed on a coordinate plane is called a relation. The x values of a
relation are its domain, and the y values are its range. For example, the domain of the
relation R={(2, 4), (–5, 3), (1, –2)} is {2, –5, 1}, and the range is {4, 3, –2}.
You can graph a linear equation in two ways: plug and chug or use slope-intercept form:
y = mx + b. At this point in math, you should definitely know how to use the slopeintercept form, but we give you a quick review of the plug and chug method, because
as the equations become more complex, you can use this old standby method to get
some key pieces of information.
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
Graphing using the plug and chug method
Start by picking domain (x) values. Plug them into the equation to solve for the range
(y) values. For linear equations, after you plot these points (x, y) on the coordinate
plane, you can connect the dots to make a line. The process also works if you choose
range values first, then plug in to find the corresponding domain values. This is a
helpful method to find intercepts, the points that fall on the x or y axes. To find the
x-intercept (x, 0), plug in 0 for y and solve for x. To find the y-intercept (0, y), plug in
0 for x and solve for y. For example, to find the intercepts of the linear equation
2x + 3y = 12, start by plugging in 0 for y: 2x + 3(0) = 12. Then, using properties of
numbers, solve for x: 2x + 0 = 12, 2x = 12, x = 6. So the x-intercept is (6, 0). For the
y-intercept, plug in 0 for x and solve for y: 2(0) + 3y = 12, 0 + 3y = 12, 3y = 12, y = 4.
Therefore, the y-intercept is (0, 4). At this point, you can plot those two points and
connect them to graph the line (2x + 3y = 12), because, as you learned in geometry,
two points make a line. See the resulting graph in Figure 1-1.
y
8
4
–8
Figure 1-1:
Graph of
2x + 3y = 12.
–4
0
4
8 x
–4
–8
Graphing using the slope-intercept form
The slope-intercept form of a linear equation gives a great deal of helpful information
in a cute little package. The equation y = mx + b immediately gives you the y-intercept
(b) that you worked to find in the plug and chug method; it also gives you the slope
(m). Slope is a fraction that gives you the rise over the run. To change equations that
aren’t written in slope-intercept form, you simply solve for y. For example, if you use
the same linear equation as before, 2x + 3y = 12, you start by subtracting 2x from each
. Now that the equaside: 3y = –2x + 12. Next, you divide all the terms by 3:
tion is in slope-intercept form, you know that the y-intercept is 4. You can graph this
point on the coordinate plane. Then, you can use the slope to plot the second point.
From the slope-intercept equation, you know that the slope is
. This tells you that
the rise is –2 and the run is 3. From the point (0, 4), plot the point 2 down and 3 to the
right. In other words, (3, 2). Lastly, connect the two points to graph the line. Note that
this is the exact same graph, just plotted a different way — the resulting graph in
Figure 1-2 is identical to Figure 1-1.
13
14
Part I: Foundation (And We Don’t Mean Makeup!)
y
8
4
–8
–4
0
4
8 x
–4
Figure 1-2:
Graph of
.
–8
Similar to graphing equalities, graphing inequalities begins with plotting two points by
either method. However, because inequalities are used for comparisons — greater
than, less than, or equal to — you have two more questions to answer after two points
are found:
⻬ Is the line dashed: < or > or solid: or ?
⻬ Do you shade under the line: y < or y
Q.
Sketch the graph of the inequality:
3x – 2y > 4.
A.
Begin by putting the equation into slopeintercept form. To do this, subtract 3x from
each side of the equation: –2y > –3x + 4.
.
Then divide each term by –2: y <
or above the line: y > or y ?
From the resulting equation, you can find
the y-intercept, –2, and the slope, ( ).
Using this information, you can graph two
points using the slope-intercept form
method. Next, you need to decide the
nature of the line (solid or dashed).
Because the inequality is not also an equality, the line is dashed. Graph the dashed
line, and then you can decide where to
shade. Because the inequality is less than,
shade below the dashed line, as you see in
Figure 1-3.
Remember that when you multiply
or divide an inequality by a
negative, you need to reverse
the inequality.
10
y
5
–10
Figure 1-3:
Graph of 3x
– 2y > 4.
–5
0
–5
–10
5
10
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
11.
Sketch the graph of
12.
.
Solve It
13.
Sketch the graph of
.
Solve It
Sketch the graph of
Solve It
.
14.
Sketch the graph of x – 3y = 4 – 2y – y.
Solve It
Using Graphs to Find Information
(Distance, Midpoint, Slope)
Graphs are more than just pretty pictures. From a graph, it’s possible to determine two
points. From these points, you can determine the distance between them, the midpoint of the segment connecting them, and the slope of the line connecting them. As
graphs become more complex in both pre-calculus and calculus, you’ll be asked to find
and use all three of these pieces of information. Aren’t you lucky?
Finding the distance
Distance is how far two things are apart. In this case, you’re finding the distance
between two points. Knowing how to calculate distance is helpful for when you get to
conics (Chapter 12). To find the distance between two points (x1, y1) and (x2, y2), you
can use the following formula:
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Part I: Foundation (And We Don’t Mean Makeup!)
Calculating the midpoint
The midpoint, as you would think, is the middle of a segment. This concept also comes
up in conics (Chapter 12) and is ever so useful for all sorts of other pre-calculus calculations. To find the midpoint of those same two points (x1, y1) and (x2, y2), you just need
to average the x and y values and express them as an ordered pair:
Discovering the slope
Slope is a key concept for linear equations, but it also has applications for trigonometric functions and is essential for differential calculus. Slope describes the steepness
of a line on the coordinate plane (think of a ski slope). To find the slope of two points
(x1, y1) and (x2, y2), you can use the following formula:
Positive slopes move up and to the right
slopes move down and to the right
or down and to the left
or up and to the left
have a slope of 0, and vertical lines have an undefined slope.
Q.
Find the distance, slope, and midpoint of
A (5, 3)
B (–2, –1)
Figure 1-4:
Segment
AB.
in Figure 1-4.
. Negative
. Horizontal lines
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
A.
The distance is
, the slope is 4⁄7, and the midpoint is M = (3⁄2, 1). First, plug the x and y values
into the distance formula. Then, following the order of operations, simplify the terms under the
radical. (Keep in mind those implied parentheses of the radical itself.) It should look something
like this:
=
=
=
=
Because 65 doesn’t contain any perfect squares as factors, this is as simple as you can get.
To find the midpoint, plug the points into the midpoint equation. Again, simplify using order of
operations.
=
=
To find the slope, use the formula and plug in your x and y values. Using order of operations,
simplify:
m=
15.
Find the distance of segment CD,
where C is (–2, 4) and D is (3, –1).
Solve It
16.
Find the midpoint of segment EF,
where E is (3, –5) and F is (7, 5).
Solve It
17
18
Part I: Foundation (And We Don’t Mean Makeup!)
17.
Find the slope of line GH, where G is
(–3, –5) and H is (–3, 4).
Solve It
18.
Find the perimeter of triangle CAT.
Solve It 0
C (4, 6)
A (5, –1)
T (2, –4)
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
19.
Find the center of the rectangle NEAT.
Solve It
20.
Determine whether triangle DOG is a right
triangle.
Solve It
N (7, 8)
D (–3, 4)
T (3, 4)
E (15, 0)
G (9, 0)
A (11, –4)
O (1, –4)
19
20
Part I: Foundation (And We Don’t Mean Makeup!)
Answers to Problems on Fundamentals
a
Simplify
. The answer is
.
Start by simplifying everything in the parentheses. Next, simplify the exponents. Finally, add
the remaining terms. It should look something like this:
=
b
Simplify
=
=
=
. The answer is 0.
Recognizing that the absolute value in the denominator acts as parentheses, add the –7 and 2
inside there first. Then, you can rewrite the absolute value of each. Next, add the terms in the
numerator. Finally, recognize that
=
c
equals zero.
=
=
=0
Simplify (23 – 32)4(–5). The answer is –5.
Begin by simplifying the exponents in the parentheses. Next, simplify the parentheses by subtracting 9 from 8. Then, simplify the –1 to the 4th power. Finally, multiply the resulting 1 by –5.
(23 – 32)4(–5) = (8 – 9)4(–5) = (–1)4(–5) = 1(–5) = –5
d
Simplify
. The answer is undefined.
Start by simplifying the parentheses. To do this, subtract 4 from 1 in the numerator and find a
common denominator for the fractions in the denominator in order to add them. Next, multiply
the terms in the numerator and denominator. Then, add the terms in the absolute value bars
in the numerator and subtract the terms in the denominator. Take the absolute value of –9 to
simplify the numerator. Finally, remember that you can’t have 0 in the denominator; therefore,
the resulting fraction 9⁄0 is undefined.
=
e
=
=
= Undefined
Solve 3 – 6[2 – 4x(x + 3)] = 3x(8x + 12) + 27. The answer is x = 1.
Lots of parentheses in this one! Get rid of them by distributing terms. Start by distributing
the –4x on the left side over (x + 3) and, on the right side, 3x over (8x + 12). This gives you
3 – 6[2 – 4x2 – 12x] = 24x2 + 36x + 27. Then distribute the –6 over the remaining parentheses on
the left side of the equation: 3 – 12 + 24x2 + 72x = 24x2 + 36x + 27. Combine like terms on the left
side: –9 + 24x2 + 72x = 24x2 + 36x + 27. To isolate x onto one side, subtract 24x2 from each side to
get –9 + 72x = 36x + 27. Subtracting 36x from each side gives you –9 + 36x = 27. Adding 9 to both
sides results in 36x = 36. Finally, dividing both sides by 36 leaves you with your solution: x = 1.
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
f
Solve
. The answer is x = 10.
Don’t let those fractions intimidate you! Start by multiplying through by the common denominator, 4. This eliminates the fractions altogether. Now, just solve like normal, combining like
terms, and isolating x. It should look something like this:
; 2x + x – 2 = 2x + 8; 3x – 2 = 2x + 8; 3x = 2x + 10; x = 10
;
g
Solve
. The answer is x =
,
.
Okay, this one is really tricky! Two absolute value terms, oh my! Relax. Just remember that
absolute value means distance from 0, so you have to consider all the possibilities to solve this
problem. In other words, you have to consider and try four different possibilities: both absolute
values are positive, both are negative, the first is positive and the second is negative, and the
first is negative and the second is positive.
Not all these possibilities are going to work. As you calculate these possibilities, you may
create what math people call extraneous solutions. These aren’t solutions at all — they’re false
solutions that don’t work in the original equation. You create extraneous solutions when you
change the format of an equation, as you’re going to do here. So to be sure a solution is real and
not extraneous, you need to plug your answer into the original equation to check.
Now, try each of the possibilities:
Positive/positive: (x – 3) + (3x + 2) = 4, 4x – 1 = 4, 4x = 5, x = 5⁄4. Plugging this back into the original equation, you get 30⁄4 = 4. Nope! You have an extraneous solution.
Negative/negative: –(x – 3) + –(3x + 2) = 4, –x + 3 – 3x – 2 = 4, –4x + 1 = 4, –4x = 3, x =
back into the original equation and you get 4 = 4. Voilà! Your first solution.
. Plug it
Positive/negative: (x – 3) + –(3x + 2) = 4, x – 3 – 3x – 2 = 4, –2x – 5 = 4, –2x = 9, x =
. Put it back
into the original equation and you get 12 = 4. Nope, again — another extraneous solution.
Negative/positive: –(x – 3) + (3x + 2) = 4, –x + 3 + 3x + 2 = 4, 2x + 5 = 4, 2x = –1, x =
original equation it goes, and you get 4 = 4. Your second solution.
h
. Into the
Solve 3 – 4(2 – 3x) = 2(6x + 2). The answer is no solution.
To solve, distribute over the parentheses on each side: 3 – 8 + 12x = 12x + 4. Combine like terms:
–5 + 12x = 12x + 4. Subtract 12x from each side and you get –5 = 4, which is false. So there is no
solution.
i
Solve
. The answer is no solution.
Start by isolating the absolute value:
,
,
. Because an
absolute value must be positive, there is no solution that would satisfy this equation.
j
Solve 3(2x + 5) + 10 = 2(x + 10) + 4x + 5. The answer is all real numbers.
Begin by distributing over the parentheses on each side: 3(2x + 5) + 10 = 2(x + 10) + 4x + 5,
6x + 15 + 10 = 2x + 20 + 4x + 5. Next, combine like terms on each side: 6x + 25 = 6x + 25.
Subtracting 6x from each side gives you 25 = 25. This is a true statement, indicating that all
real numbers would satisfy this equation.
k
Sketch the graph of
. See the graph for the answer.
Using slope-intercept form, you start by multiplying both sides of the equation by the inverse
. This leaves you with 6x + 2y = 12. Next, solve for y by
of 4⁄3, which is 3⁄4:
subtracting 6x from each side and dividing by 2: 2y = –6x + 12, y = –3x + 6. Now, because it’s in
slope-intercept form, you can identify the slope (–3) and y intercept (6). Use these to graph the
21
22
Part I: Foundation (And We Don’t Mean Makeup!)
equation. Start at the y intercept (0, 6) and move down 3 units and to the right 1 unit. Connect
the two points to graph the line.
y
8
4
–8
–4
4
0
8 x
–4
–8
l
Sketch the graph of
. See the graph for the answer.
Start by multiplying both sides of the equation by 2:
. Next, isolate y by subtracting
,
. Now that it’s in slope-intercept
5x from each side and dividing by 4:
form, you can graph the inequality. Because it’s greater than or equal to, draw a solid line and
shade above the line.
10
y
5
–10
–5
0
5
10
–5
–10
m
Sketch the graph of
. See the graph for the answer.
Again, start by getting the equation into slope-intercept form. To do this, distribute the 2 on the
left side. Next, isolate y by subtracting 4x from each side, subtracting y from each side, and
then dividing by –1 (don’t forget to switch your inequality sign!):
;
;
;
;
Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus
Because there’s no x term, this indicates that the slope is 0 (0x). Therefore, the resulting line is
a horizontal line at –8. Because the inequality is less than, you shade below the line.
10
y
5
–10
–5
0
5
10
–5
–10
n
Sketch the graph of x – 3y = 4 – 2y – y. See the graph for the answer.
Again, simplify to put in slope-intercept form. Combine like terms and add 3y to each side.
x – 3y = 4 – 2y – y; x – 3y = 4 – 3y; x = 4
Here, the resulting line is a vertical line at 4.
y
8
4
–8
–4
4
0
8
–4
–8
o
Find the distance of segment CD, where C is (–2, 4) and D is (3, –1). The answer is d =
Using the distance formula, plug in the x and y values:
using order of operations:
,
.
. Then, simplify
,d=
,d=
.
23
24
Part I: Foundation (And We Don’t Mean Makeup!)
p
Find the midpoint of segment EF, where E is (3, –5) and F is (7, 5). The answer is M = (5, 0).
Using the midpoint formula, you get
q
. Simplify from there: M = (10⁄2, 0⁄2), M = (5, 0).
Find the slope of line GH, where G is (–3, –5) and H is (–3, 4). The answer is m = undefined.
Using the formula for slope, plug in the x and y values for the two points:
simplifies to
r
. This
, which is undefined.
Find the perimeter of triangle CAT. The answer is
.
To find the perimeter, you need to calculate the distance on each side, which means you have
to find CA, AT, and TC. Plugging the values of x and y for each point into the distance formula,
you find that the distances are as follows: CA =
, AT =
, and TC =
. Adding like
terms gives you the perimeter of
s
.
Find the center of the rectangle NEAT. The answer is (9, 2).
Ah! Think we’re being tricky here? Well, the trick is to realize that if you find the midpoint of
one of the rectangle’s diagonals, you will have identified the center of it. Easy, huh? So, by using
the diagonal NA, you can find the midpoint and thus the center:
. This
simplifies to m = (9, 2).
t
Determine whether triangle DOG is a right triangle. The answer is yes.
We had to end it with another challenging one. Here you need to remember that right triangles
have one set of perpendicular lines (forming that right angle). Also, you need to remember that
perpendicular lines have negative reciprocal slopes. In other words, if you multiply their slopes
together, you get –1. So, all you have to do to answer this question is find the slopes of the lines
that appear to be perpendicular, and if they multiply to equal –1, then you know you have a
right triangle. Okay? Then let’s go!
Start by finding the slope of DO:
OG:
,
, m = –8⁄4, m =
, or –2. Next, find the slope of
, m = 1⁄2. Multiplying the two slopes together, you find that, indeed, it does
equal –1, indicating that you have perpendicular lines: (–2)(1⁄2) = –1. Therefore, triangle DOG is a
right triangle.
Chapter 2
Get Real!: Wrestling with Real Numbers
In This Chapter
䊳 Finding solutions to equations with inequalities
䊳 Using interval notation to express inequality
䊳 Simplifying radicals and exponents
䊳 Rationalizing the denominator
W
hen you build a house, you start by preparing your site and laying your foundation.
In Chapter 1, we found and graded the site and started the foundation, but now it’s
time to make sure that the foundation is set in place before we start building the frame. Precalculus, like a sturdy house, has to be based on a solid foundation. In this case, our house
is based on Algebra I and II skills. Consider algebra the mortar between your pre-calc bricks.
We’re going to refresh your memory and cement you with some of those basic skills.
In this chapter, we assume that you know most of your algebra skills well, so we review only
the tougher concepts in algebra — the ones that give a lot of our students trouble if they
don’t review them. In addition to reviewing inequalities, radicals, and exponents, we also
introduce a purely pre-calculus idea: interval notation. If you feel confident with the other
review sections in this chapter, feel free to skip ahead, but make sure you practice some of
the interval notation problems before moving on to Chapter 3. For those of you who aren’t
sure how solid your cemented foundation is, let’s get brickin’!
Solving Inequalities
Solving inequalities is very similar to solving basic equations, which we assume you know
solidly by now. There are a few subtle differences, which we’ll take the time to review and
practice here.
First, remember that an inequality is a mathematical sentence indicating that two expressions
aren’t equal. Inequalities are expressed using the following symbols:
Greater than: >
Greater than or equal to: ≥
Less than: <
Less than or equal to: ≤
Solving equations with inequalities is exactly the same as solving equations with equalities,
with one key exception: multiplying and dividing by negative numbers.
When you multiply or divide each side of an inequality by a negative number, you must
switch the direction of the inequality symbol. In other words, < becomes > and vice versa.
26
Part I: Foundation (And We Don’t Mean Makeup!)
This is also a good time to put together two key concepts: inequalities and absolute
values, or absolute value inequalities. With these, you need to remember that absolute
values have two possible solutions: one when the quantity in the absolute value bars
is positive, and one when it’s negative. Therefore, you have to solve for these two possible solutions.
The easiest way to do this is to drop the absolute value bars and apply this simple rule:
becomes ax ± b < c AND ax ± b > –c
becomes ax ± b > c OR ax ± b < –c
Need an easy way to remember this? Notice the pattern: < is AND, while > is OR. Just
think: “less thAND” and “greatOR than.”
The solutions for these absolute value inequalities can be expressed graphically, as follows in Figure 2-1.
Figure 2-1:
Graphical
solution for
.
One more trick those pesky pre-calculus professors may try and pull on you has to do
with absolute value inequalities involving negative numbers. You may encounter two
possible scenarios:
⻬ If the absolute value is less than or equal to a negative number, a solution
doesn’t exist. Because an absolute value must be positive, it can never be less
than a negative number. For example,
doesn’t have a solution.
⻬ If the absolute value is greater than or equal to a negative number, there are
infinite solutions, and the answer is all real numbers. Here, because an absolute
value indicates a positive solution and a positive number is always greater than a
negative number, an absolute value is always greater than a negative number. For
instance, it doesn’t matter which number you plug into the equation
,
you always get a true statement. Therefore, the solution to the statement is all
real numbers.
Q.
Solve for x in 5 – 2x > 4.
Q.
Solve for x in
A.
x < 1⁄2. Start by subtracting 5 from each side,
giving you –2x > –1. Next, divide both sides
by –2 (don’t forget to switch that inequality!), giving you x < 1⁄2.
A.
x ≥ 2 or x ≤ –4. First, you have to isolate
the absolute value. To do this, add 3 to
both sides. Next, drop the absolute value
bars and set up your two equations:
4x + 4 ≥ 12 OR 4x + 4 ≤ –12. Solving each
algebraically, you get x ≥ 2 or x ≤ –4.
.
Chapter 2: Get Real!: Wrestling with Real Numbers
1.
Solve for x in
2.
.
Solve It
3.
Solve for x in
Solve It
Solve for x in x2 – 5x – 20 > 4.
Solve It
.
4.
Solve for x in x3 – 5x > 4x2.
Solve It
27
28
Part I: Foundation (And We Don’t Mean Makeup!)
Expressing Inequality Solutions
in Interval Notations
Pre-calc newbie, listen up. This is something you probably didn’t experience in algebra. Interval notation, although scary sounding, is simply another way of expressing a
solution set. Why have another way to write the same thing? Well, this notation is
important to know because it’s the one most often used in pre-calculus and calculus.
And because we know you’re incredibly dedicated to becoming a math wizard (uhhuh), you need to know how to cast this spell.
The key to writing a solution in interval notation is to locate the beginning and end of a
set of solutions. You can do this by using inequality notation or by visualizing the solution by graphing it. After you locate your key points, you need to decide which type of
interval you’re dealing with: open interval (> or <) or closed interval (≥ or ≤).
⻬ Open interval is indicated by an open circle at a point on a graph and by parentheses in interval notation.
⻬ Closed interval is indicated by a solid circle at a point on a graph and by brackets for interval notation.
For example, the solution set –4 ≤ x < 3, shown in Figure 2-2, can be rewritten in interval notation as [–4, 3). Another way to think of this solution set is x ≥ –4 AND x < 3.
Figure 2-2:
Graph of
–4 ≤ x < 3.
–4
3
To indicate a solution set that includes non-overlapping sections (also known as disjointed sets), you need to state all the intervals of the solution separated by the word
OR. For example, to write the solution set of x < 2 or x ≥ 5 (as shown in Figure 2-3), you
need to write both intervals in interval notation: (–⬁, 2)艛[5, ⬁). The symbol between the
two sets is the union symbol (艛). It means that the solution can belong in either interval.
You always use open interval notation (parentheses) for ⬁ or – ⬁ because they’re not
real numbers.
Figure 2-3:
Graph of
x < 2 or
x ≥ 5.
2
5
Chapter 2: Get Real!: Wrestling with Real Numbers
Q.
Write the solution for 5 – 2x > 4 in interval
notation.
Q.
Graph the interval set (–2, 3]艛(5, ⬁) on a
number line.
A.
(1⁄2, ⬁). We solve this in the previous section, so we just have to write the answer in
interval notation.
A.
Figure 2-4. Start by putting your key points
on the number line. Then draw either solid
or open circles on your key points, depending on whether they’re closed or open
intervals. Last, shade.
6.
Write the solution for the solution of
x3 – 5x > 4x2 in interval notation, and graph
the solution on a number line.
Figure 2-4:
Graph of
(–2, 3]艛
(5, ⬁).
5.
–2
3
5
Write the solution for the solution of
in interval notation.
Solve It
7.
Graph the interval set (– ⬁, –7)艛[–5, 2)艛
(4, ⬁) on a number line.
Solve It
Solve It
8.
Graph the solution of
Solve It
.
29
30
Part I: Foundation (And We Don’t Mean Makeup!)
Don’t Get Drastic with Radicals and
Exponents — Just Simplify Them!
Radicals and exponents (also known as roots and powers) are fundamental algebra concepts. Remember our house? If you don’t have a solid foundation of these concepts,
you may blow, blow, blow your house down! Here’s one area where the foundation
needs to be absolutely pre-calc-cemented in place.
To start, we review the definitions of radicals and exponents and their relationship to
each other:
).
A radical signifies the root of a number. It’s indicated by the radical symbol (
A root of a number is a value that must be multiplied by itself to equal that number.
For example, the second root (or square root) of 9 is 3 because 3 multiplied by itself is
9 (3 · 3 = 9). Similarly, the third root (or cube root) of 8 is 2 because 2 multiplied by itself
two times is 8 (2 · 2 · 2 = 8).
An exponent is the power of a number. It indicates the number of times a number (the
base) is multiplied by itself. For example, 2 to the power of 3 is the same as 23 = 2 · 2 · 2 = 8.
Radicals and exponents are closely related to each other. In fact, they’re inverse operations. To solve an equation in which the variable is under a radical, simply take the
power of both sides. For example, to solve
, you need to square both sides,
giving you x = 16. Similarly, you can often solve an equation in which the variable is
raised to a power (or has an exponent) by taking the root of both sides. For instance,
, to get x = 3. You
to solve x3 = 27, you can take the cube root of each side,
can now use this simple fact to solve equations with radicals and exponents. Who’s
got the power now, huh?
Sometimes it’s easier to solve expressions with radicals and exponents by rewriting
them as rational exponents, or exponents written as fractions. To do this, remember
that the numerator (top number) of the rational exponent is the power, and the
denominator (bottom number) is the root.
x
m
n
= n xm =
( x)
m
n
For example, you can rewrite
or
2
as 8 3 .
Chapter 2: Get Real!: Wrestling with Real Numbers
Q.
Solve for x in x 2 − 3 x
A.
x = 0, 1, 16. Don’t let this one scare you!
Just remember your basic fraction rules
and look for a pattern for factoring. Start by
factoring out the GCF (x) from each term.
3
2
− 4x = 0 .
(
That leaves you with x x − 3 x
3
1
1
2
Q.
Solve for x in
A.
x = 5. Start this one by subtracting
4 from each side, isolating the radical:
. Next, square each side to
)
get rid of the square root:
−4 =0
(
x x
1
)(
−4 x
2
1
2
2
A common mistake is
to forget that (x – 2)2 is
(x – 2)(x – 2), not x2 + 4!
Multiply this out using FOIL. A
common method for remembering
the steps for multiplying two binomials, FOIL stands for F(first),
O(outside), I(inside), and L(last).
This helps you remember to multiply all the terms.
)
+ 1 = 0 . Next, set each factor
equal to 0 to find your solutions. It should
look something like this:
(
x x
1
2
)(
−4 x
x=0
x
1
2
1
2
)
(x ) = 4
1
x
2
2
That leaves you with 2x – 1 = x2 – 4x + 4.
Bring all terms to one side: 0 = x2 – 6x + 5,
then factor: 0 = (x – 5)(x – 1). Setting each
factor equal to 0, you get two possible
solutions: x = 5 or x = 1. Next, plug both
solutions back into the original equation
to check for extraneous roots (remember
those from Chapter 1?). Plugging them
in, you find:
+1 = 0
=4
2
, which
becomes 2x – 1 = (x – 2)(x – 2).
(remember x − x = x , just basic fraction
stuff here, nothing fancy). Now, recognize
that what’s left over in the parentheses is
merely a polynomial: y2 – 3y – 4 = 0, then
factor this polynomial: (y – 4)(y + 1) = 0.
When you recognize this, you need only
deal with the fractions as the exponents. You
can factor the polynomial into
1
2
.
1
2
= −1
( x ) = ( −1)
1
2
2
2
x = 16
x =1
If you need a refresher of exponential rules, skip ahead to Chapter 5
for a quick review. And for a quick
review of solving quadratics, skip
ahead to Chapter 4.
Therefore, x = 1 is an extraneous root,
and x = 5 is the solution!
9.
Simplify 27
Solve It
4
3
.
10.
Solve for x in x
Solve It
5
3
− 6x = x
4
3
.
31
32
Part I: Foundation (And We Don’t Mean Makeup!)
11.
Solve for x in
12.
x − 3 − 5 = 0.
Solve It
13.
Solve for x in
Solve It
Solve for x in x
8
9
= 16 x
3
+ 7x
2
9
.
Solve It
.
14.
Solve for x in x
Solve It
2
1
3
+ 10 = 0 .
Chapter 2: Get Real!: Wrestling with Real Numbers
Getting Out of a Sticky Situation, or Rationalizing
Ever find yourself justifying why you deserve a day off after a pre-calc test? It may
sound irrational, but rationalizing can come in handy sometimes. To truly simplify a
radical expression, you must rationalize your denominators. In other words, you can’t
leave a radical in the denominator of an expression. In this section, we review and
practice rationalizing the denominator.
We begin with monomials (or one term) in the denominator. For these, it’s important to
remember that you’re dealing with an expression and not an equation, so you need to
remember equivalent fractions.
Keep in mind that a monomial is an expression, NOT an equation. You can’t simply
square the term to find a solution, because you can’t counterbalance that action.
Instead, you need to multiply the numerator and denominator by the same term (which
is the same as multiplying by 1). For example, if you need to rationalize the expression
, you can multiply the expression by
, which equals 1. You then get
.
The same idea works for other radicals, but it requires a little more thinking. For example,
if you need to rationalize the expression
denominator by
, you need to multiply the numerator and
to the second power, or by
, because raising a cube root to the
third power cancels the root. After multiplying, you get
.
To rationalize expressions with binomials in the denominator, you must multiply both
the numerator and denominator by the conjugate. A conjugate is a fancy name for the
binomial that, when multiplied by the first binomial, gives you the difference of two
squares. It’s found by changing the sign of the second term of the binomial. For example, the conjugate of x + y is x – y. We know this for sure because when we multiply the
two conjugates (x + y)(x – y), we get x2 – y2, or the difference of two squares.
So to rationalize a denominator with a binomial, start by multiplying the numerator and
denominator by the conjugate and simplify. For example, to simplify
numerator and denominator by
=
, multiply the
. The steps look like this:
6+3 3
4−2 3 +2 3 − 9
=
=
33
34
Part I: Foundation (And We Don’t Mean Makeup!)
Q.
Simplify
A.
x=
.
.
To minimize the amount of work you
need to do for rationalizing denominators, it’s a good idea to rewrite
your denominator in factored form
to identify the base numbers you’re
dealing with. So for this problem, we
begin by rewriting the denominator:
.
Now, we can multiply the numerator and
denominator by
, giving us:
Simplifying, we get:
.
=
.
Q.
Simplify
A.
x=
.
. First, multiply the numerator
and denominator by the conjugate of the
denominator:
.
Be sure to correctly multiply your
terms, especially when you’re multiplying binomials. A common mistake
is to simply distribute the second
term to your conjugate in the numerator, but you need to remember that
you’re multiplying two binomials
together:
, not
. Correctly multiplied
out using FOIL, you should get:
. Combining like
terms, the final answer is
.
Chapter 2: Get Real!: Wrestling with Real Numbers
15.
Simplify
Solve It
17.
5
Simplify 3 2 .
5
2 18
Solve It
.
16.
Simplify
Solve It
18.
Simplify 82 .
4 3
Solve It
.
35
36
Part I: Foundation (And We Don’t Mean Makeup!)
Answers to Problems on Real Numbers
a
Solve for x in
. The answer is x > 8 or x < –4.
Start by dropping the absolute value bars and setting up your two equations: 4 – 2x > 12
or 4 – 2x < –12. Solve algebraically (careful when you divide by that negative!): –2x > 8 or
–2x < –16; x < –4 or x > 8.
b
Solve for x in x2 – 5x – 20 > 4. The answer is x > 8 or x < –3.
First, you need to recognize that we’re dealing with a quadratic here. To solve a quadratic, you
need to isolate it. (If you need to review quadratics, skip ahead to Chapter 4). Start by subtracting 4 from each side: x2 – 5x – 24 > 0. Next, factor your quadratic: (x – 8)(x + 3) > 0. Now what?
Don’t worry; we’ll lead you through this! Setting each factor to 0 gives you your key points: 8
and –3. If you put these on a number line, you can see that you have three possible solutions:
less than –3, between –3 and 8, or greater than 8. All you have to do is plug in numbers in each
interval to see if you get a positive or negative number. Because you’re looking for a solution
that’s greater than 0, you need a positive result when you multiply your factors, (x – 8) and
(x + 3), together. In other words, you want both of your factors to be positive or both of them to
be negative. Looking at the number line in the following figure, you see that your solutions are
x > 8 or x < –3. Whew!
(x – 8) negative (x – 8) negative (x – 8) positive
(x + 3) negative (x + 3) positive (x + 3) positive
–3
c
Solve for x in
8
. The answer is all real numbers.
Begin by isolating the absolute value by subtracting 15 from each side, giving you
.
Remember that absolute values are positive and therefore greater than any negative. No matter
what you plug in for x, you get a positive number. So, the solution is all real numbers!
d
Solve for x in x3 – 5x > 4x2. The answer is x > 1 or 0 > x > –5.
First, if you need a refresher on solving polynomials and quadratics, skip ahead to Chapter 4.
For this problem, start by gathering all your variables to one side of the equation by subtracting 4x2 from each side: x3 + 4x2 – 5x > 0. Next, factor out x from each term: x(x2 + 4x – 5) > 0.
Then factor the quadratic: x(x – 5)(x – 1) > 0. Setting your factors equal to 0, you can find your
key points. Put these points on a number line. Plug in test numbers from each possible section
to determine whether the factor would be positive or negative. Then, given that you’re looking
for a positive solution, think about the possibilities: (+)(+)(+) = (+),(+)(+)(–) = (–),(–)(+)(–) =
(+),(–)(–)(–) = (–). Therefore, your solution is x > 1 or 0 > x > –5.
e
in interval notation. The answer is (–⬁, ⬁ ).
Write the solution for the solution of
Recognize this one? We solved it in practice problem 3. The solution is all real numbers, and
you write that in interval notation by writing it as infinity to negative infinity. Cool, huh?
f
Write the solution of x3 – 5x > 4x2 in interval notation, and graph the solution on a number line.
The answer is (–5, 0)艛(1, ⬁ ).
Oops, we did it again! This one came from practice problem 4. The graph looks like this:
–5
0
1
Chapter 2: Get Real!: Wrestling with Real Numbers
g
Graph the interval set (–⬁,–7)艛[–5, 2)艛(4, ⬁ ) on a number line.
–7
h
–5
Graph the solution of
2
4
.
Start by dropping the absolute value sign and setting up your two equations: 2x – 1 ≤ 3 and
2x – 1 ≥ –3. Then solve each to find your solution: 2x ≤ 4 and 2x ≥ –2; x ≤ 2 and x ≥ –1. These can
also be rewritten as –1 ≤ x ≥ 2, which can be graphed as follows:
–1
i
Simplify 27
4
2
. The answer is 81.
3
First, recognize that you can think of this problem in two ways:
and
. Either way
gives you the correct answer, but one is easier to deal with than the other. Starting with
,
order of operations tells us to take the 27 to the power of 4 first, giving us:
If we choose to deal with the problem written like this:
. Ew! No fun!
, then it’s much easier. Order of
operations here tells us to take the cube root of 27, which is 3, then take 3 to the 4th power,
which is 81. Ah . . . much better. By the way, the cube root of 531,441 is 81 as well, but we won’t
make you memorize it.
j
Solve for x in x
5
3
− 6x = x
4
3
. The answer is x = 0, –8, 27.
Begin by bringing all the terms to one side in descending order:
(
out an x from each term: x x
2
3
−x
1
3
)
. Next, factor
− 6 = 0 . Then, you can see that the resulting quadratic is
(
2
similar to y(y – y – 6), which factors into y(y + 2)(y – 3). Similarly, you can factor x x
(
into x x
1
3
)(
+2 x
1
3
)
2
3
−x
1
3
)
−6 =0
− 3 = 0 . Then, setting each factor equal to 0 and simplifying, you can find
your three solutions:
(
x x
1
3
)(
+2 x
1
x=0
3
)
−3 =0
x
x
1
1
3
+2=0
x
3
= −2
x
( x ) = ( −2 )
1
3
3
x = –8
k
Solve for x in
3
1
1
3
−3=0
3
=3
( x ) = (3)
1
3
3
3
x = 27
x − 3 − 5 = 0 . The answer is x = 28.
Start by isolating the radical by adding 5 to each side, giving you
sides to get rid of the square root. This gives you
which is the same as x = 28.
. Next, square both
, which simplifies to x – 3 = 25,
37
38
Part I: Foundation (And We Don’t Mean Makeup!)
l
Solve for x in x
8
9
= 16 x
2
9
. The answer is x = 64.
8
First, divide both sides by x
2
9
, giving you:
(see Chapter 5 for a refresher!): x
(x )
2
3
3
2
8 −2
9 9
2
x 9 = 16 x 9
. Simplify using your exponent rules
2
2
x 9
x 9
= 16 , x
2
3
= ( 16 ) 2 . Then, simplify the radical
3
= 16 . Next, raise each side to the power of 3⁄2:
by taking the square root of 16 and raising
it to the power of 3. You get 64.
m
Solve for x in
. The answer is x = 8, 16.
Begin by isolating one of the radicals:
that radical:
. Then, square both sides to get rid of
. Make sure that you multiply your binomials correctly!
. Multiplying the terms on the right side of the equation gives
you
. Next, isolate the remaining radical using basic algebra:
. Then, you can square both sides again to remove the remaining radical:
. Using algebra, multiply the two binomials and combine like terms:
16(2x – 7) = (x + 4)(x + 4); 32x – 112 = x2 + 8x + 16; 0 = x2 – 24x +128. This quadratic factors into:
0 = (x – 8)(x – 16). Setting both factors equal to 0, you get two possible solutions: x = 8 and
x = 16. Plug both back into the original equation and you’ll find that both solutions work.
n
Solve for x in x
2
3
+ 7x
1
3
+ 10 = 0 . The answer is x = –8, –125.
Start by recognizing that this trinomial is similar to y2 + 7y + 10 = 0, which factors to
2
1
. Setting each one
(y + 5)(y +2) = 0. Similarly, x 3 + 7 x 3 + 10 = 0 factors into
equal to 0, you can easily solve for the solutions by taking each side to the power of 3. In other
words,
o
Simplify
, so x = –125, and
becomes
. The answer is
becomes
1
3
, so x = –8.
.
First, you need to separate the fraction into two radicals: one in the numerator and one in the
denominator:
. Now, multiply the numerator and denominator by the square root in the
denominator:
. Did we trick you here? This one doesn’t require the use of a
conjugate because there isn’t another term added to the radical. Simplify the numerator by
multiplying the radicals:
=
.
39
Chapter 2: Get Real!: Wrestling with Real Numbers
p
Simplify
. The answer is
.
Start by multiplying the numerator and denominator by the conjugate of the denominator:
. Next, multiply the binomials in the numerator and denominator using
FOIL:
. Then, simplify each radical:
. Finally, because
each term in the numerator and denominator is divisible by 2, divide both by 2:
q
5
Simplify 3 2 . The answer is
2 5 18
.
.
Begin by factoring the denominator:
=
. Notice the
in both the numerator
and denominator? Cancel them! Yea — one less term to worry about! Next, multiply the numerator
and denominator of
through:
final answer:
r
=
by
to eliminate the radical in the denominator:
. Multiply
and then cancel the 3 from the numerator and denominator to get your
.
8
2
. The answer is
.
4 3
Start by changing the fractional exponent into a radical:
Simplify
denominator by one more cube root of 4:
. Multiply:
. Then, multiply the numerator and
, and simplify the fraction:
.
40
Part I: Foundation (And We Don’t Mean Makeup!)
Chapter 3
Understanding the Function of Functions
In This Chapter
䊳 Determining whether a function is even or odd
䊳 Introducing parent functions and how to graph them
䊳 Graphing rational functions and piece-wise functions
䊳 Performing operations on functions
䊳 Finding the domain and range of functions
䊳 Working with inverses of functions
Y
ou’ve seen the coordinate plane, where two number lines meet at a 90° angle. You know
that the horizontal axis is called the x-axis, and the vertical one is called the y-axis. You
also know that each point, or ordered pair, on the plane is named (x, y). But did you know
that a relation is a set of ordered pairs? The domain of the relation is the set of all the x
values, and the range is the set of all the y values. Note: If you ever run into different variables (like m and n), domain and range will be based alphabetically.
A function is a relation where every x in the domain pairs with one (and only one) y in the
range. The symbol for a function is f(x), read “function of x,” or simply “f of x.” Think of a function as a computer. Domain is input and range is output. You can’t put input in a computer
and get out different outputs; otherwise, your computer would be broken. If you had the
world’s simplest computer and all it did was multiply by 3, if you input 2 in the computer, you
better get an output of 6. Domain and range have a similarly correspondent relationship. We
explore the idea of functions and some properties of them in this chapter. Now get computing!
Battling Out Even versus Odd
If you’ve ever taken an art class, you’ve probably heard the term symmetry. It means that the
picture is balanced, with equal or similar parts on both sides of the painting. A graph can be
symmetrical as well. Algebra has three different types of symmetry:
⻬ Y-axis symmetry: Each point on the left side of the y-axis is mirrored by a point on the
right side, and vice versa.
⻬ X-axis symmetry: Each point above the x-axis is mirrored by a point below it, and vice
versa.
⻬ Origin symmetry: If you turn the graph upside down, it looks exactly the same.
In pre-calculus, functions take this idea of symmetry and use different terms to describe the
same idea. A function whose graph is symmetrical with respect to the y-axis is called an even
function. Basically, each input x and the opposite input –x give the same y value. In symbols,
textbooks write that f(x) = f(–x). A function whose graph is symmetrical with respect to the
origin is called an odd function. In plain English, each x value gives a y value, and its opposite
–x gives the opposite –y. This means that f(x) = –f(x).
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Part I: Foundation (And We Don’t Mean Makeup!)
Q.
Determine whether f(x) = x4 – x2 is even,
odd, or neither.
A.
This function is even. Replace x with –x
in the equation and see what happens:
f(–x) = (–x)4 – (–x)2. A negative number to
an even power is a positive number. So
f(–x) = x4 – x2. Because you get the same
exact function as the original one, this
function is even.
1.
Is f(x) = x3 – 1 even, odd, or neither?
2.
Determine whether the given graph is
even, odd, or neither.
Solve It
3.
Sketch half the graph of f(x) =
Solve It
(x
2
−4
)
4.
Sketch half the graph of f(x) = 4x3 and use
symmetry to complete the graph.
and use symmetry to complete the graph.
Solve It
Solve It
Chapter 3: Understanding the Function of Functions
Leaving the Nest: Transforming Parent Graphs
You see certain functions over and over again in pre-calc, and sometimes you’ll be
asked to graph them. The plug and chug method for graphing works for any equation
(including functions), but the more complicated the function, the longer it takes you to
graph it using this method. The basic graph, in any case, is called the parent graph.
Common parent graphs include quadratic functions, square roots, absolute values,
cubics, and cube roots. Moving these basic graphs around the coordinate plane is
known as transforming the function and is easier than the plug and chug method.
Several types of transformations of functions exist:
⻬ Horizontal transformations
⻬ Vertical transformations
⻬ Reflections
⻬ Horizontal translations
⻬ Vertical translations
In this section, we take a look at each parent function and then show you how to transform them. Note: Even though in most sections we take a look at only one function
when discussing the transformations, the rules apply to all functions in the same way.
So if we talk about a quadratic function in the section on vertical transformations,
that’s not the only function that has vertical transformations — they all do.
Quadratic functions
Quadratic functions are second degree equations. The highest exponent on any one
variable is two. The parent quadratic function is f(x) = x2. Its graph is known as a
parabola; we talk in depth about parabolas in Chapter 12. Begin the graph at the vertex
(0, 0), and to get to the next point, move over 1, up 1; the next point from there is over
1, up 3; the next point is over 1, up 5. You always move over 1 and up the next odd
number. See Figure 3-1 for the graph of the parent quadratic function. The parent quadratic function is an even graph — it’s symmetrical with respect to the y-axis.
Figure 3-1:
Graphing
the parent
quadratic
function.
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Part I: Foundation (And We Don’t Mean Makeup!)
Square root functions
The parent square root function is g(x) = x . This graph looks like half a parabola,
turned on its side. You can’t have x values that are negative because you can’t square
root a negative number, and f(x) is also positive because x represents the principle,
or positive root. The graph begins at the origin (0, 0) and then moves up 1, over 1; up
1, over 3; up 1, over 5; and so on. The graph of the parent square root function can be
seen in Figure 3-2.
Figure 3-2:
The parent
square root
function.
Absolute value functions
The absolute value parent graph is determined by h(x) = |x|. You should recognize the
absolute value bars and know that this figure represents distance, so it always gives a
positive output. This parent function also starts at (0, 0) and then always moves over
1, up 1. Its graph can be seen in Figure 3-3. The parent absolute value function is also
an even function.
Figure 3-3:
The
absolute
value parent
function.
Cubic functions
A cubic function is one where the highest degree on a variable is three: p(x) = x3 is the
parent function in this case. The parent cubic function is an odd function; if you turn it
upside down it looks exactly the same. If you start at the origin (0, 0) and mark a point,
the point immediately to the left is over 1 and down 1, and the point immediately to
the right is over 1 and up 1. This graph is shown in Figure 3-4.
Chapter 3: Understanding the Function of Functions
Figure 3-4:
The cubic
function’s
parent
graph.
Cube root functions
Cube root functions are related to cubic functions in a similar way that quadratic and
. The graph also starts at the
square root functions are. The parent graph is r(x) =
origin and moves right 1, up 1 and left 1, down 1, but the graph is longer than it is tall.
Take a look at Figure 3-5 for this parent graph.
Figure 3-5:
The cube
root parent
function.
Vertical transformations
In any function, the method of moving up to get to the next point is known as the vertical transformation. Some teachers may explain this concept as amplitude, but that’s
technically incorrect because the functions we’ve discussed so far keep going up forever, so they don’t technically have an amplitude or height. Multiplying any function
by a constant changes a graph’s vertical transformation. This is written as a · f(x).
Think of this as a vertical stretch or shrink. A coefficient between 0 and 1 is a shrink,
and a coefficient greater than 1 is a stretch.
For example, f(x) = 2x2 multiplies each up value by 2. From the vertex, you used to
move over 1, up 1, but now you move over 1, up 2. After that, you used to move over 1,
up 3, but now you move over 1, up 6. This keeps going for each point, which is why
some teachers call it amplitude — the height of each individual point is affected.
For another example, if g(x) = 1⁄4 x2, you move in this manner: From the vertex, go over 1,
up 1⁄4; over 1, up 3⁄4; over 1, up 5⁄4; and so on. The graphs of f(x) and g(x) are shown in
Figure 3-6.
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Part I: Foundation (And We Don’t Mean Makeup!)
Figure 3-6:
Graphing
vertical
transformations.
a.
b.
Note: A negative coefficient is actually a reflection, so we discuss that type of transformation in the upcoming “Reflections” section.
Horizontal transformations
If the vertical transformation affects the up values in the function, then a horizontal
transformation affects the over values. Horizontal transformations stretch or shrink the
graph along the x-axis. This time, the coefficient is inside the function: f[c(x)].
h(x) = 3|x| is a vertical transformation; h(x) = |3x| is a horizontal transformation.
f(x) = 4x2 is a vertical transformation; f(x) = (4x)2 is a horizontal transformation.
Set the expression inside the function equal to the parent function’s normal horizontal
transformation and solve for x to find the new value. If g(x) = |3x|, then set 3x = 1 and
solve for x = 1⁄3. From the origin, you move over 1⁄3, up 1; over 1⁄3, up 1; over 1⁄3, up 1.
Translations
Moving a graph horizontally or vertically on the coordinate plane is called a translation.
Every point on the parent graph is moved right, left, up, or down. Here we take a closer
look at each kind of translation, or shift.
Horizontal shifts
Adding or subtracting a number inside the function’s grouping device is a horizontal
shift. A horizontal shift is always written in the form f(x – h) such that the horizontal
shift is the opposite of what it appears to be.
h(x) =
h(x) =
( x − 2)
( x + 3)
moves the parent square root function to the right by 2.
moves the parent square root function to the left by 3.
Chapter 3: Understanding the Function of Functions
Vertical shifts
Adding or subtracting a constant outside of the function’s grouping device is a vertical
shift. These types of transformations are written as f(x) + v, where v is the vertical shift.
p(x) = x3 – 1 moves the parent cubic function down by 1.
p(x) = x3 + 4 moves the parent cubic function up by 4.
Reflections
Reflections take the parent function and reflect it over a horizontal or vertical line.
When the vertical transformation coefficient is negative, the function is flipped upside
down over a horizontal line. For example, f(x) = –5x2 affects the vertical transformation
by a factor of 5 and turns the graph upside down at the same time. If the horizontal
transformation is negative, the function is flipped backwards over a vertical line:
h(x) =
( −x )
turns the function to the left instead of to the right.
Combinations of transformations
Putting some or all the transformations into one function is itself a transformation.
Putting all of them together into one expression looks like this: a · f[c(x – h)] + v where
a is the vertical transformation
c is the horizontal transformation
h is the horizontal translation
v is the vertical translation
We recommend doing the translations first and then doing the transformations.
Q.
A.
Graph the function f(x) = (x – 3)2 by transforming the parent graph.
See the graph. This transformation is done
in one step. Because the constant is subtracting inside the quadratic function, you
recognize it as a horizontal shift to the
right by 3. Take the parent quadratic function and move each point to the right by 3,
as shown in the graph.
f(x) = (x – 3)2
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Part I: Foundation (And We Don’t Mean Makeup!)
Q.
Sketch the graph of g(x) =
transforming the parent function.
A.
See the graph. This one takes some work
before you can begin graphing it. It must be
rewritten in the proper form to recognize
the various transformations to the parent
square root function. First, rewrite the
stuff inside the square root so that it’s in
. Next,
the right order: g(x) =
factor out the leading coefficient to get
the horizontal transformation: g(x) =
. This means that the graph
is flipped horizontally. Notice that factoring
out the coefficient affects the horizontal
translation — it’s to the right by 3, and the
vertical translation is up 1. These transformations can be seen in the final graph.
5.
Graph the function a(x) = –2(x – 1)2.
Solve It
by
g(x) = 3 – x +1
6.
Graph the function b(x) = |x + 4| – 1.
Solve It
Chapter 3: Understanding the Function of Functions
7.
Graph the function c(x) = x + 3 .
Solve It
8.
Graph the function f(x) = –x2 – 6x.
Solve It
Given the graph of the function g in Figure 3-7, sketch the graph of the functions in Problems 9 and 10.
Figure 3-7:
The given
parent function g(x).
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Part I: Foundation (And We Don’t Mean Makeup!)
9.
g(x) – 3.
Solve It
10.
2g(x – 1).
Solve It
Lucid Thinking? Graphing Rational Functions
A rational function is one where the variable is in a fraction’s denominator. You know
by now that when the denominator of a fraction is 0, the result is undefined. The same
is true for rational functions. Because the denominator has a variable, it may be possible that certain values of x will make the denominator 0. If the function has values that
make it undefined, the graph will have a vertical asymptote.
To find the vertical asymptote, if there is one, set the denominator equal to 0 and solve.
This is a vertical line that the graph will never cross. Some rational functions also have
a horizontal asymptote as well. A graph shouldn’t ever cross a horizontal asymptote, but
in some cases it will, so don’t freak out if that happens. (The second upcoming example
is a rational function where the graph crosses its horizontal asymptote.)
To find the horizontal asymptote, take a look at both the numerator’s degree and the
denominator’s degree. (If you’ve forgotten how to find the degree of a polynomial, see
Chapter 4.) Here are the three possibilities for horizontal asymptotes:
⻬ The degree of the denominator is greater: This means that the bottom of the
fraction is getting bigger, faster, and the fraction will go to 0 as x gets larger. Your
horizontal asymptote is the x-axis, or y = 0.
⻬ The degree of both is the same: This means that the top and bottom of the fraction are moving at the same rate. The quotient of the leading coefficients gives
you the horizontal asymptote.
⻬ The degree of the numerator is greater: This means that the top of the fraction
is getting bigger, faster. A really big number divided by a smaller number is still a
pretty big number. In short, as x gets larger, so will y, and there’s no horizontal
asymptote. Instead, there’s an oblique asymptote. To find it, find the quotient by
dividing the denominator into the numerator. The function that you find is the
one that you graph as an oblique asymptote.
51
Chapter 3: Understanding the Function of Functions
Q.
Graph the function f(x) =
A.
See the graph. First, find the vertical
asymptote (if there is one) by setting
the denominator equal to 0 and solving.
If 4 – x = 0, then x = 4. Draw a coordinate
plane and add in a dotted vertical line at
x = 4 to mark your vertical asymptote. Now,
look at the numerator and the denominator;
the degree on each is one. Divide the leading coefficients to find the horizontal
asymptote. In this case, the numerator’s
leading coefficient is 3 and the denominator’s is –1. This means your horizontal
asymptote is y = 3⁄–1 = –3. Now that you have
both asymptotes, use them to help you get
the graph. The vertical asymptote divides
your domain into two intervals: (–⬁, 4) and
(4, ⬁). Pick a couple of x-values on each
interval and plug them into the function to
determine whether the graph lives above or
below the horizontal asymptote. For example, if x = –5, then y = –1.77; and if x = 0, then
y = –0.25. If you graph those two points, you
see that they’re both above the horizontal
asymptote. Keep checking points until you
have a good idea of what the graph looks
like.
.
Q.
Graph the function g(x) =
A.
See the graph. This time, when you try to
find the vertical asymptote, you notice that
x2 + 4 = 0 doesn’t have a solution because
x2 = –4 has no solution (in the real numbers
anyway). Also notice that because the
denominator has a bigger degree, the horizontal asymptote is the x-axis, or y = 0.
However, also notice that by setting the
numerator equal to 0, you do get a solution:
2x – 6 = 0; 2x = 6; x = 3. This means the graph
crosses the x-axis at x = 3 even though it’s
not supposed to. Because there’s no vertical
asymptote, use this value to give you the
intervals to look at to get the graph. On the
first interval (–⬁, 3), y is negative and the
whole graph is below the horizontal asymptote. On the next interval (3, ⬁), y happens
to be positive and the function is above the
horizontal asymptote. If you pick x values
bigger than 3 that keep getting bigger, you
see y increase slowly and then decrease
again and get closer and closer to 0. This
gives you the graph of this function.
.
y=0
y = –3
x=4
52
Part I: Foundation (And We Don’t Mean Makeup!)
11.
Graph the function q(x) =
.
Solve It
13.
Graph the function t(x) =
Solve It
12.
Graph the function r(x) =
.
Solve It
.
14.
Graph the function u(x) =
Solve It
.
Chapter 3: Understanding the Function of Functions
Picking Up the Pieces: Graphing
Piece-Wise Functions
A piece-wise function is called that because it’s broken into pieces. A piece-wise function actually contains several functions, each defined on a restricted interval. The
output depends on what the input is. The graphs of these functions may look like
they’ve literally been broken into pieces. Because of this broken quality, a piece-wise
function that jumps is called discontinuous.
Q.
Graph f(x) =
A.
See the graph. This function has been
broken into two pieces: When x ≤ 1, the
function follows the graph of the quadratic
function, and when x > 1, the function follows the graph of the linear function.
Notice the hole in this second piece of the
graph to indicate that the point isn’t actually there. The graph is shown in Figure 3-8.
15.
Graph g(x) =
Figure 3-8:
The graph
of one
piece-wise
function.
16.
Graph h(x) =
Solve It
Solve It
.
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Part I: Foundation (And We Don’t Mean Makeup!)
17.
18.
Graph m(x) =
Solve It
Graph n(x) =
Solve It
Operating on Functions:
No Scalpel Necessary
You’ve come to know (and maybe to love, too?) four basic operations in math: addition, subtraction, multiplication, and division. Well, in pre-calc, you take functions and
add, subtract, multiply, and divide them. By doing this, you create a brand new function. This is sometimes called combining functions. In general, you probably won’t be
asked to graph a combined function because it usually won’t be based on any of the
parent functions. If you were asked to graph a combined function, you’d have to plug
and chug your way through it by picking plenty of x-values to make sure you get an
accurate representation of the graph. You may also be asked to find one specific value
for a combined function — you get an x value and you just plug it in and see what happens. Exciting, we know!
For all questions in this section, you use three functions:
f(x) = x2 – 6x + 2
g(x) = 2x2 – 5x
h(x) =
3x + 2
Q.
Find (f – g)(x).
A.
(f – g)(x) = –x2 – x + 2. Because these two
functions are both polynomials, solving this
is really about collecting like terms and subtracting them. Just be sure to watch your
negative signs! (f – g)(x) = (x2 – 6x + 2) –
(2x2 – 5x) = x2 – 6x + 2 – 2x2 + 5x = –x2 – x + 2.
No problem!
Chapter 3: Understanding the Function of Functions
19.
Find (f + h)(x).
Solve It
21.
Find (h/g)(x). Does this new function have
any undefined values?
Solve It
20.
Find (fg)(x).
Solve It
22.
Find (g + h)(2).
Solve It
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Part I: Foundation (And We Don’t Mean Makeup!)
Evaluating Composition of Functions
Placing one function inside of another (or, actually, in itself) is called a composition of
functions. If you have two functions f(x) and g(x), then the composition f(g(x)) takes g
and places it inside of f. This is also written as
left; the g function goes into the f function.
, and it’s basically read right to
You use the same three functions from the last section here as well:
f(x) = x2 – 6x + 2
g(x) = 2x2 – 5x
h(x) =
3x + 2
Q.
Find f(h(x)).
A.
. Start by substituting
the entire h function for every x in the f
function:
. A square
23.
Find
Solve It
.
root and a square cancel each other:
+ 2. Then simplify
by combining any like terms:
.
24.
Find
Solve It
.
Chapter 3: Understanding the Function of Functions
25.
Find h(f(x)).
Solve It
27.
Find f(g(–1)).
Solve It
26.
Find
Solve It
28.
Find g(h(3)).
Solve It
.
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Part I: Foundation (And We Don’t Mean Makeup!)
Working Together: Domain and Range
Now that you’ve been combining and composing functions in the last two sections,
you may be wondering what’s happening with the domain and the range of the new
function. Remember that domain is input, usually x, and that range is output, usually y.
The truth is that the domain of the given function totally (like, for sure!) depends on
the operation being performed and the original functions. It’s possible that something
changed, and it’s also possible that nothing did. Typically, you’ll be asked to find the
domain of a combined function and not the range.
Pre-calc teachers and textbooks talk a lot about two functions whose domains are not
all real numbers:
⻬ Rational functions: The denominator of any fraction can’t be 0, so it’s possible
that some rational functions are undefined because of this fact. Set the denominator equal to 0 and solve to find the restrictions on your domain.
⻬ Square root functions (or any even root): The radicand (what’s under the root
sign) can’t ever be negative. This affects domain; to find out how, set the radicand greater than or equal to 0 and solve. The solution to this inequality is your
domain.
Undefined values are also called excluded values, so be on the lookout for your textbook to use that terminology as well. When you’re asked to find the domain of a combined function, take your time. We can’t put it into a nice, neat package and give you
one rule that works all the time for finding a combined function’s domain. Take a look
at both of the original functions and ask yourself if their domains have any restrictions. These restrictions carry through and combine together to the new combined
function.
You use those same three functions you’ve been using for the last two sections:
f(x) = x2 – 6x + 2
g(x) = 2x2 – 5x
h(x) =
Q.
3x + 2
Find the domain of f(h(x)).
A.
The domain is all numbers greater than
or equal to –2⁄3. Take a look at the original
two functions first. f(x) is a polynomial;
there are no restrictions on the domain.
However, h(x) is a square root function, so
the radicand has to be positive. 3x + 2 ≥ 0;
3x ≥ –2; x ≥ –2⁄3. The new combined function
must honor this domain as well.
Chapter 3: Understanding the Function of Functions
29.
Find the domain of
.
Solve It
31.
Find the domain of (f + h)(x).
Solve It
30.
Find the domain of h(f(x)).
Solve It
32.
Find the domain of (h/g)(x).
Solve It
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Part I: Foundation (And We Don’t Mean Makeup!)
Finding the Inverse of a Function
(Who Knew It Was Lost?)
An inverse function undoes what a function does. You’ve seen inverse operations
before: Addition undoes subtraction and division undoes multiplication. It shouldn’t
surprise you, then, that functions have inverses. If f(x) is the original function, then
f–1(x) is the symbol for the inverse. This notation is only used for the inverse function
.
and is never meant to represent
You’ll be asked to do three main things with inverses:
⻬ Given a function, graph its inverse
⻬ Find the inverse of a given function
⻬ Show that two functions are inverses of each other
In any case, all you need to know is that it’s all about input and output. If (a, b) is a
point in the original function, then (b, a) is a point in the inverse function. Domain and
range swap places from a function to its inverse. If asked to graph the inverse function,
graph the original and then swap all x and y values in each point to graph the inverse.
To find the inverse of a given function, literally take x and y (or f(x)) and switch them.
After the swap, change the name to the symbol for an inverse function, f–1(x), and solve
for the inverse. Lastly, to show that two functions f(x) and g(x) are inverses of each
other, place one inside the other using composition of functions, f(g(x)), and simplify
to show that you get x. Then do it the other way around with g(f(x)) to make sure it
works both ways.
Q.
Find the inverse of f(x) = 5x – 4.
A.
f–1(x) =
Q.
Determine whether f(x) = 3x – 1 and
are inverses of each other.
g(x) =
A.
These two functions are inverses. First,
. First, switch x and f(x):
x = 5f(x) – 4. Name the new function by
its correct name, the inverse function:
x = 5f–1(x) – 4. Now solve for the inverse:
x + 4 = 5f–1(x);
= f–1(x).
find
: 3(
) – 1. Simplify this
expression: x + 1 – 1 = x. That’s what it’s
supposed to be, so move onto the next
one:
:
= x. That one
works, too, so these two functions are
inverses of each other.
Chapter 3: Understanding the Function of Functions
33.
Graph the inverse of g(x) =
x−2 .
Solve It
35.
Determine whether f(x) = x3 – 1 and
are inverses of each other.
g(x) =
Solve It
34.
Find the inverse of k(x) =
.
Solve It
36.
Determine whether f(x) =
and
g(x) = 1 – 2x are inverses of each other.
Solve It
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Part I: Foundation (And We Don’t Mean Makeup!)
Answers to Questions on Functions
a
Is f(x) = x3 – 1 even, odd, or neither? The answer is neither.
Find f(–x) = (–x)3 – 1 = –x3 – 1. This isn’t the same function as the original, so the answer isn’t
even. It’s also not the exact opposite of the original, so the answer isn’t odd. The answer is that
it’s neither.
b
Determine whether the given graph is even, odd, or neither. The answer is odd.
If you look at the graph upside down, it looks exactly the same — that means it’s odd.
c
Sketch half of the graph of f(x) =
graph for the answer.
x 2 − 4 and use symmetry to complete the graph. See the
Find f(–x) first and discover that the function doesn’t change at all, which means you’ve got an
even function. If you plug and chug some negative values for x, you know that the positive
values for each corresponding x will be the same.
For example, f(–2) = 0, so you know that f(2) is also 0. f(–3) =
5 , and so does f(3). f(–5) =
21 ,
and so does f(5). Knowing these points gives you the graph.
f(x) = x2 – 4
d
Sketch half the graph of f(x) = 4x3 and use symmetry to complete the graph. See the graph for
the answer.
If you find f(–x), you get –4x3, which is the exact opposite of the original function, meaning that you
have an odd graph. Each x gives you a value f(x), and each opposite –x gives the opposite –f(x).
Plug and chug some values to get the graph: f(–3) = –108, so f(3) = 108. f(2) = 32, so f(–2) = –32.
f(–1) = –4, so f(1) = 4. Put these and as many other points as you’d like on the graph.
f(x) = 4x3
e
Graph the function a(x) = –2(x – 1)2. See the graph for the answer.
This function takes the parent quadratic graph and moves it to the right by 1. The vertical
transformation is 2, making each point twice as tall. The negative sign is a reflection, turning
the graph upside down. Put all these pieces together to get the graph.
Chapter 3: Understanding the Function of Functions
a(x) = –2(x – 1)2
f
Graph the function b(x) = |x + 4| – 1. See the graph for the answer.
This absolute value function has a vertical shift of 4 to the left and a horizontal shift of 1 down.
The coefficient in the front is 1, so it doesn’t have a vertical transformation — the graph has
just been moved.
b(x) = x + 4 – 1
g
Graph the function c(x) =
x + 3 . See the graph for the answer.
This square root function is shifted horizontally to the left by 3. Don’t forget that those horizontal shifts are always the opposite of what they appear to be.
c(x) = x + 3
h
Graph the function f(x) = –x2 – 6x. See the graph for the answer.
This one doesn’t look like any of the others that you’ve dealt with so far. That’s kind of unfair of
us, but it brings up the topic of conic sections, which we talk about later in depth in Chapter 12.
A parabola is one of these fancy types of curves. To get this parabola into its graphable form,
you have to follow a procedure known as completing the square (see Chapters 4 and 12 for more
information on this procedure). We do it for you here and tell you that the function will become
f(x) = –1(x + 3)2 + 9. We include it here because a few textbooks (though not many) teach completing the square early so that you can graph these types of problems. If your teacher is mean
enough to include one of these without teaching you how to complete the square, you have to
plug and chug this type of problem in order to graph it — pick x-values to find the corresponding y-values. Just be sure that your final graph is a parabola, as is ours.
63
64
Part I: Foundation (And We Don’t Mean Makeup!)
f(x) = –x2 – 6x
f(x) = –1(x + 3)2 + 9
i
g(x) – 3. See the graph for the answer.
This is just a graph that we made up. Because it’s a function, it still follows all the rules of transforming functions. Take every single point on the given g(x) function and shift each down by 3,
and you end up with a graph that looks like ours.
g(x) – 3
j
2g(x – 1). See the graph for the answer.
This time, g(x) is shifted to the right by 1. Take the height of each point in the original function
and multiply it by 2 to get the new height. For example, the original function passes through the
point (2, 4). The height of this point is 4, so when you double that in the new graph, you make
the height 8. Do this for every single point and end up with the graph.
2g(x – 1)
k
Graph the function q(x) =
. See the graph for the answer.
The vertical asymptote comes from the denominator: (x – 4)(x + 5) = 0. This equation is already
neatly factored, so all you have to do is use the zero product property and set each factor equal
to 0 and solve. If x – 4 = 0, then x = 4, and if x + 5 = 0, then x = –5. Put both of these on the graph as
vertical asymptotes. The horizontal asymptote is the x-axis again because the denominator has
the greater degree. The intervals you need to take a closer look at are (– ⬁, –5), (–5, 4), and (4, ⬁).
Pick a couple of x-values from each interval to get an idea of what the graph is doing. When
x = –7, y = –0.09, and when x = –6, y = –0.2 — both below the horizontal asymptote. When x = –4,
y = 0.25; when x = –1, y = 0.1; when x = 1, y = 0.11; and when x = 3, y = 0.25. These are all above
the horizontal asymptote. On the final interval, when x = 5, y = –0.2, and when x = 6, y = –0.09.
These are both below the horizontal asymptote. Put all the pieces together in the final graph.
Chapter 3: Understanding the Function of Functions
q(x) = –2
(x – 4)(x + 5)
l
Graph the function r(x) =
. See the graph for the answer.
First up, vertical asymptote(s). Set x2 – x – 6 = 0 and factor to (x – 3)(x + 2) = 0. Set each factor
equal to 0 and solve. If x – 3 = 0, then x = 3, and if x + 2 = 0, then x = –2. Add these two vertical
asymptotes to your graph. Next up is the horizontal asymptote. Because the denominator has
the greater degree, the horizontal asymptote is the x-axis again. Notice, however, that now that
the variable is in the numerator as well, there may be an x-intercept. Set the numerator equal
to 0 and solve. x + 3 = 0 tells you that x = –3 is an intercept. The graph crosses the x-axis even
though it isn’t supposed to. Use this fact to set up the intervals: (– ⬁, –3), (–3, –2), (–2, 3), and
(3, ⬁). Each interval is, respectively, below, above, below, and above the horizontal asymptote.
The graph looks a little weird, but then, which of these problems doesn’t look weird?
r(x) = x + 3
x2 – x – 6
m
Graph the function t(x) =
. See the graph for the answer.
Find the vertical asymptotes for this one by factoring the denominator. If x2 – 4x – 21 = 0, then
(x – 7)(x + 3) = 0. This gives you two solutions: x = 7 and x = –3. The degrees are the same again,
so the horizontal asymptote this time is y = 1. Put the asymptotes onto the graph and then pick
x-values to get the graph.
65
66
Part I: Foundation (And We Don’t Mean Makeup!)
n
Graph the function u(x) =
. See the graph for the answer.
This is the toughest of all the problems here because there’s no horizontal asymptote — the
numerator has the greater degree. Use long division to find the quotient x – 11; graph this as an
equation, y = x – 11, with a dotted line to mark your oblique asymptote. Next, add the vertical
asymptote by solving the equation x + 1 = 0 to get x = –1. Finally, plug and chug some values on
each interval to get the graph.
o
Graph g(x) =
. See the graph for the answer.
And now you get to graph piece-wise functions together. Take a look at each interval of the
domain to determine the graph’s shape. For this function, the top piece is only defined when
x ≤ –1. This part of the graph looks like a square root graph shifted 3 to the left. The bottom piece
is defined when x > –1. This part of the graph is a parabola, shifted 3 to the left. If it helps you to
lightly sketch the whole graph and then erase the part you don’t need, we highly recommend it.
Chapter 3: Understanding the Function of Functions
x + 3 if x < – 1
(x + 3)2 if x > – 1
p
Graph h(x) =
. See the graph for the answer.
The first piece is a linear function that’s defined only when x ≤ –2. The second piece is also a
linear function, defined between –2 and 2. The third piece is another linear function, defined
when x ≥ 2.
½x – 4 if x < 2
3x + 3 if –2 < x < 2
4 – x if x > 2
q
Graph m(x) =
. See the graph for the answer.
The first piece is a cubic shifted up by 2 — its right endpoint should be open. However, when
you graph the second piece, it’s a parabola that’s shifted up by 2. Its left endpoint overlaps the
right endpoint of the first piece. This fills the hole that was there, and the graph carries on until
x = 2, where it gets broken again. The third piece follows the linear function to the right of x = 2.
The graph comes together to look like this:
67
68
Part I: Foundation (And We Don’t Mean Makeup!)
x3 + 2 if x < 0
x2 + 2 if 0 < x < 2
x + 2 if x > 2
r
Graph n(x) =
. See the previous graph for the answer.
This piece-wise function is different because the middle piece is defined only at one point.
When x = 3, y = –3; that’s it. The first piece follows the absolute value graph that has been
shifted to the right by 1. The third piece is also an absolute value graph, but it has been shifted
down 1. Here’s the not-really-last graph.
x – 1 if x < –3
–3 if x = –3
x – 1 if x > –3
s
Find (f + h)(x). The answer is x 2 − 6 x + 2 + 3 x + 2 .
Take the f function and add the h function to it. Because one is a polynomial and the other is a
2
square root, there are no pesky like terms. The answer is (f + h)(x) = x − 6 x + 2 + 3 x + 2 .
t
Find (fg)(x). The answer is 2x4 – 17x3 + 34x – 10x.
Start off by writing out what you’ve been asked to find — the product of f and g: (fg)(x) =
(x2 – 6x + 2)(2x2 – 5x). Distribute each term of the left polynomial to each term of the right
polynomial to get 2x4 – 12x3 + 4x2 – 5x3 + 30x2 – 10x. Next, combine the like terms and get
(fg)(x) = 2x4 – 17x3 + 34x2 – 10x.
Chapter 3: Understanding the Function of Functions
u
Find (h/g)(x). Are there any undefined values for this new function? The answer is
yes, there are undefined values: x = 0 and x = 5⁄2.
and
This time you’re asked to find the quotient of h and g, with h on the top and g on the bottom.
Because they’re different types of functions, they won’t simplify at all. This means your answer
is, simply,
. Because the denominator now has a polynomial, there may be undefined
values. Set the denominator equal to 0 to start: 2x2 – 5x = 0. Solve by factoring out the GCF:
x(2x – 5). This has two solutions: x = 0 and x = 5⁄2. These are the undefined values.
v
Find (g + h)(2). The answer is 2 x 2 − 5 x + 3 x + 2 .
Because you’re adding another polynomial and a square root, there are no like terms. The
answer is 2 x 2 − 5 x + 3 x + 2 .
w
Find
. The answer is 4x4 – 20x3 + 13x2 + 30x + 2.
Take the g function and start plugging into f everywhere it says x: (2x2 – 5x)2 – 6(2x2 – 5x) + 2.
Multiply everything out first: 4x4 – 20x3 + 25x2 – 12x2 + 30x + 2. Now combine like terms to get
the answer:
= 4x4 – 20x3 + 13x2 + 30x + 2.
x
Find
. The answer is 2x4 – 24x3 + 75x2 – 18x – 2.
This time, place f into g where it says x: 2(x2 – 6x + 2)2 – 5(x2 – 6x + 2). Square the polynomial
on the left first by multiplying x2 – 6x + 2 by itself and distributing each term by each term.
This is quite long, so we show you the steps so you can follow along. 2(x4 – 6x3 + 2x2 – 6x3 + 36x2 –
12x + 2x2 – 12x + 4) – 5(x2 – 6x + 2). Combine like terms: 2(x4 – 12x3 + 40x2 – 24x + 4) – 5(x2 – 6x + 2).
Distribute the coefficients next: 2x4 – 24x3 + 80x2 – 48x + 8 – 5x2 + 30x – 10. Combine the like
terms to end up with the final answer:
y
Find h(f(x)). The answer is
= 2x4 – 24x3 + 75x2 – 18x – 2.
3 x 2 − 18 x + 8 .
Substitute f in for x in the h function:
. Distribute that 3 inside the root:
. Combine those like terms to end up with the answer: h(f(x)) =
A
Find
3 x 2 − 18 x + 8 .
. The answer is x4 – 12x3 + 34x2 + 12x – 6.
This time, place f into itself everywhere it says x: (x2 – 6x + 2)2 – 6(x2 – 6x + 2) + 2. You went
through the process of squaring that polynomial once before in question 24, so we won’t do
it again here. When you multiply everything out, you get x4 – 12x3 + 40x2 – 24x + 4 – 6x2 + 36x –
12 + 2. Combine like terms to end up with the answer:
B
= x4 – 12x3 + 34x2 + 12x – 6.
Find f(g(–1)). The answer is 9.
You already found f(g(x)) in question 23 — it’s 4x4 – 20x3 + 13x2 + 30x + 2. Now, substitute –1
in for x: 4(–1)4 – 20(–1)3 + 13(–1)2 + 30(–1) + 2. Simplify by dealing with all the exponents first:
4(1) – 20(–1) + 13(1) + 30(–1) + 2. Simplify further by multiplying: 4 + 20 + 13 – 30 + 2. Add and
subtract to finally end up with 9.
C
Find g(h(3)). The answer is 22 – 5 11 .
You didn’t find g(h(x)) in any other problem, but that doesn’t mean you have to. Remember
that these are read right to left. This question is asking you to plug 3 into h and then plug that
answer into g. Start with h(3) =
= 22 – 5 11 .
. Now plug this value in g and find g
( 11 ) :
69
70
Part I: Foundation (And We Don’t Mean Makeup!)
D
Find the domain of
. The domain is all real numbers.
You found the composition of
polynomial:
4
in question 23, and you ended up with a longish
3
2
= 4x – 20x + 13x + 30x + 2. Because it’s a polynomial, there’s nothing
weird about the domain. It’s all real numbers.
E
Find the domain of h(f(x)). The domain is x < 0.48 and x > 5.52.
2
h(f(x)) = 3 x − 18 x + 8 , which you found in question 25. This puts a polynomial under a
square root. A square root’s radicands have to be positive. Find where 3x2 – 18x + 8 is positive by
setting it greater than or equal to 0 and solving using the quadratic formula. This gives you two
critical values:
. Place these test values on a number line and look at the intervals that
are determined by them: x < 0.48, 0.48 < x < 5.52, and x > 5.52. If you plug test values from each
interval into the inequality, you discover which intervals work and which don’t. In this case,
x has to be less than 0.48 and bigger than 5.52, thus the answer x < 0.48 and x > 5.52.
F
Find the domain of (f + h)(x). The domain is x ≥ –2⁄3.
(f + h)(x) = x 2 − 6 x + 2 + 3 x + 2 . This adds a square root to a polynomial, so the new combined function must follow all the rules that the square root function did by itself.
G
Find the domain of (h/g)(x). The domain is x ≥ –2⁄3, except x = 0 and x = 5⁄2.
(h/g)(x) =
. The square root in the numerator restricts the domain to x ≥ –2⁄3. The
polynomial in the denominator has undefined values x = 0 and x = 5⁄2. These are both in the
restricted domain, so they become part of the answer. You express it as one neat sentence:
“The domain is x ≥ –2⁄3, except x = 0 and x = 5⁄2.”
H
Graph the inverse of g(x) =
( x − 2 ) . See the previous graph for the answer.
No more graphing! We mean it this time . . . in this chapter anyway. Start off by graphing the
square root function shifted to the right by 2. Points on this graph include (2, 0), (3, 1), and
(6, 2). Flip them to get (0, 2), (1, 3), and (2, 6) — all points on the inverse function graph. We
show both in the graph.
f-1(x)
f(x)
Chapter 3: Understanding the Function of Functions
I
Find the inverse of k(x) =
. The answer is f–1(x) =
.
Switch x and f(x) and name the new inverse by its real name:
. Notice the inverse
is in the numerator and the denominator. The only way you can solve for it is to get rid of the
fraction first by multiplying both sides by the denominator and getting x(f–1(x) – 1) = 3f–1(x).
Distribute the x and get x f–1(x) – x = 3 f–1(x). Get everything with the inverse in it to one side
and everything else to the other side: xf–1(x) – 3f–1(x) = x. Factor out the GCF on the left, the
.
inverse function: f–1(x)(x – 3) = x. Now divide the leftovers to solve for the inverse: f–1(x) =
J
Determine whether f(x) = x3 – 1 and g(x) =
they’re not inverses.
First, find the composition
.
you can stop. They’re not inverses.
K
Determine whether f(x) =
they are inverses.
=
=
inverses.
are inverses of each other. The answer is that
=
. This doesn’t simplify to get x, so
and g(x) = 1 – 2x are inverses of each other. The answer is yes,
. One down, one to go.
. Okay, you checked both; it’s official, they’re
71
72
Part I: Foundation (And We Don’t Mean Makeup!)
Chapter 4
Go Back to Your Roots
to Get Your Degree
In This Chapter
䊳 Solving quadratic equations by factoring
䊳 Exploring methods to solve quadratic equations that don’t factor
䊳 Figuring out how many roots a polynomial has
䊳 Finding the roots of a polynomial
䊳 Using roots and the leading coefficient test to graph polynomials
A
polynomial is any expression with more than one term in it. The highest exponent on
any term in a polynomial is its degree. In this chapter, we review solving polynomial
equations to find the solutions, which are also called roots or zeros. We start with a review
of solving quadratic equations — polynomials where the highest exponent is two. Then we
move into equations with higher degrees and show you how to solve them. We also take a
look at using roots to factor polynomials and how to graph polynomials.
Reason Through It: Factoring a
Factorable Polynomial
Before getting started on the nitty-gritty, here’s some vocabulary you should know to be successful in this chapter (and after):
Standard form: What most textbooks use to write a quadratic equation: ax2 + bx + c = 0
Quadratic term: The term with the second degree: ax2
Linear term: The term with the linear degree: bx
Constant: The term with zero degree: c
Leading coefficient: The number multiplying the term with the highest degree: a
74
Part I: Foundation (And We Don’t Mean Makeup!)
In math, the process of breaking down a polynomial into the product of two polynomials with a smaller degree is called factoring. In general, factoring works best on quadratic equations and is always the first thing you should try when asked to solve
second-degree polynomials. Some types of factoring (like the difference of cubes or
grouping — more on those later in this section) may work on higher degree polynomials, and you should always check them to see if they’re factorable first. When presented with a polynomial and asked to solve it, you should always try the following
methods of factoring, in order:
⻬ Greatest common factor: The greatest common factor, or GCF, is the biggest
expression that will divide into all the other terms. It’s a little like doing the distributive property backwards.
Break each term down into prime factors, look at all those factors to see what
they share in common (that’s your GCF), factor the GCF out from every term by
putting it in front of a set of parentheses, and leave the factors that aren’t the
GCF inside the parentheses.
⻬ The polynomial is a binomial: If the polynomial has two terms, check to see
whether it’s a difference of squares or the sum or difference of cubes.
Difference of squares a2 – b2 always factors to (a – b)(a + b)
Difference of cubes a3 – b3 always factors to (a – b)(a2 + ab + b2)
Sum of cubes a3 + b3 always factors to (a + b)(a2 – ab + b2)
⻬ The polynomial is a trinomial: Try using the FOIL method backwards.
Some teachers teach the “guess and check method,” where you keep trying different pairs of binomials until you happen to stumble on the right one. This isn’t
fun by any means, and you could try all day long and never figure it out (or
maybe the polynomial is prime and won’t factor).
We recommend using the British method (also known as the FOIL method backwards) instead. Follow these steps to use this method:
1. Multiply the quadratic term and the constant term. You only do this in
your head (or somewhere else on your paper) and you only do it to proceed to the next step.
2. Write down all the factors of the result of Step 1, in pairs. Again, you do
this for you only and also to make sure that you list every possibility —
that’s why it’s not guess and check. If you list them all and none of them
work (see Step 3), you know your trinomial is prime.
3. Find the pair from the list in Step 2 that adds to produce the linear term.
Only one of them will work, and if none of them do, it’s prime.
4. Break up the linear term into two terms — the winning pair from Step 3.
You’ve now created a polynomial with four terms. Proceed to the next type of
factoring — a polynomial with more than three terms.
⻬ The polynomial has more than three terms: Try grouping the polynomial.
Group the polynomial into two sets of two. Find the GCF for each set and factor it
out. Find the GCF of the two remaining expressions and factor it out. You end up
with two binomials, exactly what you were looking for!
After you have the polynomial factored, you can use the zero product property to
solve it by setting each factor equal to 0 and solving.
Chapter 4: Go Back to Your Roots to Get Your Degree
Q.
Solve the equation 3x2 + x – 2 = 0.
Q.
Solve the equation 3x3 – 3x = 0.
A.
x = 2⁄3, x = –1. Multiply the quadratic term
and the constant term: (3x2)(–2) = –6x2.
Write down all factors of this, in pairs: –x
and 6x, x and –6x, –2x and 3x, 2x and –3x.
The pair that adds up to the linear term is
–2x and 3x. Split the middle term into two
using this pair: 3x2 – 2x + 3x – 2 = 0. Now
that you have four terms instead of three,
use grouping to factor it: x(3x – 2) + 1(3x –
2) = 0. Notice that the second two factors
only have a GCF of one; you still factor it
out. Now there’s a GCF again — both sets
of terms share (3x – 2) so that can factor
out to the front: (3x – 2)(x + 1) = 0. Finally,
use the zero product property to solve
the equation. If 3x – 2 = 0, then x = 2⁄3; and if
x + 1 = 0, then x = –1.
A.
x = 0, 1, and –1. Always check for the GCF
first and factor it out: 3x(x2 – 1) = 0. Now
recognize the “leftovers” as a difference of
squares which factors again: 3x(x – 1)(x +
1) = 0. Set each factor equal to 0 and solve:
3x = 0, x = 0. x – 1 = 0, x = 1. x + 1 = 0, x = –1.
1.
Solve the equation 2y2 + 5y = 12.
2.
Solve the equation 16m2 – 8m + 1 = 0.
Solve It
3.
Solve the equation x3 + x2 = 9x + 9.
Solve It
Solve It
4.
Solve the equation
Solve It
.
75
76
Part I: Foundation (And We Don’t Mean Makeup!)
Get Your Roots Done while Solving
a Quadratic Polynomial
What happens when a quadratic equation doesn’t factor? You’re done, right? Well, not
quite. You have two more methods you can use. One we always recommend (the quadratic formula) and the other we don’t (completing the square). However, your teacher
may require you to use both, so we include both here. When you get to graph quadratics in Chapters 3 and 12, it’s easiest to do by completing the square and then using the
rules of transforming a parent function to get the graph.
Completing the square
Completing the square is the technique to use when you’re specifically told to do so.
Other than that, save it for graphing. Here are the steps:
1. Make sure the quadratic is written in standard form: ax2 + bx + c = 0.
2. Add (or subtract) the constant term from both sides: ax2 + bx = –c.
3. Factor out the leading coefficient from the quadratic term and the linear term:
.
4. Divide the new linear coefficient by two:
; square this:
and add this inside the parentheses:
;
.
5. Keep the equation balanced by multiplying the leading coefficient by the
term you just added in Step 4:
, and adding it to the other side:
.
6. Divide the leading coefficient from both sides:
.
7. Factor the trinomial on the left side of the equation:
.
8. Take the square root of both sides:
9. Solve for x:
.
.
Quadratic formula
Of course, those of you who know the quadratic formula should vaguely recognize the
steps above — they’re the derivation of the quadratic formula. All you have to do is
find the common denominator of both the fractions inside the square root, add them
together, and watch the square root simplify. Ultimately, you end up with the quadratic
formula:
Chapter 4: Go Back to Your Roots to Get Your Degree
Q.
Solve the equation 5x2 – 12x – 2 = 0.
A.
x=
. Don’t forget to
check your square roots and simplify them
. This equation doesn’t factor,
as well:
so you use the quadratic formula to
solve it. a = 5, b = –12, c = –2. Plug these
values into the quadratic formula:
. Finally, 2 goes into
every coefficient and constant in the answer,
so it simplifies even further to
. Now simplify it:
5.
Solve x2 – 10 = 2x.
Solve It
7.
Solve x2 – 4x – 7 = 0 by completing the
square.
Solve It
77
6.
Solve 7x2 – x + 2 = 0.
Solve It
8.
Solve –2.31x2 – 4.2x + 6.7 = 0.
Solve It
.
78
Part I: Foundation (And We Don’t Mean Makeup!)
Climb the Mountains by Solving
High Order Polynomials
The greater the degree of your given polynomial, the harder it is to solve the equation by
factoring. You should always still try that first because you never know . . . it may actually work! When factoring fails, however, you begin anew with a longer and more complicated process for finding the roots. We walk you through each step, one by one.
Always begin by finding the degree of the polynomial because it gives you some very
important information about your graph. The degree of the polynomial tells you the
greatest maximum number of roots — it’s that easy. A fourth-degree polynomial will have
up to, but no more than, four roots.
Determining positive and negative roots:
Descartes’ Rule of Signs
When you know the total number of roots, you can use Descartes’ Rule of Signs to determine how many of the roots are positive and how many are negative. This literally tells
you how many times your graph crosses the x-axis on the negative side as well as on the
positive side. These will all be real roots represented as points on the real number line
that is the x-axis. All you have to be able to do is count!
Make sure that the polynomial f(x) is written in descending order first, from highest degree
to lowest. Look at the sign of each term and count how many times the sign changes from
positive to negative and vice versa. The number of sign changes represents the maximum
number of positive real roots. The rule also says that this number decreases by 2 over and
over again until you end up with 1 or 0 (more on this in the section on imaginary roots).
This gives you the list of the possible number of real positive roots.
Descartes also figured out that if you take a look at f(–x) and count again, you discover
the maximum number of negative real roots. Remember that negative numbers raised to
even powers are positive, and negative numbers raised to odd powers are negative. This
means that f(–x) changes from f(x) only on the odd degrees. Each odd exponent becomes
the opposite of what it was in f(x). Count the number of times the sign changes in this
function, subtract 2 over and over until you end up at 1 or 0, and end up with a list of the
possible number of real negative roots.
Counting on imaginary roots
Imaginary roots happen in a quadratic equation when the radicand is negative. Remember
from Algebra II that you should look at the discriminant (the part of the quadratic formula
under the root sign: b2 – 4ac). If the discriminant is negative, the roots are imaginary. The ±
sign also tells you that there are two of these roots, always in pairs. This is why in the previous section you subtract by 2; you have to account for the fact that the roots may be in
pairs of imaginary numbers. In fact, the pairs will always be complex conjugates of each
other — if one root is a + bi, for example, then the other one is a – bi.
The Fundamental Theorem of Algebra says that every polynomial has at least one root in
the complex number system. Chapter 11 explains complex numbers in depth, but for
now all you need to know is that a complex number has both a real and an imaginary
Chapter 4: Go Back to Your Roots to Get Your Degree
part. That’s why there has to be at least one complex root — it has to be either real, imaginary,
or both at the same time. Kinda cool, huh?
Because you know the total number of possible roots and the list of possible positive and negative roots, you can use all that information to determine how many imaginary roots a polynomial
has. Pair up every possible number of positive roots with every possible number of negative
roots. The remaining number of roots in each situation represents the imaginary roots.
Getting the rational roots
The Rational Root Theorem helps you narrow down the possibilities even further. Right now, if
you’ve gone through all the steps, you only know the total number of roots, how many are positive real, how many are negative real, and how many are imaginary. That still leaves an infinite
number of possibilities! The Rational Root Theorem helps you because it finds the possible roots
that are rational (those that can be written as a fraction). The problem with the theorem? Not all
roots are rational. Keep in mind that some (or all) of the roots are irrational.
To use the Rational Root Theorem, take all the factors of the constant term and divide by all the
factors of the leading coefficient. This produces a list of fractions that are all possibilities for
roots. You could try plugging each one of these possibilities into the original function in the
hopes of finding a root (remember, they’re also called zeros because the value of the function
will be 0). This process is long and tedious because, each time, you’re dealing with the original
function. If there are 50 roots, you’re not helping yourself by plugging and chugging. Instead,
move on to the next step.
Synthetic division finds some roots
With the list from the last section in front of you, pick one fraction and try it to see if it works. If it
does, the quotient is a depressed polynomial. No, it’s not sad — its degree will be less than the
one you started with. You use this quotient to find the next one, each time lessening the degree,
which narrows down the roots you have to find. At some point, your polynomial will end up as a
quadratic equation, which you can solve using factoring or the quadratic formula. Now that’s
clever! If the root you try doesn’t work, you should always try it again to see if it’s a root with
multiplicity — that is, roots that are used more than once.
Here are the steps to use for synthetic division:
1. Make sure the polynomial is written in descending order. If any degrees are missing, fill
in the gaps with zeros.
2. Write the number that’s the root you’re testing outside the synthetic division sign. Write
the coefficients of the polynomial in descending order and include any zeros from Step
1 inside the synthetic division sign.
3. Drop the first coefficient down.
4. Multiply the root on the outside and this coefficient. Write this product above the synthetic division line.
5. Add the next coefficient and the product from Step 4. This answer goes below the line.
6. Multiply the root on the outside and the answer from Step 5.
7. Repeat over and over again until you use all the coefficients.
This process is easier to see with an example. Hold on and we’ll show you. Just know that when
you do synthetic division, you end up with a list of roots that actually work in the polynomial.
79
80
Part I: Foundation (And We Don’t Mean Makeup!)
Q.
Find the roots of the equation x3 + x2 – 5x +
3 = 0.
A.
x = 1 (double root), x = –3. We go through
the whole process described in this section
for this example question.
The number of roots: First, this equation
is third degree, so it may have up to
three different roots.
Descartes’ Rule of Signs: Next, by looking at f(x) = x3 + x2 – 5x + 3, you notice
that the sign changes twice (between the
second and third terms and the third and
fourth terms). This means there could be
two or zero positive real roots. Next, look
at f(–x) = –x3 + x2 + 5x + 3 and notice the
sign only changes once, giving you only
one negative real root.
Imaginary roots: So if two roots are positive and one is negative, that leaves none
leftover that are imaginary. But if zero
are positive and one is negative, that
leaves two imaginary roots.
Rational Root Theorem: Take all the factors of 3 (the constant term) and divide
by all the factors of 1 (the leading coefficient) to determine the possible rational
roots — ± 1⁄1, ± 3⁄1. Reduce the fractions
and discard any duplicates to get the
final list: ± 1, ± 3.
Synthetic division: Pick a root, any root,
and use synthetic division to test and
see if it actually is a root. Because we
know the answers (we did write the
question), we have you start with x = 1:
1 1 1 –5 –3
1 2 –3
1 2 –3 0
The last column on the right is the
remainder; because it’s 0, you know you
have one root: x = 1. Also notice that the
other numbers are the coefficients of the
depressed polynomial you’re now working with: x2 + 2x – 3 = 0. Because this is a
quadratic, we recommend shifting gears
and factoring it to (x + 3)(x – 1) = 0 to be
able to use the zero product property to
solve and get x = –3 and x = 1 (again —
making it a double root!).
Q.
Solve the equation x3 + 8x2 + 22x + 20 = 0.
A.
x = –2, x = –3 ± i. This equation is also a
third degree, so it will have a maximum of
three roots. Looking at f(x) = x3 + 8x2 + 22x
+ 20 reveals that none of them are positive.
Looking at f(–x) = –x3 + 8x2 – 22x + 20 reveals
that either three or one of them are negative. If zero are positive and three are negative, there can’t be any imaginary roots.
However, if zero are positive and only one
is negative, two of them have to be imaginary. The Rational Root Theorem generates
this list of fractions (and we’re only looking
at the negatives because we know there
aren’t any positive roots): –1⁄1, –2⁄1, –4⁄1, –5⁄1,
–10⁄1 and –20⁄1. These all reduce, respectively,
to –1, –2, –4, –5, –10, and –20. Start off with
x = –2 to discover one of your roots:
–2 1 8 22 20
–2 –12 –20
1 6 10 0
The reduced polynomial you’re now working with is x2 + 6x + 10. This quadratic doesn’t factor, so you use the quadratic formula
to find that the last two roots are indeed
imaginary: x = –3 ± i.
Chapter 4: Go Back to Your Roots to Get Your Degree
9.
Solve the equation 2x3 + 3x2 – 18x + 8 = 0.
Solve It
11.
Solve the equation 12x4 + 13x3 – 20x2 + 4x = 0.
Solve It
Solve the equation x3 + 7x2 + 13x + 4 = 0.
Solve It
10.
12.
Find the roots of the equation x4 + 10x3 +
38x2 + 66x + 45.
Solve It
Strike That! Reverse It! Using Roots
to Find an Equation
The factor theorem says that if you know the root of a polynomial, then you also know a
factor of the polynomial. These two go back and forth, one to the other — roots and factors are interchangeable. Your textbook may ask you to factor a polynomial with a degree
higher than two, and it just won’t factor using any of the techniques we describe in the
earlier sections. In this case, you must find the roots and use them to find the factors.
If x = c is a root, then x – c is a factor and vice versa. It always works, and that’s something you can count on. Nice, huh?
81
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Part I: Foundation (And We Don’t Mean Makeup!)
Q.
Use the roots of x3 + x2 – 5x + 3 = 0 to factor
the equation.
A.
(x – 1)2(x + 3) = 0. This is the question from
the first example in the last section. You
found that the roots are x = 1 (double root)
13.
If the roots of a polynomial are x = –3, –2, 4,
and 6, what’s the polynomial?
Solve It
15.
and x = –3. Using the factor theorem, if
x = 1 is a root, then x – 1 is a factor (twice);
and if x = –3 is a root, then x – (–3), or x + 3,
is a factor. This means that x3 + x2 – 5x + 3 = 0
factors to (x – 1)2(x + 3) = 0.
14.
If the roots of a polynomial are x = 2 and
4 ± 3i, what’s the polynomial?
Solve It
Factor the polynomial 6x4 – 7x3 – 18x2 + 13x
+ 6 = 0.
Solve It
16.
Factor the polynomial x4 + 10x3 + 38x2 + 66x
+ 45.
Solve It
Graphing Polynomials
Now that you have your list of the roots of your polynomial, you’ve done the hard
work to graph the polynomial. Remember that roots or zeros are x-intercepts — you
now know where the graph crosses the x-axis. Follow these steps to get to the graph:
1. Mark the x-intercepts on your graph.
2. Find the y-intercept by letting x = 0. The shortcut? It will always be the constant
term.
Chapter 4: Go Back to Your Roots to Get Your Degree
83
3. Use the leading coefficient test to determine which of the four possible ways
the ends of your graph will point:
a. If the degree of the polynomial is even and the leading coefficient is positive, both ends of the graph will point up.
b. If the degree of the polynomial is even and the leading coefficient is negative, both ends of the graph will point down.
c. If the degree of the polynomial is odd and the leading coefficient is positive,
the left side of the graph will point down and the right side will point up.
d. If the degree of the polynomial is odd and the leading coefficient is negative,
the left side of the graph will point up and the right side will point down.
4. Figure out what happens in between the x-intercepts by picking any x-value
on each interval and plugging it into the function to determine if it’s positive
(and, therefore, above the x-axis) or negative (below the x-axis).
5. Plot the graph by using all the information you’ve determined.
Q.
Graph the equation f(x) = x3 + x2 – 5x + 3.
A.
See the graph in Figure 4-1. This is the
first example from the section on solving
higher order polynomials again. You found
that the roots are x = 1 (double root) and
Figure 4-1:
The graph of
f(x) = x3 + x2
– 5x + 3.
x = –3. The y-intercept is the constant y = 3.
The leading coefficient test tells you the
graph starts by pointing down and ends by
pointing up. The double root at x = 1 makes
the graph “bounce” and not cross there.
84
Part I: Foundation (And We Don’t Mean Makeup!)
Q.
Graph the equation f(x) = x3 + 8x2 + 22x + 20.
20
Figure 4-2:
The graph of
f(x) = x3 + 8x2
+ 22x + 20.
A.
See the graph in Figure 4-2. This is the
second example from the section on
solving higher order polynomials. You
found one real root of x = –2, as well as the
complex conjugates x = –3 ± i.
Chapter 4: Go Back to Your Roots to Get Your Degree
17.
Graph f(x) = x4 + 2x3 – 13x2 – 14x + 24.
Solve It
19.
Graph f(x) = 12x4 + 13x3 – 20x2 + 4x.
Solve It
18.
Graph f(x) = 6x4 – 7x3 – 18x2 + 13x + 6.
Solve It
20.
Graph f(x) = x4 + 10x3 + 38x2 + 66x + 45.
Solve It
85
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Part I: Foundation (And We Don’t Mean Makeup!)
Answers to Questions on Finding Roots
a
Solve the equation 2y2 + 5y = 12. The answer is y = 3⁄2, –4.
Begin with any quadratic equation by getting 0 on one side of the equation. In this case, subtract 12 from both sides: 2y2 + 5y – 12 = 0. Now, begin factoring by multiplying the leading
term (2y2) and the constant term (–12) to get –24y2. List all the factors of this, in pairs: –y
and 24y, y and –24y, –2y and 12y, 2y and –12y, –3y and 8y, 3y and –8y, –4y and 6y, 4y and –6y.
The correct pair that adds up to the linear term is –3y and 8y. Split up the trinomial
into a polynomial using this magic pair: 2y2 – 3y + 8y – 12 = 0. Factor by grouping:
y(2y – 3) + 4(2y – 3) = 0. Look at both terms and notice that each contains (2y – 3) —
that’s a greatest common factor! Factor out the GCF: (2y – 3)(y + 4) = 0. Use the zero
product property: 2y – 3 = 0, 2y = 3, y = 3⁄2 and y + 4 = 0, y = –4.
b
Solve the equation 16m2 – 8m + 1 = 0. The answer is m = 1⁄4.
How come there’s only one answer? Oh right, it’s a double root, probably. Let’s factor it and
find out: (16m2)(1) = 16m2, whose factors are m and 16m, –m and –16m, 2m and 8m, –2m and
–8m, 4m and 4m, and –4m and –4m. The winning pair is the last one. Now, create the polynomial 16m2 – 4m – 4m + 1 = 0 and group it to get 4m(4m – 1) – 1(4m – 1) = 0. Next, factor
out the GCF: (4m – 1)(4m – 1) = 0. Notice that both factors are the same. Your answer is the
same root twice! 4m – 1 = 0, 4m = 1, m = 1⁄4.
c
Solve the equation x3 + x2 = 9x + 9. The answer is x = –3, –1, and 3.
You need to get 0 on one side first: x3 + x2 – 9x – 9 = 0 will do. If you group the polynomial
into two sets of two, you get two greatest common factors: x2(x + 1) – 9(x + 1) = 0. This also
has a GCF in it: (x + 1)(x2 – 9) = 0. Notice that the right factor is a difference of squares and
will factor again: (x + 1)(x – 3)(x + 3) = 0. Set each factor equal to 0 and solve: x + 1 = 0,
x = –1; x – 3 = 0, x = 3; x + 3 = 0, x = –3.
d
Solve the equation
. The answer is x = 2 and x = –6.
We decided to make things different and get 0 on one side first. We’re kidding, of course!
You always have to get 0 on one side to solve polynomials that are second degree or higher.
. Next, we multiply every term by the least common multiple of 6 to get rid of
those pesky fractions. This gives you the polynomial x2 + 4x – 12 = 0. This factors to
(x + 6)(x – 2) = 0. The zero product property gets you to the two solutions: x = –6 and x = 2.
e
Solve x2 – 10 = 2x. The answer is
.
2
Get 0 on one side first: x – 2x – 10 = 0. This equation doesn’t factor, so use the quadratic
formula to solve.
x=
f
=
.
Solve 7x2 – x + 2 = 0. The answer is no solution.
This equation also doesn’t factor, so use the quadratic formula to solve.
x=
.
That negative sign under the square root tells you that you can stop — no solution exists.
87
Chapter 4: Go Back to Your Roots to Get Your Degree
g
Solve x2 – 4x – 7 = 0 by completing the square. The answer is
.
This time you’re asked to complete the square. Make sure you always follow your teacher’s directions. Begin by adding the 7 to both sides: x2 – 4x = 7. Now factor out the leading coefficient:
1(x2 – 4x) = 7. Take half of –4 and square it, and add that inside the parentheses to get 1(x2 – 4x +
4) = 7. You need to keep the equation balanced by multiplying the coefficient and the new term you
just added inside the parentheses and adding that to the opposite side. Because the coefficient
is 1, that’s not that hard: 1(x2 – 4x + 4) = 7 + 4. Now factor the trinomial: 1(x – 2)2 = 11, and divide
the leading coefficient: (x – 2)2 = 11. Square root both sides: x – 2 = ± 11 . Add the 2: x =
h
.
2
Solve –2.31x – 4.2x + 6.7 = 0. The solutions are approximately –2.36 and 6.56.
Those ugly decimals should make you reach immediately for a calculator and plug away at the
quadratic formula.
x=
.
Take your time through these types of problems. Simplify to get the final two answers: –2.36
and 6.56.
i
Solve the equation 2x3 + 3x2 – 18x + 8 = 0. The zeros are x = –4, 1⁄2, and 2.
This third-degree equation has at most three real roots. The two changes in sign in f(x) show
two or zero positive roots, and the one change in sign in f(–x) shows one negative root. The list
of possible rational zeros is: ± 1, ± 1⁄2, ± 2, ± 4, ± 8.
Start off by testing x = 2.
2 2 3 –18 8
4 14 –8
2 7 –4 0
The depressed polynomial is 2x2 + 7x – 4, which factors to (2x – 1)(x + 4), which tells you that
the other two roots are x = 1⁄2 and –4.
j
Solve the equation 12x4 + 13x3 – 20x2 + 4x = 0. The roots are x = 0, 1⁄4, –2, and 2⁄3.
Factor out the GCF in all the terms first: x(12x3 + 13x2 – 20x + 4) = 0. The first factor gives you
one solution immediately: x = 0. Now concentrate on the leftover polynomial inside the parentheses and solve: 12x3 + 13x2 – 20x + 4 = 0. This has three real roots, two or zero of which are
positive and one of which is negative. The list of possibilities is: ± 1, ± 1⁄2, ± 1⁄3, ± 1⁄4, ± 1⁄6, ± 1⁄12, ± 2,
± 2⁄3, ± 4, and ± 4⁄3.
Start off by testing x = –2.
–2 12 13 –20 4
–24 22 –4
12 –11 2 0
The depressed polynomial this time is 12x2 – 11x + 2. This factors to (4x – 1)(3x – 2), which gets
you to the last two roots: x = 1⁄4 and x = 2⁄3.
k
Solve the equation x3 + 7x2 + 13x + 4 = 0. The answers are x = –4 and
.
This cubic has a maximum of three real roots. None of them are positive and three or one of
them are negative. The list of possibilities this time (ignoring all the positives) is: –1, –2, and –4.
88
Part I: Foundation (And We Don’t Mean Makeup!)
Start off by testing –4.
–4 1
7 13 4
–4 –12 –4
1 3 1 0
The depressed polynomial x2 + 3x + 1 doesn’t factor, but the quadratic formula reveals that the
last two solutions are x =
l
.
Find the roots of the equation x4 + 10x3 + 38x2 + 66x + 45. The roots are x = –3 (double root) and
x = –2 ± i.
This fourth-degree polynomial also has no positive roots; 4, 2, or 0 are negative roots. The list
of possibilities to pick from is: –1, –3, –5, –9, 15, –45.
Start off with x = –3.
–3 1 10 38 66 45
–3 –21 –51 –45
1 7 17 15 0
This time, when you test it again, it works.
–3 1 7 17 15
–3 –12 –15
1 4 5 0
You’re left with the depressed polynomial x2 + 4x + 5, which doesn’t factor, but you can use the
quadratic formula to find that the last two roots are imaginary: x = –2 ± i.
m
If the roots of a polynomial are x = –3, –2, 4, and 6, what’s the polynomial? The answer is x4 –
5x3 – 20x2 + 60x + 144.
Use the factor theorem to help you figure this one out. If x = –3, then x + 3 is one of the factors.
Similarly, if x = –2, then x + 2 is a factor; if x = 4, then x – 4 is a factor; and if x = 6, then x – 6 is a
factor. If you take all the factors and multiply them, you get (x + 3)(x + 2)(x – 4)(x – 6). FOIL the
first two binomials to get x2 + 5x + 6 and the second two binomials to get x2 – 10x + 24. Multiply
your way through those two polynomials:
x2 + 5x + 6
x2 – 10x + 24
24x2 + 120x + 144
3
–10x – 50x2 – 60x
x4 + 5x3 + 6x2
x4 – 5x3 – 20x2 + 60x + 144
You end up with the polynomial x4 – 5x3 – 20x2 + 60x + 144.
n
If the roots of a polynomial are x = 2, 4 ± 3i, what’s the polynomial? The answer is x3 – 10x2 +
41x – 50.
This time the factors are x – 2, x – 4 – 3i, and x – 4 + 3i. In cases like these, it’s easier to multiply
the imaginary numbers first. When you do that you end up with the trinomial x2 – 8x + 25. Now
multiply that by the binomial to end up with the polynomial: x3 – 10x2 + 41x – 50.
Chapter 4: Go Back to Your Roots to Get Your Degree
o
Factor the polynomial 6x4 – 7x3 – 18x2 + 13x + 6 = 0. The answer is (x – 2)(x – 1)(3x + 1)(2x + 3).
You’re still using the factor theorem, but this time you have to find the roots first. The roots are
x = 2, 1, –1⁄3, and –3⁄2. This means that x – 2, x – 1, x + 1⁄3, and x + 3⁄2 are your factors. You can get rid
of those fractions by multiplying each term of the factor by the LCD. In other words, multiply
x + 1⁄3 by 3 and x + 3⁄2 by 2. This finally gives you (x – 2)(x – 1)(3x + 1)(2x + 3).
p
Factor the polynomial x4 + 10x3 + 38x2 + 66x + 45. The answer is (x2 + 4x + 5)(x + 3)2.
This is problem 12. It has two imaginary roots: x = –2 ± i and x = –3, a double root. This means
your factors are (x + 2 + i)(x + 2 – i)(x + 3)(x + 3). You multiply out the two imaginary roots to
come up with a polynomial factor and get (x2 + 4x + 5)(x + 3)2.
q
Graph f(x) = x4 + 2x3 – 13x2 – 14x + 24. See the following graph for the answer.
The x-intercepts are: x = –4, –2, 1, and 3. Mark those on the graph first. Then find the y-intercept: y = 24. The leading coefficient test tells you that both ends of this graph point up. Here’s
the graph:
y
24
x
r
Graph f(x) = 6x4 – 7x3 – 18x2 + 13x + 6. See the following graph for the answer.
You found the roots for this polynomial in problem 15: x = –3⁄2, –1⁄3, 1, and 2. Mark those on the
graph first. Then find that y = 6 is the y-intercept. The leading coefficient test tells you that both
ends of this graph point up. Here’s the graph:
89
90
Part I: Foundation (And We Don’t Mean Makeup!)
s
Graph f(x) = 12x4 + 13x3 – 20x2 + 4x. See the following graph for the answer.
This is problem 10, where you found that the solutions are x = –2, 0, 1⁄4, and 2⁄3. This polynomial
has no constant, so y = 0 is the y-intercept. This graph crosses at the origin. The leading coefficient test tells you that both ends of the graph point up. Here’s the graph:
t
Graph f(x) = x4 + 10x3 + 38x2 + 66x + 45. See the following graph for the answer.
Problem 12 has roots of x = –3 (as a double root); the other roots are imaginary. The graph will
bounce at this point. The y-intercept is 45. The leading coefficient test tells you that both ends
of the graph point up. Here’s the graph:
45
Chapter 5
Exponential and Logarithmic Functions
In This Chapter
䊳 Figuring out exponential functions
䊳 Looking at logarithmic functions
䊳 Using exponents and logs to solve equations
䊳 Working with exponential word problems
E
xponential growth is simply the idea that something gets bigger and bigger (or smaller
and smaller) very fast. Exponential and logarithmic functions can be used to describe
growth or decay. They have many practical applications, such as determining population
growth, measuring the acidity of a substance, and calculating financial growth. In addition,
they’re central to many concepts in calculus (a good reason to master them in pre-calculus!).
They’re different from the other functions we’ve been dealing with so far because their variables are no longer in the base of the expression (more on that later).
In this chapter, you practice solving equations, simplifying expressions, and graphing exponents and logarithms. In addition, you can practice manipulating functions to solve equations
and practically applying the concepts to word problems.
Things Get Bigger (Or Smaller) All the Time —
Solving Exponential Functions
Exponential functions are functions in which the variable is in the exponent. When the base of
the exponent is greater than 1, the function gets really big really fast, and when it’s less than
1, it gets really small really fast.
In exponential functions, the variable is in the power of the expression. The base can be any
constant, including a special constant that mathematicians and scientists define as e. This
irrational constant, e, has a value that’s approximately 2.7182 and is extremely useful in exponential expressions (and in logarithms, but we’re getting ahead of ourselves).
Solving exponential equations requires that you recall the basic exponent rules:
a
b
a+b
c ·c =c
–a
c =
1
ca
a b
a·b
(c ) = c
a
a
(c · d) = c · d
a
92
Part I: Foundation (And We Don’t Mean Makeup!)
0
c =1
a
b
if c = c then a = b
While graphing exponential equations, it’s important to recall the tricks for transforming graphs (see Chapter 3 for a refresher).
Q.
Solve for x in 8
A.
x = –4. First, in order to utilize our exponential rules, it’s helpful if both expressions are the same base. So, knowing that
3
4
8 = 2 and 16 = 2 , by factoring and rewriting using exponents, you can rewrite both
sides of the equation with a base of 2:
3(4x + 12)
4(2x + 5)
=2
. Now that your bases are
2
the same, you can set your exponents
equal to each other (using properties of
exponents): 3(4x + 12) = 4(2x + 5). Next,
you can simplify using the distributive
property of equality: 12x + 36 = 8x + 20.
Finally, you can solve algebraically:
4x + 36 = 20; 4x = –16; x = –4.
4x + 12
2x + 5
= 16
.
16
12
8
D
4
C
–12
–8
–4
A
E
–4
Figure 5-1:
Transformations of the
exponential
equation
with base 2.
B
–8
–12
–16
4
8
12
Q.
Sketch the graphs of (A) y = 2 , (B) y = 2 + 1,
x+3
–x
x
(C) y = 2 , (D) y = 2 , and (E) y = –2 , all
on the same set of axes.
A.
Graphs B–E are all transformations of the
first graph A (see Chapter 3 for a review
of transformations of graphs). By adding 1
to graph A, the result is graph B, a shift up
of 1 unit. By adding 3 to the exponent of
graph A, the result is graph C, shifted 3
units to the left. Graph D is the result of
making the exponent negative, which
results in a reflection over the y-axis, and
graph E, created by negating the base,
results in the reflection of the graph over
the x-axis. See the resulting graphs in
Figure 5-1.
x
x
Chapter 5: Exponential and Logarithmic Functions
1.
Solve for x in 27
x+3
3x – 9
= 81
.
Solve It
3.
2x
x
Solve for x in (5 – 1)(25 – 125) = 0.
6x + 8
=e
.
4.
x
Solve for x in 3 · 9 – 8 = –7.
Solve It
x
Sketch the graph of y = –3 – 4.
Solve It
2x – 4
Solve for x in e
Solve It
Solve It
5.
2.
6.
x–2
Sketch the graph of y = –3e
.
Solve It
The Only Logs You Won’t Cut:
Solving Logarithms
Just as multiplying by the reciprocal is another way to write division, logarithms are
simply another way to write exponents. Exponential and logarithmic functions are
inverses of each other. In other words, logarithmic functions are really just another
way to write exponential functions. So you may ask, “Why do you need both?” Well,
logarithms are extremely helpful for an immense number of practical applications. In
fact, before the invention of computers, logarithms were the only way to compute
many complex computations in physics, chemistry, astronomy, and engineering.
93
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Part I: Foundation (And We Don’t Mean Makeup!)
For solving and graphing logarithmic functions (logs), remember this inverse relationship and you’ll be sawing . . . er, solving logs in no time! Here’s this relationship in
equation form:
y
y = logbx b = x.
Just as with exponential functions, the base can be any number, including e. In fact, a
base of e is so common in science and calculus that loge has its own special name: ln.
Thus, logex = lnx.
Similarly, log10 is so commonly used that it’s just written as log (no base written).
Remember our review of domain from Chapter 3? Well, here’s one of those times when
domain can be tricky. The domain for the basic logarithm y = logbx is x > 0. Therefore,
when you’re solving logarithmic functions, it’s important to check for extraneous roots
(review Chapter 1).
Here are more properties that are true for any logarithm:
logb1 = 0
logbb = 1
The product rule: logb(a · c) = logba + logbc
The quotient rule:
c
The power rule: logba = c · logba
x
logbb = x
If logba = logbc, then a = c
Using these properties, simplifying logarithmic expressions and solving logarithmic
functions is a snap (we did say logs, not twigs, right?).
Q.
Rewrite the following logarithmic expression to a single log: 3log5x + log5(2x – 1) –
2log5(3x + 2).
A.
Using the properties of logs, begin by
rewriting the coefficients as exponents:
3
3log5x = log5x and 2log5(3x + 2) =
2
log5(3x + 2) . Next, rewrite the addition of
the first two logs as the product of logs:
3
3
log5x + log5(2x – 1) = log5x (2x – 1).
Last, rewrite the final log as a quotient:
3
2
log5x (2x – 1) – log5(3x + 2) =
Chapter 5: Exponential and Logarithmic Functions
Q.
A.
7.
Sketch the graphs of (A) y = log2x,
(B) y = 1 + log2x, (C) y = log2(x + 3), and
(D) y = –log2x, all on the same set of axes.
Solve It
9.
15
10
First, in Figure 5-2, you can see that graphs
B–D are transformations of graph A. Graph
B is a shift of 1 up, graph C is a shift of 3 to
the left, and graph D is a reflection of graph
A over the x-axis. Second (nifty trick here),
these are all inverses of graphs A–D in the
exponential section (refer to Figure 5-1).
Another way to graph logarithms is to
change the log to an exponential using the
properties of logarithms, find the inverse
function by switching x and y, graph the
inverse, and reflect every point over the
line y = x. For a review of inverses, see
Chapter 3. Here, we stick with transforming
the parent graph.
Rewrite the given expression as a single
logarithm: ln4x + 3ln(x – 2) – 2(ln2x +
ln(3x – 4)).
Solve for x in lnx + ln(2x – 1) = ln1.
Solve It
95
5
C
–10
–5
B A
0
5
10
D
Figure 5-2:
Transformations
of the logarithmic
equation
with base 2.
8.
–5
–10
–15
Solve for x in
.
Solve It
10.
Find logb(48b) if logb2 = 0.36 and logb3 = 0.56.
Solve It
96
Part I: Foundation (And We Don’t Mean Makeup!)
11.
Sketch the graph of y = –3 + log(x + 2).
12.
Sketch the graph of y = ln(x – 2) + 4.
Solve It
Solve It
Putting Them Together: Solving Equations
Using Exponents and Logs
Now, we show you how to put these two lovely functions together. By keeping in mind
y
the inverse relationship (y = logbx b = x), you can solve even more complex problems. Aren’t you excited?!
A helpful key to remember when solving equations using exponents and logs is that if
the variable is in the exponent, convert the equation into logarithmic form. This is
especially helpful if you use natural log (ln) or the common log (log10x, often referred
to as just logx) because you can plug the variable into your calculator to get a decimal
approximation of the solution.
One pitfall to avoid when manipulating logs relates to the products and quotients
of logs. Remember:
not
. These are entirely different
expressions. In fact, if you plug them into your calculator, you can see that
while
. The same can be said for products and
logs: log6 + log7 = log(6 · 7), not log(6 + 7).
Q.
A.
log(50x + 250) – log x = 2.
means log10):
. Because
2
x = 5. Start by combining the logs as a
quotient:
. Next, rewrite
in exponential form (remember that log
10 = 100, you can rewrite the equation as
. After cross-multiplying,
you can then solve algebraically:
50x + 250 = 100x; 250 = 50x; x = 5.
Chapter 5: Exponential and Logarithmic Functions
Q.
x
(x + 2)
Solve for x in 3 = 2
A.
algebra, you can distribute the ln2 across
(x + 2): x · ln3 = x · ln2 + 2 · ln2. Still using
algebra, get the terms with the variable on
the same side by subtracting x · ln2 to the
opposite side: x · ln3 – x · ln2 = 2 · ln2.
Then, using distributive property again,
remove the x as a greatest common factor:
x(ln3 – ln2) = 2 · ln2. Finally, isolate x by
dividing ln3 – ln2 from both sides:
.
. First, recognize that the
variable is in the exponent of each term, so
you can easily remedy that by taking either
log or ln of both sides. We’re going to use
ln, but it really doesn’t make a difference.
x
(x + 2)
x
(x + 2)
becomes ln3 = ln2
. Then,
So 3 = 2
you can use properties of logarithms to
solve. Start by changing the exponents
to coefficients: x · ln3 = (x + 2)ln 2. Using
13.
Solve for x in log(x – 6) – log(x + 3) = 1.
Solve It
15.
to simplify and get
14.
.
x
Solve for x: 3 = 5.
Solve It
x
x
Solve for x in 4 – 4 · 2 = –3.
Solve It
. Last, use the quotient rule
16.
x
(2x – 3)
Solve for x: 3 = 5
Solve It
.
97
98
Part I: Foundation (And We Don’t Mean Makeup!)
Solving Exponential Word Problems . . . Argh!
When will I ever use this? Well, in addition to using exponential functions in a mathematics course, they actually have many practical applications. Common uses of exponential functions include figuring compound interest, computing population growth,
and doing radiocarbon dating (no, not some new online matchmaking system). In fact,
these are so common, many teachers make you memorize their formulas. If you need
nonstandard formulas to do a problem, they’ll be provided in the question itself.
Here are formulas for interest rate and half-life:
⻬ Compound interest formula:
where A is the amount after t time
compounded n times per year if P dollars are invested at interest rate r.
r·t
⻬ Continuous compound interest formula: A = Pe , where A is the amount after t
time if P dollars are invested at interest rate r continuously throughout the year.
⻬ Formula for the half-life of a radioactive element: M ( x ) = c ⋅ 2 h where M(x) is the
mass at the time x, c is the original mass of the element, and h is the half-life of the
element.
−x
Q.
If you deposit $600 at 5.5% interest compounded continuously, what will your
balance be in 10 years?
Q.
How old is a piece of bone that has lost
60% of its carbon-14 (half-life of carbon-14
is 5,730 years)?
A.
$1,039.95. Because this is continuous compound interest, you use the formula A =
rt
Pe when you’re solving for A. A =
(0.055)(10)
. Plugging this into a calcula$600e
tor, you get approximately $1,039.95.
A.
Approximately 7,575 years old. We can
figure out this problem using the formula
for half-life. First, because 60% of the
carbon-14 is gone, the mass of carbon
remaining is 40%, so we can write the present mass as .40c. Therefore, the equation
−x
will be: 0.40c = c ⋅ 2 5,730 . We can start solving
this by cancelling c from both sides:
−x
0.40 = 2 5,730 . Taking the natural log of both
sides allows us to move the variable− x from
the exponent position: ln 0.40 = ln 2 5,730 ,
then
. From here, we
can solve algebraically:
;
.
Chapter 5: Exponential and Logarithmic Functions
17.
If you deposit $3,000 at 8% interest per
year compounded quarterly, in approximately how many years will the investment
be worth $10,500?
20.
The deer population in a certain area in
year t is approximately
.
When will the deer population reach 2,000?
Solve It
The half-life of Krypton-85 is 10.4 years.
How long will it take for 600 grams to decay
to 15 grams?
Solve It
Solve It
19.
18.
If you deposit $20,000 at 6.5% interest compounded continuously, how long will it take
for you to have $1,000,000?
Solve It
99
100
Part I: Foundation (And We Don’t Mean Makeup!)
Answers to Problems on Exponential
and Logarithmic Functions
a
Solve for x in 27
x+3
3x – 9
= 81
. The answer is x = 5.
3
4
3(x + 3)
4(3x – 9)
First, rewrite 27 as 3 and 81 as 3 . Simplify the power to get 3
=3
. Now that the bases
are the same, set the two exponents equal to each other: 3(x + 3) = 4(3x – 9), and then solve for x:
3x + 9 = 12x – 36; –9x = –45; x = 5.
b
2x – 4
Solve for x in e
6x + 8
=e
. The answer is x = –3.
Start by setting the exponents equal to each other: 2x – 4 = 6x + 8; then solve algebraically:
4x = –12 ; x = –3.
c
2x
x
Solve for x: (5 – 1)(25 – 125) = 0. The answer is x = 0, 3⁄2.
2
x
2x
2x
2x
Using the fact that 25 = 5 , replace 25 with 5 to get (5 – 1)(5 – 125) = 0. Next, set each factor
2x
equal to 0 using the zero product property (see Chapter 4 for a review) and solve: First, 5 – 1 =
2x
2x
0
0, 5 = 1, and, because anything to the power of 0 equals 1, 5 = 5 . Therefore, 2x = 0, x = 0.
2x
2x
3
Second, 5 – 125 = 0, 5 = 125, and because 125 is equal to 5 , rewrite the second equation as
2x
3
5 = 5 . Set the exponents equal to each other: 2x = 3, and solve for x = 3⁄2. Both solutions work.
d
x
Solve for x in 3 · 9 – 8 = –7. The answer is x = –1⁄2.
x
x
x
2x
Start by isolating the exponential expression: 3 · 9 = 1; 9 = 1⁄3. Next, replace 9 with 3 and 1⁄3
–1
2x
–1
with 3 , so 3 = 3 . Set the exponents equal to each other: 2x = –1, and solve for x = –1⁄2.
e
x
Sketch the graph of y = –3 – 4.
10
5
–10
–5
C
0
5
10
–5
–10
–15
x
The y-intercept is (0, –5). The graph of this function is the basic exponential graph of y = 3
shifted 4 units down and reflected upside down.
f
x–2
Sketch the graph of y = –3e
.
The y-intercept is (0, –0.406). The graph of this function is the basic exponential graph of y = e
shifted 2 units to the right, reflected upside down, and contracted by a unit of 3.
x
Chapter 5: Exponential and Logarithmic Functions
101
2
1.6
1.2
0.8
0.4
–1.2 –0.8 –0.4 0
–0.4
0.4
0.8 1.2
–0.8
–1.2
–1.6
–2
g
Rewrite the given expression as a single logarithm: ln4x + 3ln(x – 2) – 2(ln2x + ln(3x – 4)). The
answer is
.
3
2
2
Begin by rewriting coefficients as exponents: ln4x + ln(x – 2) – (ln(2x) + ln(3x – 4) ). Next,
3
2
2
rewrite the first two logarithms as a single product: ln4x(x – 2) – (ln4x + ln(3x – 4) ). Then
3
2
2
use the product rule in the parentheses to get ln4x(x – 2) – (ln4x (3x – 4) . Finally, write the
difference of logarithms as a quotient and reduce the 4 in the numerator and denominator:
.
h
Solve for x in: log74 + log7(x + 4) – 2log72 = log7(x – 2) + 1⁄2 log79. The answer is x = 5.
The first step is to write the coefficients as exponents:
. Next, rewrite the sums and differences as the
products and quotients of logarithms:
set
. Solve algebraically:
. Using the rules of logarithms,
;
;x+4=
3(x – 2); x + 4 = 3x – 6; 2x = 10; x = 5.
i
Solve for x in: lnx + ln(2x – 1) = ln1. The answer is x = 1. Rewriting the sum of natural logs as a product, we get: lnx(2x – 1) = ln1. Then, using rules of logarithms, set x(2x – 1) = 1 and solve algebraically:
2
2
2x – x = 1, 2x – x – 1 = 0. Factor the quadratic: (2x + 1)(x – 1) = 0, and using the zero product property, set each factor equal to 0. The solutions are x = –1⁄2 and x = 1, but –1⁄2 < 0 so it’s extraneous, and
the only solution is x = 1.
j
Find logb(48b) if logb2 = 0.36 and logb3 = 0.56. The answer is 3.
Start by expanding the logarithm into the sum of two logs: logb48+ logbb. Next, factoring the
4
4
48 into 16 · 3, expand again: logb16 + logb3 + logbb. Using the fact that 16 = 2 , write the first log: logb2 +
logb3 + logbb. Then, write the exponent as a coefficient: 4logb2 + logb3 + logbb. Last, replace logb2 with
0.36, logb3 with 0.56, and logbb with 1 and simplify: 4(0.36) + 0.56 + 1 = 3.
102
Part I: Foundation (And We Don’t Mean Makeup!)
k
Sketch the graph of y = –3 + log (x + 2).
7.5
5
2.5
–5
–2.5
0
2.5
5
–2.5
–5
–7.5
The y-intercept is (0, –2.699). The graph of this function is the basic exponential graph of y =
logx shifted 2 units to the left and 3 units down.
l
Sketch the graph of y = ln(x – 2) +4.
15
10
5
–10
–5
0
5
10
–5
–10
–15
There’s a vertical asymptote at x = 2. The graph of this function is the basic exponential graph
of y = lnx shifted 2 units to the right and 4 units up.
m
n
Solve for x in log(x + 6) – log(x – 3) = 1. The answer is x = 4.
Begin by writing the difference of logs as a quotient: log x + 6 = 1 . Next, rewrite the logarithm as
x−3
an exponent: x + 6 = 10 1 . Then, solve algebraically: x + 6 = 10(x – 3); x + 6 = 10x – 30; 36 = 9x; x = 4.
x−3
ln 5
x
Solve for x: 3 = 5. The answer is x = ln 3 or x ≈1.464.
x
First, take the natural log of both sides: ln3 = ln5. Then use the power rule to simplify: x · ln3 = ln5.
ln 5
Last, divide both sides by ln3: x = ln 3 or x ≈1.464.
103
Chapter 5: Exponential and Logarithmic Functions
o
p
ln 3
x
x
Solve for x in: 4 – 4 · 2 = –3. The answer is x = ln 2 ≈1.585 and x = 0.
2
x
2x 2x
x
2x
Start by using the fact that 4 = 2 and rewrite 4 as 2 : 2 – 4 · 2 = –3. Add 3 to both sides: 2 – 4 ·
x
x 2
x
x 2
2 + 3 = 0. Notice that this is the same thing as (2 ) – 4 · 2 + 3 so that you can substitute y for 2 : y –
4y + 3 = 0. Now, you can factor and solve using zero product property: (y – 3)(y – 1) = 0;
x
x
x
y = 3 and y = 0. Then, resubstitute 2 for y: 2 = 3 and 2 = 1. Taking the natural log of each side, you
ln 3
x
can solve for x by using the rules of logarithms: ln2 = ln3; xln2 = ln3; x = ln 2 , your first solution.
ln 1
ln 1
x
Finally, ln2 = ln1; xln2 = ln1; x = ln 2 ; because ln1 = 0, ln 2 = 0, your second solution. Both
solutions work.
x
(2x – 3)
Solve for x: 3 = 5
≈ 2.28.
: The answer is x =
x
(2x – 3)
First, take the natural log of both sides: ln3 = ln5
. Then use properties of logarithms to solve.
Start by changing the exponents to coefficients: x · ln3 = (2x – 3)ln 5. Using algebra, you can distribute the ln5 across (2x – 3): x · ln3 = 2x · ln5 – 3 · ln5. Still using algebra, get the terms with the variable
on the same side by subtracting 2x · ln5 to the opposite side: x · ln3 – 2x · ln5 = –3 · ln5. Then, using
distributive property again, remove the x as a greatest common factor:
x(ln3 – 2ln5) = –3 · ln5. Combine the difference of logarithms as a quotient:
Next, isolate x using algebra:
nents:
q
=
.
. Finally, simplify your answer using the rules of expo-
, which equals approximately 2.28.
If you deposit $3,000 at 8% interest per year, compounded quarterly, in approximately how many
years will the investment be worth $10,500? The answer is approximately 15.82 years.
Using the equation:
, where A = $10,500, P = $3,000, r (as a decimal) = 0.08, and n = 4:
4t
4t
4t
; 10,500 = 3,000 (1.02) ; 3.5 = (1.02) ; using logarithms: log3.5 = log1.02 ;
log 3.5 = 4tlog 1.02; finally, solve algebraically:
;t=
, which equals
approximately 15.82 years.
r
The half-life of Krypton-85 is 10.4 years. How long will it take for 600 grams to decay to 15 grams?
The answer is 55.3 years.
Using the half-life formula: M ( x ) = c ⋅ 2
−x
h
, where M(x) = 15 grams, the original mass c is 600
−x
−x
grams, and the half-life h is 10.4, we get 15 = 600 ⋅ 2 10.4 . Simplifying to 0.025 = 2 10.4 , we can solve
−x
using logarithms: ln 0.025 = ln 2 10.4 ,
which approximates to 55.3 years.
s
,
, and
The deer population in a certain area in year t is approximately
,
. When will
the deer population reach 2,000? The answer is approximately 11.4 years.
Here, you simply plug in 2,000 for P(t) and solve:
–0.56t
1 + 299
–0.56t
= 1.5; 299e
approximately 11.4 years.
–0.56t
= 0.5; e
= 0.00167; –0.56t = ln0.00167;
–0.56t
; 2,000(1 + 299e
) = 3,000;
, which equals
104
Part I: Foundation (And We Don’t Mean Makeup!)
t
If you deposit $20,000 at 6.5% interest compounded continuously, how long will it take for you
to have $1,000,000? The answer is approximately 24.8 years.
r·t
Using the equation for continuous compound interest: A = Pe , where the amount A is $1,000,000,
the initial investment P is $20,000, and the interest rate r in decimal form is 0.065, we get:
0.065t
0.065t
. Simplifying and using logarithms to solve: 50 = e
; ln50 = 0.065t;
1,000,000 = 20,000e
, which equals approximately 60.2 years.
Part II
Trig Is the Key: Basic
Review, the Unit
Circle, and Graphs
Y
In this part . . .
ou should be familiar with the basics of trigonometry
from your Geometry class — right triangles, trig
ratios, and angles, for example. But your Algebra II course
may or may not have expanded on those ideas to prepare
you for the direction that pre-calc is going to take you. For
this reason, we assume that you’ve never seen this stuff
before. We don’t want to leave you behind on our journey.
This part begins with trig ratios and word problems and
then moves on to the unit circle: how to build it and how
to use it. We solve some trig equations and make and measure arcs. Graphing trig functions is a major component of
pre-calc, so we walk you through how to graph each of the
six functions.
Chapter 6
Basic Trigonometry and the Unit Circle
In This Chapter
䊳 Working with the six trigonometric ratios
䊳 Making use of right triangles to solve word problems
䊳 Using the unit circle to find points, angles, and right triangle ratios
䊳 Isolating trig terms to solve trig equations
䊳 Calculating arc lengths
A
h . . . trigonometry, the math of triangles! Invented by the ancient Greeks, trigonometry
is used to solve problems in navigation, astronomy, and surveying. Think of a sailor
lost at sea. All he has to do is triangulate his position against two other objects, such as two
stars, and calculate his position using — you guessed it — trigonometry!
In this chapter, we review the basics of right triangle trigonometry. Then we show you how to
apply it to the unit circle, a very useful tool for graphically representing trigonometric ratios
and relationships. From there, you can solve trig equations. Finally, we combine these concepts so that you can apply them to arcs. The ancient Greeks didn’t know what they started
with trigonometry, but the modern applications are endless!
It’s All Right-Triangle Trig — Finding
the Six Trigonometric Ratios
Dude! Did you see that? He just did a 2π on his board! Huh? Oh . . . we mean a 360. In geometry, angles are measured in degrees (°), with 360° describing a full circle on a coordinate
plane (or skateboard). However, in pre-calculus, you also use another measure for angles:
radians. Radians, from the word radius, are usually designated without a symbol. Because
both radians and degrees are used often in pre-calc, you see both used here.
To convert radians to degrees and vice versa, you use the fact that 360° = 2π radians, or
180° = π radians. Therefore, to convert degrees to radians, you simply multiply by the ratio
π
⁄180°. Similarly, to convert radians to degrees, you can multiply by the ratio 180°⁄π.
When solving right triangles or finding all the sides and angles (θ), it’s important to remember the six basic trigonometric ratios: sine (sinθ), cosine (cosθ), tangent (tanθ), cosecant
(cscθ), secant (secθ), and cotangent (cotθ). The first three are the most important to remember, as the second three are inverses of the first. In other words, sinθ = 1⁄cscθ, cosθ = 1⁄secθ, and
tanθ = 1⁄cotθ.
108
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Acronyms! We love ‘em . . . LOL — otherwise known as Laugh Out Loud! One of the
most famous acronyms in math is SOHCAHTOA. It helps you remember the first three
trigonometric ratios:
Opposite
Hypotenuse
Cosine = Adjacent
Hypotenuse
Tangent = Opposite
Adjacent
Sine =
Q.
Given ∆KLM in Figure 6-1, find sin∠K.
hypotenuse
opposite
adjacent
Q.
Solve ∆RST (see Figure 6-2).
M
Figure 6-1:
Finding
sin∠K in
∆KLM.
A.
R
5
L
3
K
sin∠K = 5 34 . Because sin∠K = opp⁄hyp,
34
you first need to find the hypotenuse. To
do this, you need to use the Pythagorean
Theorem, which says that (leg)2 + (leg)2 =
(hypotenuse)2. Using this, you find that
32 + 52 = (hyp)2, and therefore 34 = (hyp)2,
34 . Plugging this
5
.
into your sine ratio, you get sin∠K =
34
But if you recall what we review in Chapter 2,
you need to rationalize this fraction.
Doing so, you get sin∠K = 5 34 .
34
so the hypotenuse is
Figure 6-2:
Solving
∆RST. T
A.
RT =
2
3
S
13 , ∠T = 33.7°, and ∠R = 56.3°.
Remember, solving a triangle means finding
all the angles and sides. So you start by
using the Pythagorean Theorem to find the
hypotenuse: 22 + 32 = (hyp)2, hyp = 13 .
Next, use any trigonometric ratio to find an
angle. You can use sin∠T = 2 . To get
13
the angle by itself, you use the fact that the
inverse operation of sin is sin–1, or arcsine.
⎛
⎞
Thus, you get ∠T = sin–1 ⎜ 2 ⎟ , which
⎝ 13 ⎠
you can find using your calculator to be
33.7°. Or, if you want to use radians, ∠T is
0.59. We prefer degrees for now. Lastly,
using the fact that the angles of a triangle
add up to 180°, you can find ∠R:
180 – (90 + 33.7°) = 56.3°.
Chapter 6: Basic Trigonometry and the Unit Circle
1.
2.
Find cosA in ∆ABC.
Solve It
Solve ∆DEF.
Solve It
4
B
D
C
25˚
e
f
41
F
A
3.
Find ∠Q in ∆QRS (round to the nearest
tenth).
Solve It
4.
13
R
12
S
E
Find the six trigonometric ratios of ∠R in
∆QRS from question 3.
Solve It
Q
11
109
110
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Solving Word Problems with Right Triangles
Uh-oh! The dreaded word problems! Don’t let these scare you. We’ve got some easy
steps to help you through them. First, as with most word problems in math, we suggest you draw a picture. That way you can visualize the problem and it’s not as scary.
Second, remember that these are just right triangles. Therefore, all you have to do is
use what you already know about right triangles to solve the problems. Simple!
Angle of elevation and angle of depression (see Figure 6-3) are two terms that come up
often in right triangle word problems. They just refer to whether the angle rises from
the horizon, angle of elevation, or falls from the horizon, angle of depression.
Figure 6-3:
The ups and
downs:
angle of elevation and
angle of
depression.
Angle of elevation
Angle of depression
Q.
When the sun is at an angle of elevation of
32°, a building casts a shadow that’s 152
feet from its base. What is the height of the
building?
A.
The building is approximately 95 feet tall.
Okay, remember your steps. Step one, draw
a picture:
32˚
152
Step two, recall what you know about right
triangles. Because you want to find the
building’s height, which is opposite the
angle, and you have the shadow length,
which is adjacent to the angle, you can use
the tangent ratio. Setting up your ratio, you
get tan32° = x⁄152, or x = 152 · tan32. Using a
calculator, you find that the building height
is approximately 95 feet.
111
Chapter 6: Basic Trigonometry and the Unit Circle
Q.
A.
5.
Two boat captains whose boats are in a
straight line from a lighthouse look up to
the top of the lighthouse at the same time.
The captain of boat A sees the top of the
40-foot lighthouse from an angle of elevation of 45°, while the captain of boat B sees
the top of the lighthouse from an angle of
elevation of 30°. How far are the boats from
each other, to the nearest foot?
40
The boats are 29 feet apart. Ooh . . . this is
a tricky one! But don’t let it scare you —
it’s completely doable! First, remember to
draw a picture. In this case, you may want
to draw three: one for the lighthouse and
both boats, and two separate pictures each
with one boat:
Romero wants to deliver a rose to his girlfriend, Jules, who is sitting on her balcony
24 feet above the street. If Romero has a
28-foot ladder, at what angle must he place
the bottom of the ladder to reach his love,
Jules?
Solve It
a
O
45˚
b
A
30˚
B
From the picture, you can see that to find
the distance between the boats, you need
to find the distance that each boat is from
the base of the lighthouse and subtract boat
A’s distance from the distance of boat B.
Because the angle of elevation is 45° for
boat A, you can set up the trigonometric
ratio: tan45° = 40⁄a. Solving for a, you find that
the distance from boat A to the base of the
lighthouse is 40 feet. Similarly, you can set
up a trigonometric ratio for boat B’s distance: tan30° = 40⁄b. Solving, you get that
b = 69 feet. Subtracting these two distances,
you find that the distance between the
boats is 29 feet. Whew!
6.
Sam needs to cross a river. He spies a bridge
directly ahead of him. Looking across the
river, he sees that he’s 27° below the bridge
from the other side. How far must he walk
on his side of the river to reach the bridge
if the bridge length is 40 feet?
Solve It
112
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
7.
Paul, a 6-foot-tall man, is holding the end of
a kite string 5 feet above the ground. The
string of the kite is 75 feet long at 35° of elevation. Paulette, Paul’s 5-foot-tall daughter,
is directly underneath the kite. How far
above Paulette’s head is the kite?
Solve It
8.
To hold up a 100-foot pole, two guide wires
are put up on opposite sides of the pole.
One wire makes a 36° angle with the
ground and the other makes a 47° angle
with the ground. How far apart are the
bases of the wires?
Solve It
Unit Circle and the Coordinate Plane:
Finding Points and Angles
The unit circle is a very useful tool in pre-calculus. The information it provides can help
you solve problems very quickly. Essentially, the unit circle is a circle with a radius (r) of
one unit, centered on the origin of a coordinate plane. If you think of the trigonometric
ratios you’ve been dealing with in terms of x and y values, where x is adjacent to the
angle, y is opposite the angle, and r is the hypotenuse, that allows you to make a right
triangle by using a point on the unit circle and the x-axis. This is often called point-inplane, and it results in an alternate definition of the six trigonometric ratios:
⻬ sinθ = y⁄r
⻬ cscθ = r⁄y
⻬ cosθ = x⁄r
⻬ secθ = r⁄x
⻬ tanθ = ⁄x
⻬ cotθ = x⁄y
y
When graphing on a coordinate plane, it’s important how you measure your angles. In
pre-calculus, the angle always begins on the positive side of the x-axis, called the initial
side. Any angle in this position is in standard position. The angle can extend to anywhere on the plane, ending on what’s called the terminal side. Any angles that have different measures but have the same terminal side are called co-terminal angles. These
can be found by adding or subtracting 360° or 2π to any angle.
From the initial side, an angle that moves in the counterclockwise direction has a positive
measure, and an angle that moves in the clockwise direction has a negative measure.
Chapter 6: Basic Trigonometry and the Unit Circle
Q.
Find three co-terminal angles of 520°.
A.
Sample answers: 160°, –200°, and 880°,
but other answers are possible. To get
these, you simply add or subtract multiples of 360° from 520°. 520° – 360° = 160°;
520° – 2 · 360° = –200°; and 520° + 360° =
880°.
113
Q.
Evaluate the six trigonometric ratios of the
point (2, –3).
A.
sinθ = −3 13 , cosθ = 2 3 , tanθ = –3⁄2,
13
13
13 , secθ = 13 , cotθ = 2⁄–3.
−3
2
Start by finding the radius using the
Pythagorean Theorem: 22 +(–3)2 = r2, 4 + 9 =
cscθ =
r2,
2
13 = r , 13 = r. Then, simply plug
into the trigonometric ratios given x = 2,
y = –3, and r = 13 . Don’t forget to rationalize any radicals in the denominator!
sinθ = −3 = −3 13 , cosθ = 2 = 2 13 ,
13
13
13
13
tanθ = –3⁄2, cscθ =
13 , secθ =
−3
13 ,
2
cotθ = 2⁄–3.
9.
Find three co-terminal angles of π .
5
Solve It
10.
Find two positive co-terminal angles of
–775°.
Solve It
114
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
11.
Evaluate the six trigonometric ratios of the
point (3, 4).
Solve It
13.
Evaluate the six trigonometric ratios of the
point (–2, 2 3 ).
Solve It
12.
Evaluate the six trigonometric ratios of the
point (–5, –7).
Solve It
14.
Evaluate the six trigonometric ratios of the
point (6, −3 5 ).
Solve It
Chapter 6: Basic Trigonometry and the Unit Circle
Isn’t That Special? Finding Right Triangle
Ratios on the Unit Circle
Well, isn’t that special? Yes, it is special — special right triangles that is! Remember
your geometry teacher drilling in 30-60-90 and 45-45-90 triangles? Well, they’re back!
And with good reason, as they’re very common in pre-calculus, and they’re the foundation of the unit circle. Using these two special triangles, you can find the specific trig
values that you can see on a completed unit circle (to see one, check out the Cheat
Sheet in the front of this book).
One important point to remember about the unit circle is that the radius is 1.
Therefore, the hypotenuse of any right triangle drawn from a point to the x-axis is 1.
Thus, for any point, (x, y), you know that x2 + y2 = 1.
Recalling 30-60-90 triangles, the sides are in the ratio of 1: 3 : 2 . Therefore, if you want
the hypotenuse to be 1, as it is in the unit circle, divide each side by 2. Similarly, the
sides of 45-45-90 triangles are in the ratio of 1:1: 2 . Converting to a unit circle, the
values are
2 : 2 : 1.
2
2
Now, using the point-in-plane definition, the six trigonometric ratios are easy to find. In
fact, because the hypotenuse is now 1, sinθ = x⁄r now becomes sinθ = x. Similarly, cosθ =
y
⁄r now becomes cosθ = y. Thus, any point on the unit circle is now (cosθ, sinθ). Imagine
the possibilities!
If you don’t have a unit circle handy, you can always use reference angles ( ) to find
your solutions. A reference angle is the angle between the x-axis and the terminal side
of an angle. It’s different for each quadrant (see Figure 6-4). If the original angle is θ,
then the reference angle in quadrant I is θ. In quadrant II, the reference angle is 180° – θ.
For quadrant III, the reference angle is θ – 180°. Lastly, quadrant IV’s reference angle is
360° – θ.
Quadrant II
Figure 6-4:
Referencing
reference
angles.
Quadrant I
θ’ = 180˚– θ
θ’ = θ
θ’ = θ –180˚
θ’ = 360˚– θ
Quadrant III
Quadrant IV
115
116
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Q.
Evaluate the six trigonometric ratios of
120° on the unit circle.
A.
sinθ =
cscθ =
3
, cosθ = –1⁄2, tanθ = – 3 ,
2
Q.
What’s the value of θ when sinθ = 1⁄2 and
90° < θ < 360°?
A.
θ = 150°. On a unit circle, sinθ = 1⁄2 when
θ = 30° and 150°. Because you’re limited to
90° < θ < 360°, the answer is just 150°.
16.
Find θ when cosθ =
2 3 , secθ = –2, and cotθ = − 3 .
3
3
Start by finding the reference angle on th3e
unit circle. Knowing the angle is in quadrant II, the reference angle is 180° – 120°,
or 60°. You now have a 30-60-90 triangle!
Therefore, x = -1⁄2 and y = 3 . Now you
2
can easily find the trig ratios using the
point-in-plane definition. Keep in mind that
r = 1. You start with sine, cosine, and tangent: sinθ = y = 3 (look at the defi2
nition of y earlier in this explanation —
that’s correct, as is the sin at the beginning
of the answer); cosθ = x = –1⁄2; tanθ = y⁄x =
. Next,
do the reciprocal ratios: cscθ = 1⁄y =
2 = 2 3 ; secθ = 1⁄x =
3
3
and cotθ = x⁄y =
15.
= –2;
=– 3 .
Evaluate the six trigonometric ratios of
225°.
Solve It
Solve It
3 and 0° < θ < 360°.
2
Chapter 6: Basic Trigonometry and the Unit Circle
17.
Evaluate the six trigonometric ratios of
330°.
18.
Find θ when tanθ =
3 and 0° < θ < 360°.
3
Solve It
Solve It
Solving Trig Equations
Solving trigonometric equations is just like solving regular algebraic equations, with a
twist! The twist is the trigonometric term. Now, instead of isolating the variable, you
need to isolate the trigonometric term. From there, you can use the handy unit circle
to find your solution. For a complete unit circle, check out the Cheat Sheet in the front
of this book.
Given what you already know about co-terminal angles, you know that any given equation may have an infinite number of solutions. Therefore, for these examples, you stick
with angles that are within one positive rotation of the unit circle 0 ≤ x ≤ 2π. But be
sure that you check for multiple solutions within that unit circle!
117
118
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Q.
Solve 2sinx = 1 in terms of radians.
A.
x = π⁄6, or 5π⁄6. Because you already know
how to solve 2y = 1, you also know how to
solve 2sinx = 1 — it’s sinx = 1⁄2. The question
is, what do you do with it from there? Well,
now you need to find the angle or angles
that make the equation true. Here’s where
that unit circle comes in handy! Remembering that sinθ = y, you can look at the unit
circle to find which angles have y = 1⁄2. The
two angles are π⁄6 and 5π⁄6.
19.
Solve for θ in 3tanθ – 1 = 2.
Solve It
21.
Solve for θ in 2cos2θ – 1 = 0.
Solve It
Q.
Solve 2cos2x – cosx = 1, giving answers in
terms of degrees.
A.
x = 0°, 120°, 240°, and 360°. Don’t let this
one trip you up! Just keep in mind your
amazing basic algebra skills. If you think of
this as 2y2 – y = 1, you see that it’s a simple
quadratic that you need to try and factor,
and then use the zero product property to
solve: 2cos2x – cosx – 1 = 0 factors into
(2cosx + 1)(cosx – 1) = 0. Using the zero
product property: 2cosx + 1 = 0, so 2cosx =
–1 and cosx = –1⁄2, or cosx – 1 = 0, so cosx =
1. Now it’s time to use those reference
angles! Ask yourself, when is cosx = –1⁄2?
Well, considering that cosθ = x on the unit
circle, it’s clear that your reference angle
( ) is 60°, and your answer falls in quadrants II and III. Therefore, the resulting
angles are in quad II (180° – 60° = 120°)
and quad III (180° + 60° = 240°). For your
second equation, cosx = 1 and x = 0° or 360°,
because the only place on the unit circle
that x = 1 is on the x-axis. Therefore, your
four solutions are 0°, 120°, 240°, and 360°.
20.
Solve for θ in sin2θ = sinθ.
Solve It
22.
Solve for θ in 4sin2θ + 3 = 4.
Solve It
Chapter 6: Basic Trigonometry and the Unit Circle7
23.
Solve for θ in 4sin4 – 7sin2θ + 3 = 0.
Solve It
24.
Solve for θ in tan2θ – tanθ = 0.
Solve It
Making and Measuring Arcs
If someone asks you how far an ant on the edge of a 6-inch CD travels if the CD spins at
120°, you’d probably ask yourself, why do I care? You may even be thinking that the
ant is probably messing up your CD player! But wacky math teachers love coming up
with questions like that, so we’re here to help you solve them.
To calculate the measure of an arc, a portion of the circumference of a circle like the
path that pesky ant is taking, you need to remember that arcs can be measured in two
ways: as an angle and as a length. As an angle, there’s nothing to calculate — it’s
simply the same measure as the central angle. As a length, the measure of the arc is
directly proportional to the circumference of the circle. Here’s a nifty formula to give
you this value:
Figure 6-5:
Calculating
arc length
and the
variables
involved.
r
θ
s
119
120
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Q.
Let’s find out about that ant! A pesky ant is
on the edge of a 6-inch CD. How far does
the ant travel if the CD spins at 120°?
A.
The ant travels approximately 6.3 inches.
First, remember that the angle needs to be
25.
Find the length of an arc in a circle with a
radius of 4 feet if the central angle is π⁄6.
Solve It
27.
Find the length of an arc in a circle with a
radius of 18 feet if the angle is 210°.
Solve It
in radians, so multiply the 120° by π⁄180° to
get 2π⁄3. The diameter is 6 inches, so the
radius is 3 inches. Use the formula s = θ · r =
2π
⁄3 · 3 = 2π inches, which is approximately
6.3 inches.
26.
Find the length of an arc in a circle with a
diameter of 16 centimeters if the central
angle is 7π⁄4.
Solve It
28.
Find the radius of an arc in a circle that
has an arc length of 42 inches if the angle is
2 radians.
Solve It
Chapter 6: Basic Trigonometry and the Unit Circle
Answers to Problems
a
Find cosA in ∆ABC. The answer is cosA = 5 41 .
41
Because cosθ = adj⁄hyp, you need to find the adjacent side using the Pythagorean Theorem:
2
5
41 , x2 + 16 = 41, x2 = 25, x = 5. Therefore, cosA =
x2 + 42 =
. Rationalizing the denominator,
41
you get 5 41 .
41
( )
b
Solve ∆DEF. The answer is ∠E = 65°, side e = 23.6 cm, and side f = 26 cm.
c
First, because you know ∠D and ∠F, you can find ∠E by subtracting ∠D and ∠F from 180°:
180° – (25° + 90°) = 65°. To find side e, you can use tan65° = e⁄11. Multiplying both sides by 11,
you get e = 11 · tan65° = 23.6 cm. To find side f, you can use sin25° = 11⁄f. Multiply both sides by f:
f · sin25° = 11. Divide by sin25° to get f = 11 , which is approximately 26 cm.
sin 25
Find ∠Q in ∆QRS (round to the nearest tenth). The answer is approximately 22.6°.
Because you have the adjacent side to ∠Q and the hypotenuse, you use cosine: cosQ = (12⁄13). To
solve, take the inverse cosine of each side: Q = cos–1(12⁄13), which is approximately 22.6°.
d
Find the six trigonometric ratios of ∠R in ∆QRS from question 3. The answer is sinR = 12⁄13,
cosR = 5⁄13, tanR = 12⁄5, cscR = 13⁄12, secR = 13⁄5, and cotR = 5⁄12.
Start by using the Pythagorean Theorem to find the third side: 122 + q2 = 132, 144 + q2 = 169,
q2 = 25, q = 5. Then, plug the sides into the trigonometric ratios: sinR = 12⁄13, cosR = 5⁄13, tanR = 12⁄5,
cscR = 13⁄12, secR = 13⁄5, cotR = 5⁄12.
e
Romero wants to deliver a rose to his girlfriend, Jules, who is sitting on her balcony 24 feet
above the street. If Romero has a 28-foot ladder, at what angle must he place the bottom of the
ladder to reach his love, Jules? The answer is 59°.
To solve, draw a picture:
24
28
With the picture, you can see that you have the opposite side, 24 feet, from the angle you want
and the hypotenuse, 28 feet. Therefore, you can use the sine ratio to solve: sinθ = 24⁄28. To isolate
the angle, use inverse sine: θ = sin–1(24⁄28), which is approximately 59°.
121
122
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
f
Sam needs to cross a river. He spies a bridge directly ahead of him. Looking across the river, he
sees that he’s 27° below the bridge from the other side. How far must he walk on his side of the
river to reach the bridge if the bridge length is 40 feet? The answer is 79 more feet.
First, draw a picture:
40
Bridge
x
27˚
Considering that you have the adjacent side from the angle, 40 feet, and you’re looking for the
opposite side, you can use the tangent ratio: tan27° = 40 . Multiplying both sides by x, you get
x
40 · tan27° = x. Dividing 40 by tan27°, you get that x equals approximately 79 feet.
g
Paul, a 6-foot-tall man, is holding the end of a kite string 5 feet above the ground. The string of
the kite is 75-feet long at 35° of elevation. Paulette, Paul’s 5-foot-tall daughter, is directly underneath the kite. How far above Paulette’s head is the kite? The answer is about 43 feet.
Begin by (you guessed it!) drawing a picture:
x
75
35˚
Chapter 6: Basic Trigonometry and the Unit Circle
Because Paul is holding the end of the kite string at the same height as Paulette’s head, you
only need to consider the string of the kite, which forms the hypotenuse of the triangle and the
angle. Because you’re looking for the opposite side from the angle and you have the hypotenuse,
use the sine ratio to solve: sin35° = x⁄75. Multiplying both sides by 75, you get x = sin35° · 75,
which is approximately 43 feet.
To hold up a 100-foot pole, two guide wires are put up on opposite sides of the pole. One wire
makes a 36° angle with the ground and the other makes a 47° angle with the ground. How far
apart are the bases of the wires? The answer is about 231 feet apart.
100
h
36˚
47˚
Using your picture, you can set up two tangent ratios: tan36° = 100⁄x and tan47° = 100⁄y. Multiply
both sides by x and y respectively: x · tan36° = 100 and y · tan47° = 100. Isolate the variables
100 and y = 100 . Therefore, x is approximately 137.6 feet and y is
tan 47 tan 36 approximately 93.3. Add these together to get that the total distance apart is about 231 feet.
by dividing: x =
i
Find three co-terminal angles of π⁄5. Although there are multiple answers, three possible
answers are 11π⁄5, –9π⁄5, and 21π⁄5.
Simply add or subtract multiples of 2π: π⁄5 + 2π = 11π⁄5; π⁄5 – 2π = –9π⁄5; and π⁄5 + 2 · 2π = 21π⁄5.
j
Find two positive co-terminal angles of –775°. Two possible answers are 305° and 665°.
Here, just add multiples of 360° to –775° until you get at two positive co-terminal angles: –775° +
3 · 360° = 305°; –775° + 4 · 360° = 665°.
k
l
Evaluate the six trigonometric ratios of the point (3, 4). sinθ = 4⁄5, cosθ = 3⁄5, tanθ = 4⁄3, cscθ = 5⁄4,
secθ = 5⁄3, and cotθ = 3⁄4.
First, find the radius using the Pythagorean Theorem: 32 + 42 = r2, 9 + 16 = r2, 25 = r2, 5 = r. Using
this and x =3, y = 4, plug into the trigonometric ratios: sinθ = 4⁄5, cosθ = 3⁄5, tanθ = 4⁄3, cscθ = 5⁄4,
secθ = 5⁄3, cotθ = 3⁄4.
Evaluate the six trigonometric ratios of the point (–5, –7). sinθ = − 7 74 , cosθ = − 5 74 ,
74
74
tanθ = 7⁄5, cscθ = − 74 , secθ = − 74 , and cotθ = 5⁄7.
7
5
Find your radius: (–5)2 + (–7)2 = r2, 25 + 49 = r2, 74 = r2,
74 = r. Using this and the point
7 74
;
(x, y), plug into the trig ratios and rationalize if necessary: sinθ = −7 = −
74
74
cosθ = −5 = − 5 74 ; tanθ = –7⁄5 = 7⁄5; cscθ = − 74 ; secθ = − 74 ; and cotθ = –5⁄–7 = 5⁄7.
7
74
5
74
123
124
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
m
Evaluate the six trigonometric ratios of the point (–2, 2 3 ). sinθ =
3 , cosθ = –1⁄2,
2
3 .
tanθ = − 3 , cscθ = 2 3 , secθ = –2, and cotθ = −
3
3
(
Begin by finding your radius: (–2)2 + 2 3
)
2
= r2, 4 + 12 = r2, 16 = r2, 4 = r. Now plug into your trig
3 = 3
ratios and rationalize if necessary: sinθ = 2
; cosθ = –2⁄4 = –1⁄2; tanθ = −2 = − 3 ;
4
2
2 3
cscθ =
n
4 = 2 =
3
.
2 3 ; secθ = 4⁄–2 = –2; and cotθ = −2 = − 1 = −
3
3
2 3
3
2 3
3
5
Evaluate the six trigonometric ratios of the point (6, −3 5 ). sinθ = −
, cosθ = 2⁄3,
3
5
5
5
tanθ = −
, cscθ = −3
, secθ = 3⁄2, and cotθ = −2
.
2
5
5
(
Start by finding the radius: 62 + −3 5
)
2
= r2, 36 + 45 = r2, 81 = r2, 9 = r. Plug into your trig ratios
5 = − 5 ; cosθ = 6⁄9 = 2⁄3; tanθ = −3 5 = − 5 ;
and rationalize if necessary: sinθ = −3
9
3
6
2
cscθ =
o
9 = − 3 = −3 5 ; secθ = 9⁄6 = 3⁄2; and cotθ = 6 = − 2 = −2 5 .
5
5
−3 5
5
−3 5
5
Evaluate the six trigonometric ratios of 225°. sinθ = − 2 , cosθ = − 2 , tanθ = 1,
2
2
cscθ = − 2 , secθ = − 2 , and cotθ = 1.
Using reference angles, you can see that you’re dealing with a 45-45-90 triangle (225° – 180°).
Therefore, x = − 2 and y = − 2 . Now, by using the point-in-plane definition, you can find
2
2
the sixtrig ratios. sinθ = x = − 2 ; cosθ = y = − 2 ; tanθ = y⁄x =
2
2
cscθ = 1⁄x =
p
; secθ = 1⁄y =
; and cotθ = x⁄y = y⁄x =
= 1;
= 1.
3 and 0° < θ < 360°. The answer is 60° and 300°.
2
Looking at the special right triangles, you can see that cosθ = 3 when θ is 60°. Because
2
cosine is equal to the x value on the unit circle and x is positive in quadrants I and IV, the
answer is 60° and 300°.
Find θ when cosθ =
Chapter 6: Basic Trigonometry and the Unit Circle
q
Evaluate the six trigonometric ratios of 330°. sinθ = –1⁄2, cosθ = 3 , tanθ = − 3 , cscθ = –2,
2
3
secθ = 3 , and cotθ = − 3 .
3
Considering that 330° is in quadrant IV, using reference angles (360°–330°), you find that you’re
dealing with a 30-60-90 triangle. Using the point-in-plane definition, you get sinθ = x = –1/2;
cosθ = y =
3 ; tanθ = y⁄x =
2
; and cotθ = x⁄y =
r
; cscθ = 1⁄y =
;
secθ = 1⁄x =
.
3 and 0° < θ < 360°. The answer is θ = 30° and 210°.
3
To solve this, use special right triangles. You can see that if θ is 30°, then x = 1⁄2 and y =
Find θ when tanθ =
This means that because tanθ = y⁄x, tanθ =
3.
2
, which is what you want.
Therefore, θ = 30° and 210°.
s
Solve for θ in 3tanθ – 1 = 2. The answer is θ = 45° and 225°.
Begin by using algebra to isolate tanθ: 3tanθ – 1 = 2, 3tanθ = 3, tanθ = 1. Because tanθ = y/x,
sine and cosine must be the same value for tanθ to equal 1. This occurs at = 45°. Because the
answer is positive, both sine and cosine must be the same sign, which occurs in quadrants I
and III. Therefore, using reference angles, quadrant I: θ = 45°, and quadrant III: 180° + 45° = 225°.
t
Solve for θ in sin2θ = sinθ. The answer is θ = 0, π⁄2, π, and 2π.
To solve, think of sin2θ = sinθ as x2 = x, which can be solved by bringing both terms to the same
side and factoring: x2 – x = 0, x(x – 1) = 0. Similarly, sin2θ = sinθ, sin2θ – sinθ = 0, sinθ(sinθ – 1) =
0. Therefore, sinθ = 0, or sinθ – 1 = 0, which means sinθ = 1. Knowing that sinθ = y on the unit
circle, sinθ = 0 at 0, and π, 2π and sinθ = 1 at π⁄2, you have your answers!
u
Solve for θ in 2cos2θ – 1 = 0. The answer is θ = π⁄4, 3π⁄4, 5π⁄4, and 7π⁄4.
First, isolate the cosine term using algebra: 2cos2θ – 1 = 0, 2cos2θ = 1, cos2θ = 1⁄2. Now, take the
1 =± 2
square root of each side: cos 2 θ = ± 1 . Thus, cosθ = ±
2 . This occurs at four
2
2
angles on the unit circle: π⁄4, 3π⁄4, 5π⁄4, and 7π⁄4.
v
w
Solve for θ in 4sin2θ + 3 = 4. The answer is θ = π⁄6, 5π⁄6, 7π⁄6, and 11π⁄6.
Begin by using algebra to isolate the sine term: 4sin2θ + 3 = 4, 4sin2θ = 1, sin2θ = 1⁄4. Taking the
square root of each side, you get: sin 2 θ = ± 1 ,sin θ ± 1 . This means that θ = π⁄6, 5π⁄6, 7π⁄6, and 11π⁄6.
4
2
Solve for θ in 4sin4θ – 7sin2θ + 3 = 0 in degrees. The answer is θ = 0°, 60°, 120°, 240°, 300°, 90°,
180°, 360°, and 270°.
Start by thinking of 4sin4θ – 7sin2θ + 3 = 0 as 4x4 – 7x2 + 3 = 0, which factors into (4x2 – 3)(x2 – 1) =
0. Similarly, 4sin4θ – 7sin2θ + 3 = 0 factors into (4sin2θ – 3)(sin2θ – 1) = 0. Set each factor equal to
zero and take the square root of each side to find sinθ: 4sin2θ – 3 = 0, 4sin2θ = 3, sin2θ = 3⁄4,
3
sin 2 θ = ± 4 , sinθ = ± 3 . Therefore, θ = 60°, 120°, 240°, 300°. Or, sin2θ – 1 = 0, sin2θ = 1,
2
sin 2 θ = ± 1 , sinθ = ± 1. Therefore, sinθ also equals 0°, 90°, 180°, 360°, and 270°.
125
126
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
x
Solve for θ in tan2θ – tanθ = 0. The answer is θ = 0, π, 2π, π⁄4, and 5π⁄4.
Notice that this one is similar to question 20. You can factor the same way: tan2θ – tanθ = 0,
tanθ(tanθ – 1) = 0. Setting each factor to zero: tanθ = 0 or tanθ – 1 = 0, so tanθ = 1. These occur
at 0, π, 2π, π⁄4, and 5π⁄4.
y
Find the length of an arc in a circle with a radius of 4 feet if the central angle is π⁄6. The answer is
s = 2π⁄3 feet, or approximately 2.1 feet.
Use the formula s = θ · r = π⁄6 · 4 = 2π⁄3 feet, which is approximately 2.1 feet.
A
Find the length of an arc in a circle with a diameter of 16 centimeters if the central angle is 7π⁄4.
The answer is s = 14π cm ≈ 44cm.
Start by finding the radius by dividing the diameter by two: 8 cm. Next, plug into the arc length
formula: s = θ · r = 7π⁄4 · 8 = 14π cm ≈ 44cm.
B
Find the length of an arc in a circle with a radius of 18 feet if the angle is 210°. The answer is
s = 21π feet, which is approximately 66 feet.
Begin by changing degrees to radians by multiplying 210° by π⁄180°: 210° · π⁄180° = 7π⁄6. Now, plug in the
radius and angle into the arc length formula: s = θ · r = 7π⁄6 · 18 = 21π feet, which is approximately
66 feet.
C
Find the radius of an arc in a circle that has an arc length of 42 inches if the angle is 2 radians.
The answer is 21 inches.
Just plug ‘em in! s = θ · r, 42 = 2 · r. Dividing both sides by 2, you find r = 21 inches.
Chapter 7
Graphing and Transforming
Trig Functions
In This Chapter
䊳 Exploring period graphs
䊳 Graphing sine and cosine
䊳 Picturing tangent and cotangent
䊳 Charting secant and cosecant
G
raphing trig functions is similar to graphing any other function. You simply insert values
into the input to find the output. In this case, the input is typically θ and the output is
typically y. And, just like graphing any other function, knowing the parent trig graph — the
most basic, unshifted graph — makes graphing more complex graphs easier. In this chapter,
we show you the parent graph of each trig function and its transformations.
Getting a Grip on Period Graphs
Periodic graphs are like that everlasting bunny we all know . . . they keep going, and going,
and going. Well, when graphing trig functions, if you remember that they’re periodic graphs,
then the steps to graphing them are easy! Because they repeat their values over and over
again, you just need to figure out one period (or cycle), and you can repeat it as many times
as you like.
The key is to graph one period. To do this, you need to start by graphing the parent graph,
then make transformations (just like you do in Chapter 3 for other types of graphs). Like
other graphs, trig graphs make vertical and horizontal transformations, as well as vertical
and horizontal translations:
⻬ Presto, chango! Vertical and horizontal transformations:
• For trig functions, vertical transformations are achieved by changing the amplitude, or height. To do this, simply multiply a constant by the parent function. For
example, f(π) = 2sinπ is the same as the parent graph, only its wave goes up to a
value of 2 and down to –2.
• Multiplying a negative constant to the parent graph simply flips the graph upside
down, or reflects it over the x-axis.
• Horizontal transformations occur by changing the period of the graph. For sine
and cosine parent graphs, the period is 2π. The same is true for cosecant and
secant graphs. For tangent and cotangent, the period is π. Multiplying a constant
128
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
by the angle transforms the period, or makes a horizontal transformation.
For example, f(θ) = cos 2θ results in a graph that repeats itself twice in the
amount of space the parent graph would.
⻬ Vertical and horizontal translations — shifting the parent graph up, down, left, or
right:
• Just as we show you in Chapter 3, vertical and horizontal shifts just change
the location on the graph: up, down, left, or right.
• The general equation for these shifts for sine, for example, is sin(θ – h) + v,
where h represents the horizontal shift left or right and v represents the
vertical shift up or down.
• To find the horizontal shift of a function, simply set the inside parentheses
equal to 0. For example, the horizontal shift for sin(θ + 3) is –3 because θ +
3 = 0, so θ = –3.
We suggest that when you put the transformations together, you follow this simple
order:
1. Change the amplitude.
2. Change the period.
3. Shift the graph horizontally.
4. Shift the graph vertically.
Sine and Cosine: Parent Graphs
and Transformations
Sine and cosine graphs look like waves. These waves, or sinusoids in math speak, keep
going and going like our bunny friend. To graph these sinusoids, you need to start with
checking out the parent graphs (see Figure 7-1).
1
Figure 7-1: 0
Parent
graphs of
sine and –1
cosine.
1
π
4
π
2
3π
4
π
5π
4
3π
2
7π
4
2π
0
π
4
π
2
3π
4
π
5π
4
3π
2
7π
4
2π
–1
y = sinθ
y = cosθ
Notice that the periods of both the sine and cosine graphs are the same: 2π. Similarly,
they both have an amplitude (or height) of 1. You use this information for your transformations.
Chapter 7: Graphing and Transforming Trig Functions
129
Putting together all the transformation information into one equation, you get
f(θ) = a · sin[p(θ – h)] + v
f(θ) = a · cos[p(θ – h)] + v
where a is the amplitude, h is the horizontal shift, v is the vertical shift, and you divide
2π by p to get the period.
Q.
Graph 2sinθ + 3.
A.
Starting with amplitude, you can see that
a = 2, so your amplitude is 2. The period is
2π because the period doesn’t change from
the parent equation. The vertical shift is
positive 3 because v = 3. The graph is
shown in the following figure.
5
4
3
2
1
–2π
–3π
2
–π
−π
2
π
2
0
π
3π
2
2π
–1
1.
Graph f(θ) = −1 ⋅ cosθ .
2
Solve It
2.
Graph f(θ) = cos 1 θ .
2
Solve It
130
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
3.
Graph f(θ) = sin(θ + π⁄4).
Solve It
5.
Name the amplitude, period, horizontal
shift, and vertical shift of f(θ) =
3sin(2θ + π⁄2) – 1.
Solve It
4.
Graph f(θ) = cos1⁄3 θ + 2.
Solve It
6.
Graph f(θ) = 3sin(2θ + π⁄2) – 1.
Solve It
Chapter 7: Graphing and Transforming Trig Functions
Tangent and Cotangent: Mom,
Pops, and Children
Tangent and cotangent are both periodic, but they’re not wavelike like sine and cosine.
Instead, they have vertical asymptotes that break up their graphs. As we discuss in
Chapter 3, a vertical asymptote is where the function is undefined. Because tangent and
cotangent are rational functions where
and
, they both have
values that are undefined where their denominators are equal to 0. For tangent, this
occurs on the unit circle at π⁄2 and 3π⁄2. For cotangent, this occurs at 0, π, and 2π on the
unit circle. Therefore, these are the locations of their asymptotes (see Figure 7-2).
−π
Figure 7-2:
Parent
graphs of
tangent and
cotangent.
2
2
1
1
−π −π 0
2 4
π
4
π
2
π
−π −3π −π −π 0
4 2 4
–1
–1
–2
–2
y = tanθ
π
4
π
2
3π π
4
y = cotθ
Notice that the periods of both the tangent and cotangent graphs are the same: π. The
x-intercepts for tangent are 0, π, and 2π. For cotangent, the x-intercepts are π⁄2 and 3π⁄2.
Using this information, you can make your transformations.
Putting together all the transformation information from earlier in this chapter into
one equation, you get
f(θ) = a · tan[p(θ – h)] + v
f(θ) = a · cot[p(θ – h)] + v
where a is the vertical transformation (no amplitude with tangent and cotangent), h is
the horizontal shift, v is the vertical shift, and you divide π by p to get the period.
131
132
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Q.
Graph f(θ) = 1⁄2 · tan2θ.
A.
Starting with the vertical transformation,
you can see that it’s 1⁄2. Next, find the period
by dividing π by 2, which is π⁄2. Because
there are no horizontal or vertical shifts,
you’re ready to graph:
2
1
−π −π 0
2 4
π π
4 2
–1
–2
7.
Graph f(θ) = cot 1 θ .
2
Solve It
8.
Graph f(θ) = tanθ + 2.
Solve It
Chapter 7: Graphing and Transforming Trig Functions
9.
Graph f(θ) = 1⁄3 · cotθ.
Solve It
11.
Name the amplitude, period,
horizontal shift, and vertical shift of
f(θ) = 2 · tan(θ + π⁄4) – 1.
Solve It
10.
Graph f(θ) = tan(θ – π⁄2).
Solve It
12.
Graph f(θ) = 2 · tan(θ + π⁄4) – 1.
Solve It
133
134
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
Secant and Cosecant: Generations
To graph cosecant and secant, it’s important to remember that they’re the reciprocals
of sine and cosine, respectively:
and
. Using this fact, the
easiest way to graph cosecant or secant is to start by graphing sine or cosine — the
graphs of the reciprocals are easily found from there!
Given that cosecant and secant are reciprocal functions of sine and cosine, this also
means that they’re rational functions. And just as with tangent and cotangent, rational
functions often mean asymptotes. So, yep, you guessed it — cosecant and secant
graphs have asymptotes. These occur wherever their reciprocal functions (sine or
cosine) have a value of 0. To graph, follow these easy steps:
1. Graph the sine graph with transformations to graph a cosecant graph, or
graph the cosine graph with transformations to get the secant graph.
2. Draw asymptotes where the sine or cosine functions are equal to 0.
3. Sketch the reciprocal graph of cosecant or secant between each pair of
asymptotes.
For example, if the sine graph gets bigger, the cosecant graph would get smaller.
The parent graphs of cosecant and secant are shown in Figure 7-3. The parent sine and
cosine graphs are also left for you to see where they came from.
2
2
1
1
0
π
4
π
2
3π
4
π
5π
4
–1
3π
2
7π
4
2π
0
π
4
π
2
3π
4
π
5π
4
–1
Figure 7-3:
Parent
graphs of
cosecant –2
and secant.
–2
y = cscθ
y = secθ
3π
2
7π
4
2π
Chapter 7: Graphing and Transforming Trig Functions
Q.
Graph f(θ) = cscθ + 1.
A.
Here there’s only a vertical shift of 1, which means you shift the parent graph up one (see the
following figure).
3
2
1
0
π
4
π
2
3π
4
π
3π
2
2π
5π
2
3π
–1
13.
Graph f(θ) = –cscθ – 1.
Solve It
14.
Graph f(θ) = sec 2θ + 1.
Solve It
135
136
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
15.
Name the asymptotes from 0 to 2π,
and the horizontal and vertical shifts of
f(θ) = 1 ⋅ csc ⎛ θ − π ⎞ − 1 .
⎝
4
2⎠
16.
Graph f(θ) = 1 ⋅ csc ⎛ θ − π ⎞ − 1 .
⎝
4
2⎠
Solve It
Solve It
17.
Name the amplitude, period, horizontal shift, and vertical shift of
f(θ) = 2 ⋅ sec 1 ⋅ ⎛ θ + π ⎞ + 1 .
2 ⎝
4⎠
Solve It
18.
Graph f(θ) = 2 ⋅ sec 1 ⋅ ⎛ θ + π ⎞ + 1 .
2 ⎝
4⎠
Solve It
Chapter 7: Graphing and Transforming Trig Functions
Answers to Problems on Graphing and
Transforming Trig Functions
a
Graph f(θ) = –1⁄2 · cosθ. See the following figure.
Because the cosine function is multiplied by –1⁄2, the graph is inverted with an amplitude of 1⁄2.
The period doesn’t change, and there are no shifts.
1
0
π
2
π
3π
2
2π
–1
b
Graph f(θ) = cos1⁄2 · θ. See the following figure.
Here, the amplitude doesn’t change, but the period does. The new period is found by dividing
2π by 1⁄2, which is 4π. There are no vertical or horizontal shifts.
1
0
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
–1
c
Graph f(θ) = sin(θ + π⁄4). See the following figure.
This graph has a horizontal shift. To find it, set what’s inside the parentheses to the starting
value of the parent graph: θ + π⁄4 = 0, so θ = –π⁄4. There are no other changes from the parent graph.
2
1
−π 0
4
–1
–2
π
4
π 3π π 5π 3π 7π 2π 9π
2 4
4 2 4
4
137
138
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
d
Graph f(θ) = cos1⁄3 · θ + 2. See the following figure.
This has a change in period, which can be found by dividing 2π by 1⁄3 to get 6π. There’s also a
vertical shift of 2.
3
2
1
0
–1
e
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
9π
2
5π 11π 6π
2
Name the amplitude, period, horizontal shift, and vertical shift of f(θ) = 3sin[2(θ + π⁄2)] – 1. The
amplitude is 3, the period is π, the horizontal shift is –π⁄2, and the vertical shift is –1.
The only calculation you need to do is to find the period. Here, you divide 2π by 2 to get π.
From the equation, you can see that the amplitude is 3 and the vertical shift is –1. By setting
θ + π⁄2 = 0, you find that the horizontal shift is –π⁄2.
f
Graph f(θ) = 3sin(2θ + π⁄2) – 1. See the following figure.
Using the information you gather in problem 5, this graph comes together quickly.
2
1
0
–1
–2
–3
–4
π
2
π
Chapter 7: Graphing and Transforming Trig Functions
g
Graph f(θ) = cot1⁄2θ. See the following figure.
For this cotangent graph, the period has a change, which can be found by dividing π by 1⁄2 (you
get 2π). There are no other changes to the parent cotangent graph.
2
1
0
π
2
π
3π
2
2π
–1
–2
h
Graph f(θ) = tanθ + 2. See the following figure.
Here, there’s only a vertical shift of 2. The amplitude and period are the same as the parent
graph.
4
3
2
1
−π
−π
2
0
π
2
π
139
140
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
i
Graph f(θ) = 1⁄3 · cotθ. See the following figure.
This graph shows a vertical transformation of 1⁄3.
2
1
1
3
−π −π 0 –1 π
3 4
2 4
π
2
3π π
4
–1
j
Graph f(θ) = tan(θ – π⁄2). See the following figure.
This tangent graph has a horizontal shift of π⁄2. There are no other changes to the parent graph.
2
1
0
π
4
π
2
3π π 5π 3π
4
4 2
–1
k
Name the vertical transformation, period, horizontal shift, and vertical shift of f(θ) =
2 · tan(θ + π⁄4) – 1. The vertical transformation is 2, the period is π, the horizontal shift
is –π⁄4, and the vertical shift is –1.
The only calculation here is to find the horizontal shift by setting θ + π⁄4 = 0. You get θ = –π⁄4.
You get the rest of the info straight from the equation.
Chapter 7: Graphing and Transforming Trig Functions
l
Graph f(θ) = 2 · tan(θ + π⁄4) – 1. See the following figure.
Using the information you gather in problem 11, you get the following graph:
2
1
−π –3π −π −π
4 2 4
π
4
π
2
π
3π
4
–1
m
Graph f(θ) = –cscθ – 1. See the following figure.
For these you start by graphing the sine graph to find the asymptotes, and then sketch the
reciprocal function of cosecant. We include the sine graph for you to see that it has been
flipped because of the negative in front and shifted down 1 because of the vertical shift.
1
−π
−π
2
π
2
π
–1
n
Graph f(θ) = sec 2θ + 1. See the following figure.
Again, we include the reciprocal cosine graph. This has a change in period, which you find by
dividing 2π by 2 to get π. It also has a vertical shift of 1. Then you draw the asymptotes and
sketch the secant graph.
3
2
1
−π −π
2 4
0
–1
π
4
π
2
3π
4
π
141
142
Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs
o
Name the asymptotes from 0 to 2π, and the horizontal and vertical shifts of f(θ) = 1⁄4csc(θ – π⁄2) – 1.
The asymptotes are at π⁄2, 3π⁄2, and 5π⁄2. The horizontal shift is –π⁄2, and the vertical shift is –1.
To find the asymptotes, you need to first look for any shifts or changes in period that would
affect the parent graph. The shifts are evident from the equation, where the horizontal shift is
–π⁄2 and the vertical shift is –1. This makes the zeros of the reciprocal sine graph at π⁄2, 3π⁄2, and 5π⁄2.
This is then where the asymptotes will be.
p
Graph f(θ) = 1⁄4csc(θ – π⁄2) – 1. See the following figure.
Using the information from problem 15, sketch the sine graph, the asymptotes, and then the
reciprocal — the cosecant graph.
1
−π
−π
2
0
π
π
2
3π
2
2π
–1
–2
q
Name the amplitude, period, horizontal shift, and vertical shift of f(θ) = 2 sec1⁄2(θ + π⁄4) + 1. The
amplitude is 2, the period is 4π, the horizontal shift is –π⁄4, and the vertical shift is 1.
The amplitude and shifts are evident from the equation. To find the period, simply divide 2π
by 1⁄2 and you get 4π.
r
Graph f(θ) = 2sec(1⁄2 · θ + π⁄4) + 1. See the following figure.
Begin by using the information from problem 17 to sketch the reciprocal, the cosine graph.
From there, sketch the asymptotes where the cosine graph equals 1 (or the cosine function
equals 0). Then, sketch the reciprocal function to graph the secant graph.
4
3
2
1
-2π -3π
2
π
−π 0
2
-1
-2
-3
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
Part III
Advanced Trig: Identities,
Theorems, and Applications
T
In this part . . .
he concepts of trig keep building, and we’re here to
help you follow along.
These chapters move into identities — they’re like formulas but they’re true all the time, no matter what you put in
for the variable(s). These identities are used to simplify
expressions and solve equations, and they’re even used in
trig proofs (and you thought you were done with proofs in
geometry!).
Chapter 8
Basic Trig Identities
In This Chapter
䊳 Simplifying with reciprocal identities
䊳 Reducing using Pythagorean identities
䊳 Finding patterns with even-odd identities
䊳 Using co-function identities
䊳 Discovering periodicity identities
䊳 Solving trigonometric proofs
E
ver want to pretend you were someone else — change your identity? Well, trig expressions have the opportunity to do that all the time. In this chapter, we cover basic identities, or statements that are always true. We use these identities to simplify problems and
then to complete trigonometric proofs. Each section builds upon the previous one, so we
recommend you spend some time reviewing the identities in each section before jumping to
the end to practice proofs.
Using Reciprocal Identities to Simplify
Trig Expressions
We actually introduce some of these back in Chapter 6, but now we’re going to use reciprocal
identities to simplify more complicated trig expressions. Because these identities are all
review, we also include the ratios of tangent and cotangent — the ratio identities that we
introduce in Chapter 6. The reciprocal (and ratio) identities are:
Because each pair of expressions is mathematically equivalent, you can substitute one for
another in a given expression and watch things simplify. Typically, changing a given expression to all sines and cosines causes a whole lot of canceling! Try it and see . . . we dare you!
146
Part III: Advanced Trig: Identities, Theorems, and Applications
Q.
Use reciprocal identities to rewrite
A.
The answer is 1. Start by using reciprocal and ratio identities to rewrite secx and tanx in terms of
sine and cosine (we dared you). Next, use your knowledge of fractions to rewrite the expression
as a division problem. Then, multiply by the reciprocal and cancel where you can. Here’s what it
should look like:
1.
Simplify cotx · secx.
Solve It
3.
Simplify sin3x · csc2x + tanx · cosx.
Solve It
.
2.
Simplify sinx · secx.
Solve It
4.
Simplify
Solve It
.
Chapter 8: Basic Trig Identities
Simplifying with Pythagorean Identities
Pythagorean identities are extremely helpful for simplifying complex trig expressions.
These identities are derived from those right triangles on a unit circle (turn to Chapter 6
for a review if you need to). Remember that cosθ = the x leg of a triangle, sinθ = the y
leg of a triangle, and the hypotenuse of the triangle on that unit circle is 1. Given the
fact that leg2 + leg2 = hypotenuse2, we get the first Pythagorean identity. The other two
are derived from that (check out Pre-Calculus For Dummies if you want to see how this
works!). These identities are especially helpful when simplifying expressions that have
a term that has been squared (sin2, cos2, and so on). Here are the Pythagorean identities (and some derivatives):
sin2x + cos2x = 1
or
cos2x = 1 – sin2x
or
sin2x = 1 – cos2x
tan2x + 1 = sec2x
or
tan2x = sec2x – 1
or
1 = sec2x – tan2x
1 + cot2x = csc2x
or
cot2x = csc2x – 1
or
1 = csc2x – cot2x
Q.
Simplify (secx + tanx)(1 – sinx)(cosx).
A.
cos2x. Start by changing everything to sine
and cosine using the reciprocal and ratio
identities from the previous section. Then
add the resulting fractions (the common
denominator is cosine) and cancel the
cosine in the numerator and denominator.
This leaves you with two terms that you
can FOIL. Recognize this last term as a
Pythagorean identity? We hoped you might!
Substitute it in and you have your answer.
5.
Simplify
Solve It
7.
Simplify sinx · cot2x + sinx.
Solve It
.
The steps look like this:
6.
Simplify
.
Solve It
8.
Simplify (sin2x – 1)(tan2x + 1).
Solve It
147
148
Part III: Advanced Trig: Identities, Theorems, and Applications
Discovering Even-Odd Identities
All functions, including trig functions, can be described as being even, odd, or neither
(see Chapter 3 for review). Knowing whether a trig function is even or odd can actually
help you simplify an expression. These even-odd identities are helpful when you have
an expression where the variable inside the trig function is negative (such as –x). The
even-odd identities are:
sin(–x) = –sinx
csc(–x) = –cscx
cos(–x) = cosx
sec(–x) = secx
tan(–x) = –tanx
cot(–x) = –cotx
Q.
Simplify tan2(–x) + sec2(–x).
A.
The answer is 1. Using the even-odd identities, start by substituting for the negative
angles: –tan2x + sec2x. Using commutative
property of equality (from Chapter 1),
rewrite the expression: sec2x – tan2x.
Recognize this from our Pythagorean identities from the last section? This expression
equals 1.
9.
Simplify sec(–x) · cot(–x).
10.
Simplify sinx · [cscx + sin(–x)].
Solve It
11.
Simplify
Solve It
Solve It
.
12.
Simplify
Solve It
.
Chapter 8: Basic Trig Identities
149
Solving with Co-Function Identities
Ever notice that the graphs of sine and cosine look exactly alike, only shifted (see
Chapter 7 for a visual)? This is also the case for tangent and cotangent, as well as
for secant and cosecant. Because these functions have the same values, only
shifted, we can define them as being co-functions. We can write them as co-function
identities and use them to simplify expressions. The co-function identities are:
sinx = cos(π⁄2 – x)
cosx = sin(π⁄2 – x)
tanx = cot(π⁄2 – x)
cotx = tan(π⁄2 – x)
cscx = sec(π⁄2 – x)
secx = csc(π⁄2 – x)
Q.
Simplify
A.
sinx. Start by using the co-function identity to replace cot (π⁄2 – x) with tanx. Next, rewrite the fraction as a division problem. Then, rewrite in terms of sine and cosine using reciprocal and ratio
identities. Finally, simplify by using the multiplicative inverse, cancelling any common terms. The
steps look like this:
13.
Simplify
.
14.
.
Simplify
.
Solve It
Solve It
15.
Simplify
.
16.
Simplify
.
Solve It
Solve It
150
Part III: Advanced Trig: Identities, Theorems, and Applications
Moving with Periodicity Identities
Recall that horizontal transformations change the period of a graph or move it left or
right (see Chapter 7). If you shift the graph by one whole period to the left or right, you
end up with the same function. This is the idea behind periodicity identities. Because
the periods of sine, cosine, cosecant, and secant repeat every 2π, and tangent and
cotangent repeat every π, the periodicity identities are as follows:
sin(x + 2π) = sinx
cos(x + 2π) = cosx
tan(x + π) = tanx
cot(x + π) = cotx
csc(x + 2π) = cscx
sec(x + 2π) = secx
Q.
Simplify 1 – [sin(2π + x) · cot(π + x) ·
cos(2π + x)].
A.
sin2x. Begin by rewriting the trig terms
using periodicity identities: 1 – (sinx · cotx ·
cosx). Next, rewrite cotangent in terms of
sine and cosine: 1 – (sinx · cosx⁄sinx · cosx).
Then, cancel the sine from the numerator
and denominator, leaving you with 1 – cos2x.
Using Pythagorean identities, this is the
same as sin2x.
17.
Simplify cos(2π + x) + sin(2π + x) ·
cot(π + x).
18.
Simplify
Solve It
Solve It
19.
Simplify
Solve It
.
.
20.
Simplify [sec(x – 2π) – tan(x – π)]
[sec(x + 2π) + tan(x + π)].
Solve It
Chapter 8: Basic Trig Identities
151
Tackling Trig Proofs
Proofs?!? You thought you left those behind in geometry. Nope, sorry. Don’t worry,
though — we walk you through pre-calc’s version of proofs, which are trig proofs.
One thing to remember is that you’re just using what you’ve already practiced in this
chapter. These proofs are composed of two sides of an equation. Your job is to make
one side look like the other. Here are some hints on how to solve these:
⻬ Deal with fractions using basic fraction rules: The same rules apply to simplifying trig expressions as any other expression. Two key rules to remember:
• Dividing a fraction by another fraction is the same as multiplying by the
reciprocal.
• Use lowest common denominator (or LCD) when adding or subtracting
fractions.
⻬ Factor when you can: Keep an eye out for factorable terms, including factoring
out the greatest common factor (GCF) and factoring trinomials (see Chapter 4).
⻬ Square square roots: When you have a square root in a proof, you probably
have to square both sides of the proof.
⻬ Work on the more complicated side first: Because the goal is to make one side
look like the other, it’s generally easier to start on the more complicated side
first. If you get stuck, try working on the other side for a while. You can then
work backward to simplify the first side.
Be careful with this last hint — some teachers require the proof be done to one side
only, so take a look at the other side of the equation, but only to help you get to the
end. In Pre-Calculus For Dummies, we show you exactly how to do this if you get a
teacher who insists that you work only on one side.
Q.
Prove
A.
The left side is more complicated, so we’ll work on that side. Start by finding the common denominator and add the fractions.
.
From there, notice that you have a Pythagorean identity. Rewrite the resulting fraction in terms of
sine and cosine using ratio identities, and multiply terms to complete the proof.
152
Part III: Advanced Trig: Identities, Theorems, and Applications
21.
Prove
.
Prove
.
Solve It
25.
Prove
Solve It
26.
Prove secx – cosx = sinx · tanx.
Solve It
Prove
.
Solve It
Solve It
23.
22.
24.
Prove
Solve It
.
.
Chapter 8: Basic Trig Identities
Answers to Problems on Basic Trig Identities
a
Simplify cotx · secx. The answer is cscx.
Start by using ratio identities to show the expression in terms of sine and cosine. Cancel the
cosine in the numerator and denominator and write as a single fraction. Finally, rewrite using
reciprocal identities.
b
Simplify sinx · secx. The answer is tanx.
Begin by rewriting secx using reciprocal identities. Then multiply the terms to get a single fraction. Finally, rewrite as a single expression using ratio identities.
c
Simplify sin3x · csc2x + tanx · cosx. The answer is 2 · sinx.
Start by using reciprocal and ratio identities to rewrite the expression in terms of sine and
cosine. Next, cancel any terms you can. Finally, combine the like terms.
Careful that you write sinx + sinx = 2sinx, not sin2x; this is a double angle (which we introduce
in the next chapter).
d
Simplify
. The answer is sinx.
Here, you may want to start by rewriting the tangent term using ratio identities. Then, cancel
the like terms in the numerator and denominator, giving you your answer.
e
Simplify
. The answer is csc2x.
Begin by using ratio identities to rewrite the ratio of cosine and sine as cotangent. Then distribute the cotangent term and rewrite the tangent term using ratio identities, canceling terms in
the numerator and denominator. Notice that the resulting expression is a Pythagorean identity
that can be simplified.
153
154
Part III: Advanced Trig: Identities, Theorems, and Applications
f
Simplify
. The answer is cos2x.
Again, begin by using reciprocal identities to rewrite tangent and secant in terms of sine and
cosine. Next, rewrite the fraction as a division problem, then multiply by the reciprocal. Cancel
terms in the numerator and denominator. Finally, use Pythagorean identities to rewrite the
expression as a single term.
g
Simplify sinx · cot2x + sinx. The answer is cscx.
Factor sinx from both terms. Then, using Pythagorean identities, simplify the expression. Next,
rewrite the cosecant term using reciprocal identities. Cancel the like terms in the numerator
and denominator. Finally, use reciprocal identities to simplify the final term.
h
Simplify (sin2x – 1)(tan2x + 1). The answer is 1.
Start by replacing the sine term using Pythagorean identities. Next, distribute the cosine term
across the tangent expression. Then, using ratio identities, rewrite the tangent term. You can
then cancel cosine from the numerator and denominator. The resulting expression is another
Pythagorean identity.
i
Simplify sec(–x) · cot(–x). The answer is –cscx.
Begin by using even-odd identities to get rid of all the –x values inside the trig expressions.
Next, use reciprocal and ratio identities to rewrite the expression in terms of sine and cosine.
Then, cancel any like terms in the numerator and denominator. Last, rewrite the resulting fraction using reciprocal identities.
j
Simplify sinx · [cscx + sin(–x)]. The answer is cos2x.
Use even-odd identities to replace the –x value; then you can distribute the sinx term. Using
reciprocal identities, rewrite the cosecant term and cancel where you can. The resulting expression can be simplified using Pythagorean identities.
Chapter 8: Basic Trig Identities
k
Simplify
. The answer is –secx.
Start by replacing the –x value using even-odd identities. Then, using reciprocal and ratio identities, rewrite everything in terms of sine and cosine. You can then rewrite the big division bar
with a division sign and then invert the fraction so you can multiply. Cancel any terms you can.
Finally, simplify the resulting expression using reciprocal identities.
l
Simplify
. The answer is sec2x.
Again, you can start by replacing the –x using even-odd identities. Then you can factor out a
negative from the numerator. This results in a Pythagorean identity, which can be simplified. At
this point the negatives in the numerator and denominator cancel each other out, leaving you
with a positive expression. Next, using reciprocal and ratio identities, change everything into
terms using sine and cosine. Rewrite the big fraction as a division problem, which can then be
changed to multiplication by inverting the fraction. Cancel what you can and use reciprocal
identities to simplify. Voilà!
m
Simplify
. The answer is sinx.
Here you can start by using co-function identities to rewrite the sine and cotangent terms. Next,
use ratio identities to rewrite the tangent term as a rational function of sine and cosine. Cancel
the cosine terms, leaving you with your answer.
n
Simplify
. The answer is secx.
Begin by using co-function identities to replace any terms with π⁄2 in them. Then rewrite tangent
using ratio identities. Next, you need to find a common denominator in order to add the resulting terms. Rewriting the resulting fraction, you can see that the numerator is a Pythagorean
identity, which can be simplified. Finally, use reciprocal identities to simplify the resulting
expression.
155
156
Part III: Advanced Trig: Identities, Theorems, and Applications
o
Simplify
. The answer is tan2x.
Begin by using co-function identities to replace each term. Then use ratio identities to replace
the sine/cosine fraction with tangent. Finally, multiply the tangent terms.
p
Simplify
. The answer is sinx.
Start by replacing terms using co-function identities. Next, use reciprocal and ratio identities to
rewrite the terms using sine and cosine. Then, because the first two terms have a common
denominator of sine, you can write them as a single fraction. Next, you can multiply using FOIL.
The resulting numerator is a Pythagorean identity which can be simplified. Finally, cancel sine
from the numerator and denominator.
q
Simplify cos(2π + x) + sin(2π + x) · cot(π + x). The answer is 2cosx.
Using periodicity identities, replace each term. Then, using ratio identities, replace the cotangent with a ratio of cosine and sine. Cancel the sine from the numerator and denominator and
add the two cosines.
r
Simplify
. The answer is sinx.
Ah! Don’t let this one trick you. 4π is a multiple of 2π, so adding that to x also gives you a periodicity identity. Replacing the terms with the appropriate periodicity identity, you can easily
simplify this problem. The next step is to use ratio identities to replace the cotangent term with
a ratio of cosine and sine. Then, simplify the complex fraction by multiplying the numerator by
the reciprocal of the denominator. Cancel where you can to get a simplified expression.
s
Simplify
. The answer is tanx.
Start by replacing both terms using periodicity identities. Then, you can rewrite the fraction as
a division problem. Next, rewrite the division problem by multiplying by the reciprocal. Then
use reciprocal identities to replace both terms. Multiply to write as a single term, which can be
simplified using ratio identities.
Chapter 8: Basic Trig Identities
t
Simplify [sec(x – 2π) – tan(x – π)][sec(x + 2π) + tan(x + π)]. The answer is 1.
Simplify by replacing every term using periodicity identities. Then you can FOIL. Finally, you
have a Pythagorean identity that can be simplified.
u
Prove
.
For this proof, we start with the left side because it’s more complicated. Begin by using evenodd identities to replace the –x and use ratio identities to rewrite tangent and cotangent in
terms of sine and cosine.
Now multiply the numerator and denominator by the LCD to simplify the complex fraction.
Finally, you can pull out a common factor on both the numerator and denominator, cancel any
like terms, and simplify the resulting fraction using ratio identities.
v
Prove
.
Start by using co-function identities to replace the terms with π.
Next, you can separate the fraction into two different fractions and then cancel any terms on
both the numerator and denominator.
157
158
Part III: Advanced Trig: Identities, Theorems, and Applications
Then, use reciprocal and ratio identities to rewrite all terms using sine and cosine. You can simplify the denominator of the complex fraction using multiplication.
Next, get rid of the complex fraction by changing the large fraction bar to division and then
multiplying by the reciprocal. The resulting fraction can also be simplified using ratio identities,
giving you the answer.
w
Prove
.
In this proof, the right side is more complicated, so it’s wise to start there. Notice the squared
term? You can replace it using Pythagorean identities, which then cancels out the 1 in the
numerator. Last, simply cancel a cotangent from both the numerator and denominator and
you’re there!
x
Prove
.
For this proof, you want to start by finding a common denominator for the two fractions and
multiplying it through using FOIL for the first fraction. Next, you can simplify using a
Pythagorean identity.
After you have a single fraction, use reciprocal and ratio identities to change the numerator,
allowing you to cancel many of the terms. Again, you have a Pythagorean identity that you can
simplify.
Now that the numerator looks like your final answer, you can concentrate on the denominator.
Distribute the cosecant and use reciprocal identities to simplify.
Chapter 8: Basic Trig Identities
y
Prove
.
Start by dealing with the gigantic square root by squaring both sides.
Next, replace any terms with π⁄2 using co-function identities, and replace any terms with 2π using
periodicity identities. Multiply the resulting trig terms.
Replace the cosecant term using reciprocal identities and write as a single fraction that you can
replace using ratio identities.
Finally, you have a Pythagorean identity that can be simplified to get the answer you want.
A
Prove secx – cosx = sinx · tanx.
Starting by replacing secant with its reciprocal identity, you end up with a fraction. Adding the
fractions using a common denominator of cosine, you get a Pythagorean identity in the numerator, which can be simplified. Finally, multiplying out a sine leaves you with a ratio of sine and
cosine that you can replace using ratio identities to get the right side of the proof.
159
160
Part III: Advanced Trig: Identities, Theorems, and Applications
Chapter 9
Advanced Identities
In This Chapter
䊳 Using sum and difference identities
䊳 Simplifying with double angle identities
䊳 Discovering half-angle identities
䊳 Tapping product to sum and sum to product identities
䊳 Solving with power-reducing formulas
O
kay . . . the training wheels are off! We’re getting into the advanced stuff here —
advanced identities, that is. This chapter builds on the basic identities you practiced
in Chapter 8.
In this chapter, we give you formulas that are essential for calculating precise values of
angles that you could never get before (even your calculator only gives approximate
answers). These identities are essential for calculus and are well-loved by pre-calc teachers,
so it’s time to get friendly with advanced identities.
Simplifying with Sum and Difference Identities
Here those pesky mathematicians took simple concepts (addition and subtraction) and
related them to a more complex one (trigonometric angles). These sum and difference identities allow you to write an angle that’s not from the special triangles of 45-45-90 or 30-60-90
(see Chapter 6) as the sum or difference of those helpful angle measures. For example, you
can rewrite the measure of 105° as the sum of 45° and 60°. The problems presented here (and
in pre-calc books everywhere) can always be written using the angles you already have exact
values for, even though these identities can be used for any value.
The sum and difference identities are:
162
Part III: Advanced Trig: Identities, Theorems, and Applications
Q.
Find tan 7π⁄12 using sum or difference
identities.
A.
. Start by breaking up the fraction
into a sum of two values that can be found
on the unit circle: 7π⁄12 = 3π⁄12 + 4π⁄12 = π⁄4 + π⁄3.
Next, plug the angles into the sum
identity for
tangent:
Finally, plug in the known values for the
angles from the unit circle (see the Cheat
Sheet) and simplify. Remember to rationalize the denominator using conjugates (see
Chapter 2 for review). The steps are as
follows:
.
Keep a close eye on the order of a
plus or minus symbol in an equation.
If it’s inverted to be a minus or plus
symbol, then you perform the opposite operation than the given problem. In this case, the problem
involves addition, so we use addition
in the numerator and subtraction in
the denominator.
1.
Find cos15° using sum or difference
identities.
Prove
Solve It
Simplify tan(45° – x).
Solve It
Solve It
3.
2.
.
4.
Prove
Solve It
.
Chapter 9: Advanced Identities
5.
Find csc5π⁄12 using sum or difference
identities.
Solve It
6.
163
Simplify sec(180° + x) using sum or difference identities.
Solve It
Using Double Angle Identities
Double angle identities help you find the trig value of twice an angle. These can be used
to find an exact value if you know the original angle. They can also be used to prove
trig proofs (see Chapter 8) or solve trig equations. Cosine has three double angle identities created from the Pythagorean identities of Chapter 8. You have a choice as to
which you want to use depending on the problem.
The double angle identities are:
Q.
Solve 6cos2x – 6sin2x = 3 for π > x > 0.
A.
x = π⁄6. Start by factoring out the 6 from the
left side of the equation: 6(cos2x – sin2x) =
3. Next, substitute using the appropriate
double angle identity: 6(cos2x) = 3. Isolate
the trigonometric term: cos2x = 1⁄2. Then,
take the inverse of cosine using the unit
circle (see the Cheat Sheet): 2x = π⁄3. Finally,
solve for x by dividing both sides by 2: x =
π
⁄6. Because you’re limited to quadrants I
and II and cosine is only positive in quadrants I and IV, there’s only the one answer.
164
Part III: Advanced Trig: Identities, Theorems, and Applications
7.
Find the value of cos2x if cscx = 12⁄5.
Solve It
9.
Prove
.
Simplify tan3x using double angle identities.
Solve It
Find the value of tan2x if cotx = 1⁄2.
Solve It
Solve It
11.
8.
10.
Prove
.
Solve It
12.
Solve 6 – 12sin2x = 3 3 for π > x > 0.
Solve It
Chapter 9: Advanced Identities
Reducing with Half-Angle Identities
Similar to sum and difference identities, half-angle identities help you find exact values of
unusual angles, namely ones that are half the value of ones you already know. For example, if we want to find a trig value of 22.5°, we would use the half-angle identity of half of
45° because 22.5 is half of 45. Also, just like every other identity we’ve reviewed so far,
half-angle identities can be used for proving trig proofs and solving trig equations.
The half-angle identities are:
Q.
Find cot5π⁄12 using half-angle identities.
A.
2 − 3 . First, because there isn’t a halfangle formula for cotangent, we have to
start by recognizing that cot5π⁄12 is the
reciprocal of tangent of the same angle.
Therefore, we’re going to find the value of
tan5π⁄12, then take the reciprocal. The angle
⁄12 can be rewritten as
5π
Find tan3π⁄8 using half-angle identities.
Solve It
. This simplifies to 3 . But
wait — we’re not done! This is the value of
tan5π⁄12, and we need cot 3 . So we have to
. Plugging that
into the half-angle identity, we get
13.
Replacing the trig expressions for the
exact values from the unit circle, we get
take the reciprocal, which, after rationalizing, we find to be 2 – 3 . Whew!
.
14.
Find sin7π⁄12.
Solve It
165
166
Part III: Advanced Trig: Identities, Theorems, and Applications
15.
Prove
.
Solve It
16.
Find an approximate value of cosπ⁄24 using
half-angle identities.
Solve It
Changing Products to Sums
You know that age-old question about math — when will I ever use this stuff? Well, this
is one section in pre-calc that even stumps us, because the only time you’ll ever need
product to sum (or difference) identities is in calculus; they don’t really have any realworld uses. So, do you still need to learn about them? Yep, ‘fraid so — pre-calc and
calc teachers expect you to know them, so here we go.
You need to know three product to sum identities: sin · cos, cos · cos, and sin · sin.
Here they are:
Q.
Express 8sin3x · sinx as a sum or
difference.
A.
4(cos2x – cos4x). Start by plugging in the
appropriate product to sum identity:
. Then,
simplify using multiplication:
.
Chapter 9: Advanced Identities
17.
Express 12cos6x · sin2x as a sum or
difference.
Solve It
19.
Express 6cos3x · sin6x as a sum or
difference.
Solve It
18.
Express 2sin5x · cos2x as a sum or
difference.
Solve It
20.
Express 7sin8x · sin3x as a sum or
difference.
Solve It
167
168
Part III: Advanced Trig: Identities, Theorems, and Applications
Expressing Sums as Products
Although less frequently used than the other identities in this chapter, the sum to product
identities are useful for finding exact answers for some trig expressions. In cases where
the sum or difference of the two angles results in an angle from our special right triangles
(Chapter 6), then sum to product identities can be quite helpful. The sum (or difference)
to product identities involve the addition or subtraction of either sine or cosine.
The sum (or difference) to product identities are:
Q.
Find cos165° + cos75°.
. Simplify
− 2 . Begin by using the sum to product
2
identity to rewrite the expression:
A.
the results using unit circle values:
2cos(240⁄2) · cos(90⁄2) = 2cos120° · cos45° =
2(–1⁄2) ·
21.
Find sin195° – sin75°.
Solve It
23.
Find sin7π⁄12 + sinπ⁄12.
Solve It
22.
2 =− 2
.
2
2
Find cos375° – cos75°.
Solve It
24.
Find cos23π⁄12 + cos5π⁄12.
Solve It
Chapter 9: Advanced Identities
Powering Down: Power-Reducing Formulas
Power-reducing formulas can be used to simplify trig expressions with exponents and
can be used more than once if you have a function that’s raised to the fourth power or
higher. These nifty formulas help you get rid of exponents so you can focus on solving
for an angle’s measure.
Here are the three power-reducing formulas:
Q.
Simplify cos4x using power-reducing formulas.
A.
⁄8(3 + 4cos2x + cos4x). After rewriting (cos2x)2, you can see that you need to use the
power-reducing formula twice. The first time gives you
1
Use the formula again on the remaining squared term and reduce:
25.
Prove cos23x – sin23x = cos6x.
Solve It
27.
Express sin4x – cos4x without exponents.
Solve It
26.
Prove (1 + cos2x) · tan3x = tanx(1 – cos2x).
Solve It
28.
Express cot2(x⁄2) without exponents.
Solve It
169
170
Part III: Advanced Trig: Identities, Theorems, and Applications
Answers to Problems on Advanced Identities
a
Find cos15° using sum or difference identities. The answer is
.
Start by rewriting the angle using special angles from the unit circle. We chose 15° = 45° – 30°.
Plugging these values into the difference formula, you get cos(45° – 30°) = cos45° · cos30° –
sin45° · sin30°. Use the unit circle to plug in the appropriate values and simplify:
2 ⋅ 3 − 2 ⋅1 = 6 − 2 =
.
2 2
2 2
4
4
b
Simplify tan(45° – x). The answer is
.
Rewrite the expression using the difference formula:
its unit circle value of 1 and simplify the expression:
c
Prove
. Then, replace tan45° with
=
.
.
Begin the proof by rewriting the left side using sum formulas. Next, substitute for unit circle
values and simplify. Finally, replace the sine and cosine with tangent using ratio identities (from
Chapter 8). The steps are as follows:
d
Prove
.
Start by replacing the sin(x + y) term using sum formula. Next, separate the fraction into the
sum of two fractions and reduce terms. Last, rewrite using ratio identities from Chapter 8.
Chapter 9: Advanced Identities
e
Find csc5π⁄12 using sum or difference identities. The answer is
6− 2.
First, know that you can rewrite 5π⁄12 as 2π⁄12 + 3π⁄12, which is the same as π⁄6 + π⁄4. Substitute this back
into the problem: csc(π⁄6 + π⁄4). Because you don’t have a sum identity for cosecant, you need to
find
. To ease this calculation, you start by finding sin(π⁄6 + π⁄4), then invert it. Using
sum identities, sin(π⁄6 + π⁄4) = sinπ⁄6 · cosπ⁄4 + cosπ⁄6 · sinπ⁄4. Replace the trig expressions with the
appropriate values from the unit circle and simplify: 1 ⋅ 2 + 3 ⋅ 2 = 2 + 6 =
2 2
2 2
4
4
.
The last step is to invert this solution and simplify by rationalizing the denominator:
.
f
Simplify sec(180° + x) using sum or difference identities. The answer is secx.
Because you don’t have a sum identity for secant, you need to use the one for cosine and then
find the reciprocal. Using this identity, you get cos180 · cosx – sin180 · sinx. Plugging in values
from the unit circle and simplifying, you get (1) · cosx – (0) · sinx = cosx. The reciprocal of this
is 1⁄cosx, which by reciprocal identities (from Chapter 8) is secx.
g
Find the value of cos2x if cscx = 12⁄5. The answer is 47⁄72.
Because cos2x = 1– 2sin2x, you need to know sinx to plug it in. No problem! You have the reciprocal: cscx. Therefore, sinx = 5⁄12. Plugging this in, you get cos2x = 1 – 2(5⁄12)2 = 1 – 2(25⁄144) = 1– 25⁄72 = 47⁄72.
h
Find the value of tan2x if cotx = 1⁄2. The answer is –4⁄3.
Begin with finding tanx by taking the reciprocal of cotx: 2⁄1 = 2. Then plug it into the formula for
tan2x:
= –4⁄3.
171
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Part III: Advanced Trig: Identities, Theorems, and Applications
i
Prove
.
Working with the left side, change all the double angles using the double angle identities. After
combining like terms, cancel the 2 and cosine from the numerator and denominator. Finally,
rewrite the result using ratio identities (Chapter 8). The steps are as follows:
;
;
;
j
Prove
;
;
.
Begin by using the double angle identities for the left side of the proof:
.
Next, separate the single fraction into the difference of two fractions:
.
Then, cancel any terms in the numerator and denominator:
. Finally, use ratio identities to replace the fractions
with cotangent and tangent and factor out the GCF:
k
.
Simplify tan3x using double angle identities. The answer is
.
Start by separating out the angle into a single and double angle: tan(x + 2x). Next, use the sum
identity for tangent:
formula:
. Next, replace the double angle terms using the double-angle
. Then, to simplify the complex fraction, multiply by
the common denominator:
and combine like terms:
l
. Multiply through:
.
Solve 6 – 12sin2x = 3 3 for π > x > 0. The answer is 15°.
Begin by factoring out 6 from both terms: 6(1 – 2sin2x) = 3 3 . Next, use double angle identities
to replace the trigonometric term: 6(cos2x) = 3 3 . Isolate the trig term by dividing both sides
Chapter 9: Advanced Identities
by 6 and simplifying:
. Then, to solve for x, take the inverse cosine of
each side:
and divide by 2: 15°. Because you’re limited to quadrants I
and II and the answer is positive, there’s only one answer: 15°.
m
Find tan3π⁄8 using half-angle identities. The answer is
2 + 1.
Start by rewriting the angle to find the half angle:
identity:
. Then, plug it into the half-angle
; knowing that the angle is in quadrant II tells you that the result for tangent
is positive. Find the exact values for the trigonometric terms on the unit circle:
.
Then simplify the complex fraction:
2 +1.
n
Find sin7π⁄12. The answer is
.
Begin by rewriting the angle:
. Plug the angle into the appropriate half-angle identity:
. Because the angle is in quadrant I, sine will be positive. The next step is to replace
the trig terms with the exact values from the unit circle:
o
Prove
. Simplify the complex fraction:
.
Start with the left side by replacing the half-angle term using the appropriate identity:
. Next, use ratio identities to change everything to sine
and cosine:
. Multiply through to complete the proof:
.
173
174
Part III: Advanced Trig: Identities, Theorems, and Applications
p
Find an approximate value of cosπ⁄24 using half-angle identities. The answer is ≈ 0.99.
Rewriting the angle using half-angle identities, remember that the angle is in quadrant I,
so it’s positive:
. Replace it with the appropriate half-angle identity:
. Because
you don’t have a special right triangle value, you need to use the half-angle identities again:
. Now you can replace the cosine term using the unit circle value (finally!):
. Finish by plugging the complex fraction into your calculator: ≈ 0.99.
q
Express 12cos6x · sin2x as a sum or difference.
Start by plugging in the appropriate product to sum identity:
. Then simplify using multiplication:
.
r
Express 2sin5x · cos2x as a sum or difference.
Begin by plugging in the appropriate product to sum identity:
.
.
Then simplify using multiplication:
s
Express 6cos3x · sin6x as a sum or difference.
Again, plug in the appropriate product to sum identity:
simplify using multiplication:
t
. Then
.
Express 7sin8x · sin3x as a sum or difference.
You guessed it! Plug in the appropriate product to sum identity:
. Simplify using multiplication:
.
u
6
Find sin195° – sin75°. The answer is − 2 .
Begin by using the sum to product identity to rewrite the expression:
. Simplify the result using unit circle values:
= − 6.
2
Chapter 9: Advanced Identities
v
2
.
2
Use the sum to product identity to rewrite the expression:
Find cos375° – cos75°. The answer is
. Then, plug in the unit circle values and
simplify:
w
=
Find sin7π⁄12 + sinπ⁄12. The answer is
2
.
2
6 .
2
Rewrite the expression with the appropriate sum to product identity:
.
Use unit circle values to simplify the result:
x
=
6 .
2
6 .
2
Again, start by using the sum to product identity to rewrite the expression:
Find cos23π⁄12 + cos5π⁄12. The answer is
.
Next, simplify the result using unit circle values:
y
=
6.
2
Prove cos23x – sin23x = cos 6x.
Working on the left side, use the power-reducing formulas to rewrite both terms:
. Combine the fractions into one:
. Combine like terms:
. Reduce to complete
the proof: cos6x = cos6x.
A
Prove (1 + cos2x) · tan3x = tanx(1 – cos2x).
Start by rewriting the tangent term so that you can use a power-reducing formula:
(1 + cos2x) · tanx · tan2x = tanx(1 – cos2x). Now use the power-reducing formula:
. Cancel terms in the numerator and denominator:
, and you’re there! tanx(1 – cos2x) = tanx(1 – cos2x).
175
176
Part III: Advanced Trig: Identities, Theorems, and Applications
B
Express sin4x – cos4x without exponents. The answer is –cos2x.
For this one, we start by factoring the difference of two squares: (sin2x + cos2x)(sin2x – cos2x).
Now use Pythagorean identities (see Chapter 8) to simplify: (1)(sin2x – cos2x). If you replace the
cosine term using the same Pythagorean identity, you can combine like terms to have only one
squared term remaining: sin2x – (1 – sin2x) = sin2x – 1 + sin2x = 2sin2x – 1. From here, you just
need to replace the squared term using power-reducing formulas and simplify:
= 1 – cos2x – 1 = –cos2x.
C
Express cot2(x⁄2) without exponents. The answer is
.
Because you don’t have a power-reducing formula for cotangent, you need to start by using
ratio identities (see Chapter 8) to rewrite the expression:
tangent’s power-reducing formula:
by its reciprocal:
. Now you can use
. Simplify the complex fraction by multiplying
. Finally, make cancellations to simplify:
=
.
Chapter 10
Solving Oblique Triangles
In This Chapter
䊳 Using the Law of Sines and the Law of Cosines to solve triangles
䊳 Solving word problems using oblique triangles
䊳 Finding the area of a triangle
T
he trigonometry functions sine, cosine, and tangent are great for finding missing sides
and angles inside of right triangles. But what happens when a triangle isn’t quite right?
These types of triangles are known as oblique triangles — any kind of triangle that isn’t a
right triangle. As you can see in Chapter 6, the process of finding all the sides and angles in
a triangle is known as solving the triangle. This chapter helps you figure out that process for
oblique triangles.
As long as you know one angle and the side directly across from it, you can use the Law of
Sines. The Law of Sines can be used in three different cases: ASA, AAS, and SSA. The first two
cases have exactly one solution. The third case is known as the ambiguous case, as it may
have one, two, or no solutions. We take a look at each case to show you how to deal with
them. In fact, the ambiguous case gets its own section in this chapter.
If you don’t know an angle and the side opposite it, then you start off with the Law of
Cosines, which can be used for two cases: SSS and SAS.
Most books use standard notation to label an oblique triangle: Each vertex is labeled with a
capital letter, and the side opposite it is the same lowercase letter (across from angle A is
side a, and so on). We recommend drawing out the triangle and labeling the information
you’re given. You’ll know right away which case you’ve got on your hands and therefore
which formula to use.
Solving a Triangle with the Law of Sines:
ASA and AAS
After you draw out your triangle and see that you have two angles and the side in between
them (ASA) or two angles and a consecutive side (AAS), proceed to solve the triangle with
the Law of Sines:
In either case, because you know two angles inside the given triangle, you automatically
know the third, as the sum of the angles inside any triangle is always 180°. Remember, you’ll
find exactly one solution in either of these cases of the Law of Sines.
178
Part III: Advanced Trig: Identities, Theorems, and Applications
Q.
A.
Solve the triangle if A = 54°, B = 28°, and
c = 11.2.
b=
your calculator to give you b = 5.31.
C = 98°, a = 9.15, b = 5.31. First draw and
label the triangle, as shown in Figure 10-1.
You can see that you have a side sandwiched between two angles (ASA), so you
know you can start with the Law of Sines.
Because you know two of the angles in the
triangle, you can find the third angle first:
54° + 28° + C = 180°; C = 98°. Now substitute
all the given information and the angle
you just found into the Law of Sines:
Note: If you put trig function values
into your calculator and round as
you go, your final answer will be
affected. If, however, you wait until
the end like we did in solving for
b above, the answer will be more
precise.
Set the first and third ratios equal to each
other to get the proportion that can be
. Using the second
solved for a:
two equivalent ratios gives you a proportion you can solve:
. This goes directly into
. Cross multiply
to get a · sin98 = 11.2 · sin54. Divide the sin98
. Cross
from both sides to get a =
multiply to get b · sin98 = 11.2 · sin28.
Divide both sides by sin98 to get
or
a = 9.15.
C
Figure 10-1:
Always
draw and
label your
given
triangle. A
1.
b
a
54˚
28˚
c = 11.2
Solve the triangle if B = 46°, C = 62°, and
a = 21.
Solve It
3.
Solve the triangle if A = 49°, B = 21°, and
a = 5.
Solve It
2.
B
Solve the triangle if A = 19°, C = 100°, and
b = 4.4.
Solve It
4.
Solve the triangle if A = 110°, C = 56°, and
a = 8.
Solve It
Chapter 10: Solving Oblique Triangles
179
Tackling Triangles in the
Ambiguous Case: SSA
A triangle with two sides and a consecutive angle is the ambiguous case of the Law of
Sines. In this case, you may have one, two, or no solutions. We think it’s easiest to
assume that two solutions exist. That way, in attempting to find them both, you’ll either
Get an error message on your calculator that tells you that there are no solutions
Or you’ll find one solution and then
In finding the second one you’ll get an answer that doesn’t make sense (meaning
there really is one solution)
Or, you’ll find a second one that works (so there really are two solutions)
When using your calculator to solve these types of problems, anytime you use an
inverse trig function (like sin–1, cos–1 , or tan–1) to solve for an angle, know that the calculator gives you the reference angle, or the first quadrant answer θ (for more review
on this, check out Chapter 6). There’s also a second quadrant answer, 180 – θ.
Q.
Solve the triangle if a = 25, c = 15, and
C = 40°.
A.
No solution. Start off by substituting the
given information into the Law of Sines:
Now, cross-multiply to get 15 · sinA =
25 · sin40. Solve for the trig function with
the variable in it to get
the expression into your calculator to get
that sinA = 1.07. Even if you forget that sine
only has values between –1 and 1 inclusively and you inverse sine both sides of
the equation, you get an error message on
your calculator that tells you there is no
solution.
. Notice that the
middle ratio has absolutely no information
in it at all, so you basically ignore it for
now and work with the first and third
ratios to get the proportion:
5.
Solve the triangle if b = 8, c = 14, and
C = 37°.
Solve It
. Put
.
6.
Solve the triangle if b = 5, c = 12, and
B = 20°.
Solve It
180
Part III: Advanced Trig: Identities, Theorems, and Applications
7.
Solve the triangle if a = 10, c = 24, and
A = 102°.
Solve It
8.
Solve the triangle if b = 10, c = 24, and
B = 20°.
Solve It
Conquering a Triangle with the
Law of Cosines: SAS and SSS
The Law of Cosines comes in handy when the Law of Sines doesn’t work. Specifically,
you use the Law of Cosines in two cases:
⻬ You know two sides and the angle in between them (SAS)
⻬ You know all three sides (SSS)
The Law of Cosines is:
a2 = b2 + c2 – 2bccosA
b2 = a2 + c2 – 2accosB
c2 = a2 + b2 – 2abcosC
You may also see your textbook present three forms for each of the angles in the triangle:
You don’t have to memorize all these formulas. The ones for the angles are just the
first three, each rewritten in terms of the angle. We show the steps for one of them in
Pre-Calculus For Dummies, so if you’d like to see exactly how you arrive at the angle formulas, check it out there.
Chapter 10: Solving Oblique Triangles
Q.
Solve the triangle if A = 40°, b = 10, and
c = 7.
A.
a = 6.46, C = 44.1°, B = 95.9°. If you draw
out the triangle like the one in Figure 10-2,
you notice that this time the angle is
between two sides (SAS), and you know
that you have to use the Law of Cosines.
First, a2 = 102 + 72 – 2(10)(7)cos40. Put this
right into your calculator to get a2 = 41.75,
or a = 6.46. Next, continue using the Law of
181
Cosines to solve for angles (you’ll avoid
that pesky ambiguous case!) and get 72 =
6.462 + 102 – 2(6.46)(10)cosC, or 49 = 41.75 +
100 – 129.2cosC. Isolate for the trig function next and get .718 = cosC (this is why
you don’t have to memorize the formulas
for the angles!). Inverse cosine both sides
of the equation to get C = 44.1°. Now it’s
easy to find B because the triangle’s angles
must total 180°. In this case, B = 95.9°.
B
c=7
Figure 10-2:
Still drawing
pictures
to solve
triangles. A
9.
a
40˚
Solve the triangle if C = 120°, a = 6, and
b = 10.
Solve It
11.
Solve the triangle if a = 9, b = 5, and c = 7.
Solve It
C
b = 10
10.
Solve the triangle if A = 70°, b = 6, and c = 7.
Solve It
12.
Solve the triangle if a = 7.3, b = 9.9, and
c = 16.
Solve It
182
Part III: Advanced Trig: Identities, Theorems, and Applications
Using Oblique Triangles
to Solve Word Problems
Usually around this time, your textbook or your teacher will present word problems.
Argh! Run away, right? Wrong. Draw out a picture and discover that each and every
problem at this time is a triangle that isn’t right (an oblique triangle), where you’re
looking for one missing piece of information (as opposed to looking for all the angles
and sides, like you do in the last three sections). That means less work for you! Each
situation requires you to use the Law of Sines or the Law of Cosines exactly once to
solve for the missing information.
Q.
A.
A plane flies for 300 miles in a straight line,
makes a 45° turn, and continues for 700
miles. How far is it from its starting point?
S
e
300 miles
T
45˚
700
t
s
s
mile
Figure 10-3:
A first-class
representation of the
plane in the
problem.
Approximately 936.47 miles. First, draw
out a picture like the one in Figure 10-3.
Notice that we use S for the starting point,
T for the turning point, and E for the
ending point.
E
Now that you have the picture, you can figure out whether you need to use the Law of Sines or the
Law of Cosines. Because this is SAS, you start off with the modified Law of Cosines, using the different
variables from the picture: t2 = s2 + e2 – 2secosT, or t2 = 7002 + 3002 – 2(700)(300)cos45. This means that
t2 = 876,984.85, and t = 936.47.
Q.
Two fire towers are exactly 5 miles apart in
a forest. They both spot a forest fire, one at
an angle of 30° and the other at an angle of
42°. Which tower is closer?
A.
The first fire tower is closer. Okay, so we
lied just a little bit in the introduction to
this section. You aren’t always looking for
only one missing piece of information. In
this problem, you have to find how far both
towers are from the fire in order to know
which one is closer. But you forgive us,
right? First, draw out a figure like the one
in Figure 10-4.
Chapter 10: Solving Oblique Triangles
O
Figure 10-4:
Burnin’ up
to draw a
picture of
the forest
fire.
5 mi
30˚
t
42˚
T
o
F
This time, we use O for fire tower one, T for fire tower two, and F for the fire itself. You have a classic
case of ASA, so you can use the Law of Sines this time. Knowing two of the angles makes it possible
to find the third one easily: F = 108°. Now that you have the third angle, you can use the Law of Sines
to set up two proportions:
. Solving for o gets you o =
Now set up another proportion to solve for t:
, or o = 2.6 miles.
, which means that t =
, or
t = 3.5 miles. That means the first tower is closer.
13.
Two trains leave a station at the same time
on different tracks that have an angle of 100°
between them. If the first train is a passenger train that travels 90 miles per hour and
the second train is a cargo train that can
only travel 50 miles per hour, how far apart
are the two trains after three hours?
Solve It
14.
A radio tower is built on top of a hill. The
hill makes an angle of 15° with the ground.
The tower is 200 feet tall and located 150
feet from the bottom of the hill. If a wire is
to connect the top of the tower with the
bottom of the hill, how long does the wire
need to be?
Solve It
183
184
Part III: Advanced Trig: Identities, Theorems, and Applications
15.
A mapmaker stands on one side of a river
looking at a flagpole on an island at an
angle of 85°. She then walks in a straight
line for 100 meters, turns, and looks back
at the same flagpole at an angle of 40°. Find
the distance from her first location to the
flagpole.
Solve It
16.
Two scientists stand 350 feet apart, both
looking at the same tree somewhere in
between them. The first scientist measures
an angle of 44° from the ground to the top
of the tree. The second scientist measures
an angle of 63° from the ground to the top
of the tree. How tall is the tree?
Solve It
Figuring Flatness (Area)
In the proof of the Law of Sines, if you go in a slightly different direction you discover
a handy formula to find the area of an oblique triangle if you know two sides and the
angle between them, as shown in Figure 10-5. The area of the triangle formed is
A=
Figure 10-5:
If you know
two sides of
a triangle
and the
angle
between
them, you
can calculate the
area.
.
C
a
b
Chapter 10: Solving Oblique Triangles
185
Note: The letters in the formula aren’t important. The area of the triangle is always
one-half the product of the two sides and the sine of the angle between them.
You can also find the area when you know all three sides (a, b, and c) by using what’s
called Heron’s Formula. It says that the area of a triangle is
, where s =
The variable s is called the semiperimeter, or half of the triangle’s perimeter.
Q.
17.
Find the area of the triangle where b = 4,
c = 7, and A = 36°.
Find the area of the triangle where a = 7,
c = 17, and B = 68°.
Solve It
A.
18.
The area is about 8.23 square units. When
you have the two sides and the angle
between them, you plug the given information into the formula to solve for the area.
In this case, A =
, or ≈ 8.23.
Find the area of the triangle on the coordinate plane with vertices at (–5, 2), (5, 6),
and (4, 0).
Solve It
186
Part III: Advanced Trig: Identities, Theorems, and Applications
Answers to Problems on Solving Triangles
a
Solve the triangle if B = 46°, C = 62°, and a = 21. The answer is A = 72°, b = 15.9, and c = 19.5.
We draw out this first triangle only, then save some trees and leave the rest to you.
C
62˚
a = 21
46˚
A
B
This one is ASA, so you use the Law of Sines to solve it. Because you already know two angles,
begin by finding the third: A = 72°. Now set up a proportion from the Law of Sines to solve for
you get that b =
the two missing sides. From
you get that c =
b
, or b = 15.9. And from
, or c = 19.5.
Solve the triangle if A = 19°, C = 100°, and b = 4.4. The answer is B = 61°, a = 1.64, and c = 4.95.
Draw out the figure first. It’s ASA again, which means you use the Law of Sines to solve. Find
the missing angle first: B = 61°. Now set up the first proportion to solve for a:
a=
c=
c
;
, or a = 1.64. Set up another proportion to solve for c:
;
, or c = 4.95.
Solve the triangle if A = 49°, B = 21°, and a = 5. The answer is C = 110°, b = 2.37, and c = 6.23.
This AAS case keeps you on your toes, but you still use the Law of Sines. The missing angle is
, which gets you b =
C = 110°. The first proportion is
second proportion is
d
, which gets you c =
, or b = 2.37. The
, or c = 6.23.
Solve the triangle if A = 110°, C = 56°, and a = 8. The answer is B = 14°, b = 2.06, and c = 7.06.
This one is also AAS, so you use the Law of Sines to solve it. First, the missing angle B is 14°.
, to get that c =
Now set up the proportion,
another proportion,
e
, to get that b =
, or c = 7.06. Set up
, or b = 2.06.
Solve the triangle if b = 8, c = 14, and C = 37°. The answer is A = 122.9°, B = 20.1°, and
a = 19.53.
Notice when you draw this one that it’s the dreaded SSA, or the ambiguous case. Always
assume there are two answers when you’re dealing with these types of problems, until you
find out otherwise. Set up the proportion
or sinB =
. This means that 14 · sinB = 8 · sin37,
= 0.344. Use inverse sine to get that B1 = 20.1°. This is the first quadrant
Chapter 10: Solving Oblique Triangles
answer. The second quadrant has a second answer: B2 = 180° – 20.1° = 159.9°. However, if
you look closely you notice that we start off with C = 37°. You can’t then add a 159.9° angle
on top of that and still have a triangle, so you throw this second solution away. Only one
triangle
satisfies the conditions given. Now that you know C and (the one and only) B, it’s easy as
pi (Get it? Pi!) to find A = 122.9°. Set up another proportion,
that a =
f
, which means
= 19.53.
Solve the triangle if b = 5, c = 12, and B = 20°. The answer is A1 = 104.8°, C1 = 55.2°, and
a1 = 14.13; or, A2 = 35.2°, C2 = 124.8°, and a2 = 8.42.
Two solutions! How did we get that? Start at the beginning (a very good place to start) and
use the Law of Sines to set up the proportion
. By cross-multiplying, you get the
. If sinC ≈
equation 5 · sinC = 12 · sin20. Solve for sinC by dividing the 5:
0.821, then use the inverse sine function to discover that C1 = 55.2°. The second quadrant
answer is C2 = 180 – 55.2 = 124.8°. If you add 20° to both of these answers, you discover that
it’s possible to make a triangle in both cases (because you haven’t exceeded 180°). This
sends you on two different paths for two different triangles. We separate them out to make
sure you follow the steps.
If C1 = 55.2°, then A1 = 104.8°. Next, set up the proportion
a1 =
= 14.13.
If C2 = 124.8°, then A2 = 35.2°. Set up another proportion,
a2 =
g
. This means that
, to then get that
= 8.42.
Solve the triangle if a = 10, c = 24, and A = 102°. The answer is no solution.
Here we go again. By the time you’re done with this section of pre-calc, you’ll be an expert
(whether you like it or not) at solving triangles. If you draw this one out, you see another
ambiguous SSA case. Set up the proportion
using the Law of Sines.
Cross-multiply to get 10 · sinC = 24 · sin102, and then divide the 10 from both sides to get
sinC =
≈ 2.35. That’s when the alarms go off. Sine can’t have a value bigger than 1,
so there’s no solution.
h
Solve the triangle if b = 10, c = 24, and B = 20°. The answer is A1 = 104.8°, C1 = 55.2°, and
b1 = 28.3; or, B2 = 35.2°, C2 = 124.8°, and b2 = 16.8.
Watch as we work it out for you:
10 · sinC = 24 · sin20
sinC =
C1 = sin–1(0.821) = 55.2°
A1 = 104.8°
187
188
Part III: Advanced Trig: Identities, Theorems, and Applications
10 · sin101.25 = b1 · sin20
= b1
b1 = 28.3
C2 = 124.8° (this solution also works)
B2 = 35.2°
10 · sin35.7 = b2 · sin20
= b2
b2 = 16.8
i
Solve the triangle if C = 120°, a = 6, and b = 10. The answer is c = 14, A = 21.8°, and B = 38.2°.
This is definitely a Law of Cosines problem when you draw out the triangle (SAS). Find c first:
c2 = a2 + b2 – 2abcosC. Plug in what you know: c2 = 62 + 102 – 2(6)(10)cos120. Plug this right
into your calculator to get that c2 = 196, or c = 14. Now find A: a2 = b2 + c2 – 2bccosA. Plug
in what you know: 62 = 102 + 142 – 2(10)(14)cosA. Simplify: 36 = 296 – 280cosA. Solve for
cosA by first subtracting 296 from both sides: –260 = –280cosA. Now divide the –280 to get
0.929 = cosA. Now use inverse cosine to get A = 21.8, and then use the fact that a triangle is
180° to figure out that B = 38.2°.
j
Solve the triangle if A = 70°, b = 6, and c = 7. The answer is a = 7.50, B = 48.7°, and C = 61.3°.
By plugging what you know into the Law of Cosines, a2 = b2 + c2 – 2bccosA, you get a2 = 62 +
72 – 2(6)(7)cos70. This simplifies to a2 = 56.27, or a = 7.50.
Now switch the substituting in the second law: b2 = a2 + c2 – 2accosB; 62 = 7.502 + 72 –
2(7.50)(7)cosB. Simplify: 36 = 105.27 – 105cosB. Isolate cosB by first subtracting 105.27:
–69.27 = –105cosB, and then dividing the –105: 0.650 = cosB. This means that B = 48.7°.
From there you can figure out that C = 61.3°.
k
Solve the triangle if a = 9, b = 5, and c = 7. The answer is A = 95.7°, B = 33.6°, and C = 50.7°.
You’re solving an SSS triangle using the Law of Cosines, so let’s get crackin’ to find A:
a2 = b2 + c2 – 2bccosA
92 = 52 + 72 – 2(5)(7)cosA
81 = 74 – 70cosA
7 = –70cosA
–0.1 = cosA
95.7° = A
And again to find B:
b2 = a2 + c2 – 2accosB
52 = 92 + 72 – 2(9)(7)cosB
25 = 130 – 126cosB
–105 = –126cosB
Chapter 10: Solving Oblique Triangles
0.833 = cosB
33.6° = B
Last, but certainly not least, C = 50.7°.
l
Solve the triangle if a = 7.3, b = 9.9, and c = 16. The answer is A = 18.3°, B = 25.3°, and
C = 136.4°.
Here we go again to find A:
a2 = b2 + c2 – 2bccosA
7.32 = 9.92 + 162 – 2(9.9)(16)cosA
53.29 = 98.01 + 256 – 316.8cosA
–300.72 = –316.8cosA
0.949 = cosA
18.3° = A
And once more to find B:
b2 = a2 + c2 – 2accosB
9.92 = 7.32 + 162 – 2(7.3)(16)cosB
98.01 = 53.29 + 256 – 233.6cosB
–211.28 = –233.6cosB
0.904 = cosB
25.3° = B
And finally, C = 136.4°.
Two trains leave a station at the same time on different tracks that have an angle of 100°
between them. If the first train is a passenger train that travels 90 miles per hour and the
second train is a cargo train that can only travel 50 miles per hour, how far apart are the
two trains after three hours? The answer is approximately 330.86 miles apart.
The two trains depart (F and S) from the same station (T) in a picture like this one:
T
f=5
0mp
h
S
s = 90mph
m
t
F
Notice that you have to use the Law of Cosines to solve for how far apart the two trains are, t.
Next, you need to find how far the trains have traveled to know how far apart they are.
189
Part III: Advanced Trig: Identities, Theorems, and Applications
Using D = rt, for the first train you get D = 90[3] = 270 miles; for the second train, D = 50[3] =
150 miles. Plug these values into the equation:
t2 = f 2 + s2 – 2fscosT
t2 = 1502 + 2702 – 2(150)(270)cos100. This goes right into your calculator to give you
t2 = 109,465.50, or t = 330.86 miles.
n
A radio tower is built on top of a hill. The hill makes an angle of 15° with the ground. The tower
is 200 feet tall and located 150 feet from the bottom of the hill. If a wire is to connect the top of
the tower with the bottom of the hill, how long does the wire need to be? The answer is about
279.3 feet long.
This time our picture is:
T
200 ft
h
H
ft
15˚
150
190
Ground
B
15˚
To find ∠H in the picture, you add in a horizontal line that’s parallel to the ground. Then, using
the facts that alternate interior angles are congruent and that the tower has to be completely
vertical (or else we have a leaning tower), we know that H = 15° + 90° = 105°.
Now, jump in with the Law of Cosines:
h2 = t2 + b2 – 2tbcosH
h2 = 1502 + 2002 – 2(150)(200)cos105
h2 = 78,029.14
h = 279.3 feet.
o
A mapmaker stands on one side of a river looking at a flagpole on an island at an angle of 85°.
She then walks in a straight line for 100 meters, turns, and looks back at the same flagpole at an
angle of 40°. Find the distance from her first location to the flagpole. The answer is 78.5 meters.
Looking down on the surveyor and the flagpole, here’s the picture you use to solve this problem:
P
s
F
85˚
40˚
100 m
S
Chapter 10: Solving Oblique Triangles
Because you have two angles, you can find that P = 55° and use the Law of Sines.
tells you that s =
p
= 78.5 meters.
Two scientists stand 350 feet apart, both looking at the same tree somewhere in between them.
The first scientist measures an angle of 44° from the ground to the top of the tree. The second
scientist measures an angle of 63° from the ground to the top of the tree. How tall is the tree?
The answer is 226.53 feet tall.
This problem takes some work. You have to know the distance from either scientist to the top
of the tree (FT or TS in the following figure) to know how tall the tree (TB) really is. Here’s a
drawing of the two scientists and the tree in between them:
T
F
44˚
63˚
B
S
350 ft
Knowing two angles gets you the third one: T = 73°. Law of Sines it is, then!
gives you that FT =
= 326.10 feet.
Now, because the tree grows straight up, you have a right triangle in which you know one angle
and one side. That means you need to go back to SOHCAHTOA (see Chapter 6 for more information) to solve for the missing side, TB. Knowing that you have the hypotenuse and are
looking for the opposite side, you can use the sine function and get
that TB = 226.53 feet tall. That’s one big tree!
q
. This means
Find the area of the triangle where a = 7, c = 17, and B = 68°. The answer is 55.17.
Knowing two sides and the angle between them makes this an easy one:
A=
r
= 55.17.
Find the area of the triangle on the coordinate plane with vertices at (–5, 2), (5, 6), and (4, 0).
The answer is 28.
To figure this out you have to know how long two of the sides are and the angle between them.
The problem is, the easiest way to find the angle is to find the length of all three sides and then
use the Law of Cosines to find an angle. So you should start by drawing a picture:
191
192
Part III: Advanced Trig: Identities, Theorems, and Applications
C (5, 6)
b
a
(–5, 2) A
c
B (4, 0)
Find all three sides first (for a review of how to find the distance between two points, see
Chapter 1):
(this is the same as side a when
AC =
you use the Law of Cosines)
(this is side c)
AB =
(this is side b)
BC =
Now that you’ve found the length of all three sides, use Heron’s Formula to find the area:
s=
A=
≈ 13.04.
=
=
= ≈ 28.05.
Part IV
And the Rest . . .
T
In this part . . .
his is the section for everything else. The topics aren’t
necessarily related other than the fact that they’re all
covered in a typical Pre-Calc course. Watch out for those
unsignaled left turns!
This part begins with how to perform operations with and
graph complex numbers. We also introduce the idea of
polar coordinates, a brand new way of graphing equations!
We know you’ve been asking yourself when that was
gonna happen . . . well, here you go. Conic sections are
also a great thing to graph, so we cover them in detail.
We then move on to the systems of equations and cover
solving linear and nonlinear equations, as well as working
with matrices. Next, we’re on to sequences and series. We
discuss how to find any term in a sequence, how to calculate the sum of a sequence, and how to write the formula
that determines a given sequence. Lastly, we cover the
topics that usually constitute the end of pre-calc (and the
beginning of calc): limits and continuity.
Chapter 11
Complex Numbers and
Polar Coordinates
In This Chapter
䊳 Working with complex numbers in operations and graphs
䊳 Switching between polar and rectangular coordinates
䊳 Graphing polar coordinates and equations
O
nce upon a time, mathematicians delved into their imaginations and invented a whole
new set of numbers. They were so deep into their imaginations that they decided to
call these numbers imaginary numbers, because there was just no way these new numbers
would ever pop up in the real world.
Well, they were wrong. These imaginary numbers did eventually appear. Fields like engineering, electricity, and quantum physics all use imaginary numbers in their everyday applications. An imaginary number is basically the square root of a negative number. The imaginary
unit, denoted i, is the solution to the equation i2 = –1.
A complex number is the sum of a real number and an imaginary one. a + bi is an example of a
complex number, where a is the real part and bi is the imaginary part. Pure real numbers like
17 are considered complex numbers as well, but the imaginary part is 0 (it can be written as
17 + 0i). Pure imaginary numbers like –2i are similar, but the real part is 0 (it can be written
as 0 – 2i).
This chapter is like the bonus features on your favorite DVD, as it also includes graphing
points and equations in a whole new way, called “polar coordinates.” Hang in there for a bit
while we explore complex numbers in depth before moving onto polar coordinates.
Performing Operations with and
Graphing Complex Numbers
Complex numbers in the form a + bi can be graphed on a complex coordinate plane. Each real
number is placed on the horizontal axis (so now it’s called the real axis), and each imaginary
number is placed on the vertical one (called the imaginary axis). Each complex number a + bi
corresponds to a point (a, b) on the complex plane. Figure 11-1 shows several examples of
points on the complex plane.
196
Part IV: And the Rest . . .
4i
3 + 2i
–5 + i
1
Figure 11-1:
Graphing
complex
numbers.
5 – 2i
–1 – 4i
Adding and subtracting complex numbers is just another example of collecting like
terms: You can only add or subtract real numbers and you can only add or subtract
imaginary numbers.
When multiplying complex numbers, you FOIL the two binomials. All you have to do is
remember that the imaginary unit is defined such that i 2 = –1, so any time you see i 2 in
an expression, replace it with –1. When dealing with other powers of i, notice the pattern here:
i=
i5 = i
i 2 = –1
i 6 = –1
i 3 = –i
i 7 = –i
i4 = 1
It continues in this manner forever, repeating in a cycle every fourth power. To find a
larger power of i, rather than counting forever, realize that the pattern repeats. For
example, to find i 243, divide 4 into 243 and you get 60 with a remainder of 3. It will repeat
the pattern 60 times and then have 3 left over, so i 243 is the same as i 3, which is –i.
The conjugate of a complex number a + bi is a – bi, and vice versa. When you multiply
two complex numbers that are conjugates of each other, you always end up with a
pure real number:
(a + bi)(a – bi).
FOIL the binomials: a2 – abi + abi – b2i2.
Cancel the two middle terms: a2 – b2i2.
Replace i2 with –1: a2 – b2(–1).
Simplify: a2 + b2.
When dividing complex numbers, you end up with a root in the denominator (because
). This means that you have to rationalize the denominator. To do
if i 2 = –1, then i =
so, you must multiply the complex number in the denominator by its conjugate, and
then multiply this same expression in the numerator. (For more information on rationalizing the denominator of any fraction, see Chapter 2.)
Chapter 11: Complex Numbers and Polar Coordinates
197
We show you how to do this in variables, but you may like it better if you go to the
forthcoming example to see how to rationalize a fraction with complex numbers
involved.
.
Multiply both the numerator and the denominator by the conjugate of the
denominator:
.
FOIL the numerators and the denominators:
.
Cancel like terms in the denominator and replace the i2 with –1 in the numerator
and denominator:
.
Q.
Perform the indicated operation: (3 – 4i) +
(–2 + 5i)2.
Q.
Graph 4 – 6i on the complex coordinate
plane.
A.
–18 – 24i. Follow your order of operations
(PEMDAS) and square the second binomial
by FOILing it times itself: (3 – 4i) + (–2 +
5i)(–2 + 5i); (3 – 4i) + (4 – 10i – 10i + 25i2).
Combine the like terms: (3 – 4i) + (4 – 20i +
25i2). Substitute –1 for i2: (3 – 4i) + (4 – 20i –
25). Combine like terms: (3 – 4i) + (–21 –
20i). Combine like terms one more time —
reals with reals and imaginary with imaginary: –18 – 24i.
A.
See the following graph. Go to the right 4
units on the real axis and 6 units down on
the imaginary axis and place a point. Easy!
Q.
Perform the indicated operation:
A.
.
4 – 6i
. The conjugate of the denominator
is 2 + 9i. Multiply this on the top and bottom
of the fraction:
. Distribute the
numerator and FOIL the denominator:
. Cancel like terms in the
denominator and replace i2 with –1:
This simplifies to the answer
.
.
Q.
Find x and y: 3x + 4yi = 6 – 2i.
A.
x = 2, y = –1⁄2. The real parts of both sides of
this equation must equal each other: 3x = 6.
Divide both sides by 3 to get the solution
x = 2. Furthermore, the imaginary parts
must also equal each other: 4yi = –2i, so
that when you divide both sides by 4i and
reduce, you get y = –1⁄2.
198
Part IV: And the Rest . . .
1.
Plot the point 4 – 3i on the complex plane.
Solve It
3.
Find i22.
Solve It
2.
Find | 3 – 4i |.
Solve It
4.
Solve the equation 5x2 – 2x + 3 = 0.
Solve It
Chapter 11: Complex Numbers and Polar Coordinates
Round a Pole: Graphing Polar Coordinates
Up until now in your math career, you’ve been graphing everything based on the rectangular coordinate system. It’s called that because it’s based on two number lines perpendicular to each other. Pre-calc takes that concept further when it introduces polar coordinates.
In polar coordinates, every point is located around a central point, called the pole, and
is named (r, θ). r is the radius, and θ is the angle formed between the polar axis (think
of it as what used to be the positive x-axis) and the segment connecting the point to
the pole (what used to be the origin).
Most books use radians when measuring angles in polar coordinates, so we do the
same thing. If you need a recap on radians, see Chapter 6. Figure 11-2 shows the polar
coordinate plane.
π/2
π/3
2π/3
π/4
3π/4
4
π/6
5π/6
3
2
1
π
Figure 11-2:
Graphing
round and
round on
the polar
coordinate
plane.
1
2
3
0
4
7π/6
11π/6
7π/4
5π/4
5π/3
4π/3
3π/2
Notice that a point on the polar coordinate plane has more than one name. Because
you’re moving in a circle, you can always add or subtract 2π to any angle and end up at
the same point. This concept is so important in graphing equations in polar forms that
we dedicate this entire section to making sure that you understand.
When both the radius and the angle are positive, most of our students have no difficulties finding the point on the polar plane. If the radius is positive and the angle is negative,
the point moves in a clockwise direction, just like radians do. If the radius is negative and
the angle is positive, find the point where both are positive first and then reflect that
point across the pole. If both the radius and the angle are negative, find the point where
the radius is positive and the angle is negative and then reflect that across the pole.
199
200
Part IV: And the Rest . . .
Q.
What’s the polar coordinate of point P in the following figure?
π/2
π/3
2π/3
π/4
3π/4
4
π/6
5π/6
3
2
1
π
1
2
3
0
4
7π/6
11π/6
7π/4
5π/4
5π/3
4π/3
3π/2
A.
(2, π⁄4). First determine the radius by noticing that the point is 2 units away from the pole. Also
notice that an angle forms when you connect the given point to the pole and that the polar axis is
a π⁄4 angle. This means the point is (2, π⁄4).
Q.
Name two other points that determine the
same point in the last question.
A.
Possible answers include (2, 9π⁄4), (2, 17π⁄4),
and (2, –7π⁄4). You add 2π to the angle twice
and subtract it once to get these three
angles. It’s all about finding the common
denominator at that point: π⁄4 + 2π = π⁄4 + 8π⁄4 =
9π
⁄4, our first answer. Then take that answer
and add 2π to it to get the next one. You
can do this for the rest of your life and still
not list all the possibilities.
Chapter 11: Complex Numbers and Polar Coordinates
5.
Graph (4, –5π⁄3).
Solve It
7.
Name two other polar coordinates for the
point in question 5: one with a negative
angle and one with a positive angle.
Solve It
6.
Graph (–5, π⁄2).
Solve It
8.
Name two other polar coordinates for the
point in question 6: one with a negative
angle and one with a positive radius.
Solve It
201
202
Part IV: And the Rest . . .
Changing to and from Polar
You can use both polar and rectangular coordinates to name the same point on the
coordinate plane. Sometimes it’s easier to write an equation in one form than the
other, but usually, pre-calc books begin by having you switch between the two just to
get used to them. Figure 11-3 shows how to determine the relationship between these
two not-so-different methods.
Figure 11-3:
A right
triangle
reveals the
relationship
between
rectangular
and polar
coordinates.
(r, θ) or (x, y)
r
y
θ
x
Some simple right triangle trig and a little Pythagorean Theorem determine the
relationship:
⻬ sinθ = y⁄r or y = rsinθ
⻬ cosθ = x⁄r or x = rcosθ
⻬ x2 + y2 = r2
⻬ tanθ = y⁄x or θ = tan–1(y⁄x)
Q.
Rewrite the equation in rectangular form:
r = 2sinθ.
Q.
Rewrite the equation r = 2cscθ in rectangular form.
A.
x2 + y2 = 2y. First, convert all trig functions
to x, y, and r. If r = 2sinθ, then r = 2 · y⁄r. Then
multiply the r to the other side to get
r2 = 2y. Then you can replace r2 from the
Pythagorean Theorem and get x2 + y2 = 2y.
A.
y = 2. First, realize that cosecant is the
reciprocal of sine (see Chapter 6 for a
refresher). If sinθ = y⁄r, then cscθ = r⁄y. Substitute this into the equation to get r =
Multiply the y to the other side and get
ry = 2r. This means that y = 2.
.
Chapter 11: Complex Numbers and Polar Coordinates
9.
Rewrite r = 3sinθ + 4cosθ in rectangular
form.
Solve It
11.
Rewrite 3x – 5y = 10 in polar form.
Solve It
13.
Convert the polar coordinate (4, π⁄6) to rectangular coordinates.
Solve It
10.
Rewrite y = 2x – 1 in polar form.
Solve It
12.
Rewrite x2 + y2 = 16 in polar form.
Solve It
14.
Convert the rectangular coordinate (–1, 1)
to polar coordinates.
Solve It
203
204
Part IV: And the Rest . . .
Graphing Polar Equations
When given an equation in polar coordinates and asked to graph it, most students go
with the plug-and-chug method: Pick values for θ from the unit circle that you know so
well and find the value of r. Polar equations have various types of graphs, and we take
a closer look at each one. Be sure to also see Chapter 12 (conic sections) for information about how to graph conic sections in polar coordinates.
Archimedean spiral
r = aθ gives a graph that forms a spiral. a is a constant that’s multiplying the angle, and
then the radius is the same. If a is positive, the spiral moves in a counterclockwise
direction, just like positive angles do. If a is negative, the spiral moves in a clockwise
direction.
Cardioid
You may recognize the word cardioid if you’ve ever worked out and done your cardio.
The word relates to the heart, and when you graph a cardioid, it does look like a
heart, of sorts. Cardioids are written in the form r = a(1 ± sinθ) or r = a(1 ± cosθ). The
cosine equations are hearts that point to the left or right, and the sine equations open
up or down.
Rose
A rose by any other name is . . . a polar equation. If r = asinnθ or r = acosnθ, the graphs
look like flowers with petals. The number of petals is determined by n. If n is odd, then
there are n (the same number of) petals. If n is even, there are 2n petals.
Circle
When r = asinθ or r = acosθ, you end up with a circle with a diameter of a. Circles with
cosine in them are on the x-axis, and circles with sine in them are on the y-axis.
Lemniscate
A lemniscate makes a figure eight; that’s the best way to remember it. r2 = ± a2sin2θ
forms a figure eight in between the axes, and r2 = ± a2cos2θ forms a figure eight that lies
on one of the axes.
Chapter 11: Complex Numbers and Polar Coordinates
205
Limaçon
A cardioid is really a special type of limaçon, which is why they look similar to each
other when graphing them. The limaçon form is familiar:
r = a ± b sinθ
r = a ± b cosθ
in this manner until you end up with the
graph.
Q.
Sketch the graph of r = 1 + sinθ.
A.
See the following graph. This is a cardioid.
If you’re lucky enough to have a teacher
who lets you use a graphing calculator to
graph polar equations, make sure your calculator is set to radians, input the equation
into the grapher, and presto, you have
yourself a graph. If not, just plug and chug
the equation to get the graph. For instance,
if θ = 0, then r = 1 + sin0 = 1 + 0 = 1. If r = π⁄2,
then r = 1 + sinπ⁄2 = 1 + 1 = 2. Keep going
Q.
Sketch the graph of r = cos3θ.
A.
See the following graph. This is a rose
with three petals because the coefficient
r = 1 + sinθ
on the inside is odd. Plug and chug this one
as well, as shown in the following chart (we
show the first quadrant values only):
θ
cos3θ
r
0
cos3(0)
1
π
cosπ⁄2
0
π
cos3π⁄4
π
cosπ
–1
π
cos3π⁄2
0
⁄6
⁄4
⁄3
⁄2
r = cos 3θ
1
1
–1
–1
206
Part IV: And the Rest . . .
15.
Sketch the graph of r2 = 9cos2q.
Solve It
17.
Sketch the graph of r = 1 – 3sinθ.
Solve It
16.
Sketch the graph of r = 2θ.
Solve It
18.
Sketch the graph of r = 3 + 2sinθ.
Solve It
Chapter 11: Complex Numbers and Polar Coordinates
Answers to Problems on Complex
Numbers and Polar Coordinates
a
Plot the point 4 – 3i on the complex plane. See the following graph for the answer.
The real unit is 4 to the right and the imaginary unit is 3 down. This lands you at the point in
the figure.
4 – 3i
b
Find | 3 – 4i |. The answer is 5.
Remember that absolute value bars represent distance. In the case of complex numbers, they
represent the distance from the point to the origin. This distance is always the same as the
length of the hypotenuse of the right triangle drawn when connecting the point to the x- and
y-axes. d =
c
= 5.
22
Find i . The answer is –1.
The pattern for the powers of i repeats every four. Divide 22 by 4 and get 5 with a remainder of
2. This means that i22 = i2 = –1.
d
Solve the equation 5x2 – 2x + 3 = 0. The answer is
.
This quadratic equation is unfactorable, so you have to resort to the quadratic formula to solve
it. x =
, which simplifies to
. If
you’ve forgotten how to deal with the quadratic formula and/or how to simplify roots, see
Chapter 4.
e
Graph (4, –5π⁄3). See the following graph for the answer.
The radius is 4 and the angle is negative, which moves in a clockwise direction and ends up in
the first quadrant.
(4, –5π)
3
207
208
Part IV: And the Rest . . .
f
Graph (–5, π⁄2). See the following graph for the answer.
The radius for this one is negative, which reflects the point (5, π⁄2) over the pole.
(–5, π)
2
g
Name two other polar coordinates for the point in question 5: one with a negative angle and
one with a positive angle. Possible answers include (4, –11π⁄3), (4, π⁄3).
The radius isn’t changing, so don’t do anything to it. Subtract 2π from the angle to get the first
answer. Add 2π to the angle to get the second answer.
h
Name two other polar coordinates for the point in question 6: one with a negative angle and
one with a positive radius. Possible answers include (–5, –3π⁄2), (5, 3π⁄2).
To deal with the first situation, don’t change the radius; just subtract 2π to get a negative angle:
–3π⁄2. To change the radius to a positive 5, you have to change the angle. Because the point was
down 5 originally, this angle is the same as 3π⁄2.
i
Rewrite r = 3sinθ + 4cosθ in rectangular form. The answer is x2 + y2 = 3y + 4x.
First, change sinθ to y⁄r and cosθ to x⁄r:
. Multiply everything by r to get rid of the frac-
2
tions and get r = 3y + 4x. Use the Pythagorean substitution and get x2 + y2 = 3y + 4x.
j
Rewrite y = 2x – 1 in polar form. The answer is
.
To move the other direction, remember that x = rcosθ and y = rsinθ. Make these substitutions
first and get: rsinθ = 2rcosθ – 1. Get all terms with r to one side first: rsinθ – 2rcosθ = –1. Factor
out the common factor of r: r(sinθ – 2cosθ). Now solve for r by dividing the rest to the other
side:
k
.
Rewrite 3x – 5y = 10 in polar form. The answer is
.
Use the same substitutions as the last question to get 3rcosθ – 5rsinθ = 10. Factor: r(3cosθ –
5sinθ) = 10. Divide:
l
.
Rewrite x2 + y2 = 16 in polar form. The answer is r = 4.
Use the Pythagorean substitution first: x2 + y2 = r2, so r2 = 16. Take the square root of both sides
and get r = 4.
m
Convert the polar coordinate (4, π⁄6) to rectangular coordinates. The answer is (
If x = rcosθ, then x = 4cosπ⁄6 =
If y = rsinθ, then y = 4sinπ⁄6 = 4(1⁄2) = 2.
.
, 2).
Chapter 11: Complex Numbers and Polar Coordinates
n
Convert the rectangular coordinate (–1, 1) to polar coordinates. The answer is (
, –π⁄4).
Use the Pythagorean substitution first to find r: r2 = x2 + y2 means that r2 = (–1)2 + (1)2 =
1 + 1 = 2, or r = . Then use tangent to find the angle: tanθ = –1, or θ = –π⁄4.
o
Sketch the graph of r2 = 9cos2θ. See the following graph for the answer.
This graph is a lemniscate. If θ = 0, r = 3. If θ = π⁄6, then r ≈ 2.12, and so on. Plug and chug it
and wind up with the graph.
r2 = 9cos2θ
–3
p
3
Sketch the graph of r = 2θ. See the following graph for the answer.
This is a spiral. If θ = 0, r = 0; if θ = π⁄6, r ≈ 1.14, and so on.
r = 2θ
q
Sketch the graph of r = 1 – 3sinθ. See the following graph for the answer.
This is a limaçon. Plug and chug it and get the graph.
r = 1 – 3sinθ
209
210
Part IV: And the Rest . . .
r
Sketch the graph of r = 3 + 2sinθ. See the following graph for the answer.
This is a cardioid. If you plug and chug it, you end up with the graph.
r = 3 + 2sinθ
Chapter 12
Conquering Conic Sections
In This Chapter
䊳 Closing in on circles
䊳 Graphing parabolas, ellipses, and hyperbolas
䊳 Recognizing the different conic sections
䊳 Working with parametric form and polar coordinates
W
ho doesn’t love a good cone? We’re big fans of the ice cream cone ourselves, but
maybe you’re partial to the traffic cone because it keeps you safe on the road.
Whatever type of cone is your favorite, for mathematicians the cone is the creative fuel for
the fire of a whole bunch of ideas.
You see, about 2,200 years ago, some smart mathematician named Apollonius of Perga decided
one day to stack two cones point to point. He sliced them in different directions and came up
with four different conic sections: the circle, the ellipse, the parabola, and the hyperbola. Each
conic section has its own equation and its own parts, which you need to determine in order to
graph it. Typically, a textbook question will ask you to graph a conic section, identify certain
parts, or write its equation. To do any of these tasks, you must be able to recognize what kind
of conic section it is (more on that later) and write it in its own equation form.
When a conic section is already written in its form, you know most of the information about
the parts of that particular conic. When a conic section isn’t written in its form, well, you just
have to write it that way. How do you do that, you ask (and we’re so glad that you did)? It’s
the process called completing the square. That should sound familiar to you from previous
math classes. If not, review Chapter 4.
A Quick Conic Review
To save you some time, we quickly review here how to complete the square for conic sections. This is the only way to change a conic section’s equation that’s not written in its
proper form to one that is. The steps of the process are as follows:
1. Add/subtract any constant to the opposite side of the given equation, away from all
the variables.
2. Factor the leading coefficient out of all terms in front of the set of parentheses.
3. Divide the remaining linear coefficient by two, but only in your head.
4. Square the answer from Step 3 and add that inside the parentheses. Don’t forget that
if you have a coefficient from Step 2, you must multiply the coefficient and this number
from Step 4 and add that to both sides.
5. Factor the quadratic polynomial as a perfect square trinomial.
This chapter is dedicated to each mathematical conic superstar, one at a time — how to
write each one in its form and how to graph it.
212
Part IV: And the Rest . . .
Going Round and Round with Circles
A circle is simple, but not so plain. You can do so much with a circle. The very tires
you drive on are, after all, circular in shape. A circle has one point in the very middle
called the center. All the points on the circle are the same distance, called the radius,
from the center.
When a circle is drawn centered at the origin of the coordinate plane, the equation that
describes it is simple as well:
x2 + y2 = r2
r is the variable that represents the circle’s radius.
When the circle is moved around the coordinate plane with horizontal and/or vertical
shifts, its equation looks like this:
(x – h)2 + (y – v)2 = r2
h is the horizontal shift of the equation, v is the vertical shift, and r is still the radius. If
a circle isn’t written in this form, you’ll still recognize it as a circle because the equation will have both an x2 and a y2 and the coefficients on both will be equal.
Don’t be intimidated if you see this written another way in your pre-calc textbook.
Different books may write this equation differently. Just know that it’s always written
as (x – horizontal)2 and (y – vertical)2.
Q.
What’s the center and the radius of the
circle (x + 3)2 + y2 = 16?
A.
Center (–3, 0); radius r = 4. Because we’ve
been talking about circles, we thought we’d
start you off with training wheels. This
equation is already in the proper form of a
circle, so finding its information is easy.
The horizontal value with x is 3, so h = –3;
the vertical shift is missing, so v = 0. This
means that the center is located at the
point (–3, 0). Meanwhile, the other side of
the equation gives you the radius squared.
Setting r2 = 16 gives you the solution r = 4.
Q.
Graph x2 + y2 – 6y + 2 = 0.
A.
See Figure 12-1. Now the training wheels
come off! First subtract the constant to
move it to the other side: x2 + y2 – 6y = –2.
Because the leading coefficients are 1, you
don’t have to factor anything out and can
move onto the next step. Then, because
the x variable doesn’t have a linear term,
you don’t have to complete the square
there. Notice, however, that the y variable
does have one, so you follow the process:
= 9. This value gets added to both
sides: x2 + y2 – 6y + 9 = 7. Factor the y’s
as a perfect square trinomial to get
x2 + (y – 3)2 = 7. Voilá! You got yerself a
circle. Its center is (0, 3), and its radius is
. See Figure 12-1 for the graph of this
beautiful circle.
Chapter 12: Conquering Conic Sections
Figure 12-1:
Graphing a
circle:
Mark the
center first,
and then
mark the
radius in all
directions.
1.
x2 + (y – 3)2 = 7
Find the center and radius of the circle
2x2 + 2y2 – 4x = 15. Then graph the circle.
Solve It
2.
Write the equation of the circle with the
center (–1, 4) if the circle passes through
the point (3, 1).
Solve It
Graphing Parabolas: The Ups and Downs
When you graph a quadratic polynomial (see Chapter 4 for more information on this
type of polynomial), you always get a parabola. Typically, up until this point in precalc, the graphs of parabolas have been vertical: they open up or down. In conic sections, however, they can also open horizontally: to the left or to the right. We take a
look at each situation in the following sections.
213
214
Part IV: And the Rest . . .
Officially, a parabola is the set of all points on a plane that are the same distance from
a given point (focus) and a given line (directrix). Each parabola can be folded exactly in
half over a line called the axis of symmetry. The point where the axis of symmetry intersects the graph is called the vertex. This gives you what we like to call the martini of
conic sections: The parabola is the glass, the axis of symmetry is the stem, the directrix is the base, and the focus is the olive. Every good martini has all its parts, and
every good parabola does, too! But don’t be too shaken (or stirred) up! Figure 12-2
shows all the parts of the parabola.
Parabola
Focus
Directrix
Figure 12-2:
The
parabolatini . . . no
twist.
Axis of
symmetry
Standing tall: Vertical parabolas
The equation of a vertical parabola is:
y = a(x – h)2 + v
where h is the horizontal shift, v is the vertical shift, and a is the vertical transformation. You’ll recognize a vertical parabola when x is squared but y isn’t. We discuss
these types of transformations ad nauseum in Chapter 3, and it shouldn’t surprise you
that nothing changes just because we’re calling them conic sections. Each vertical
parabola still has the following parts:
⻬ Vertex: (h, v)
⻬ Axis of symmetry: x = h
⻬ Focus: (h, v +
)
⻬ Directrix: y = v –
Chapter 12: Conquering Conic Sections
215
Don’t memorize these as formulas to find the parts. Instead, use what you know about
a parabola to get the job done. The vertex is always the first point that you graph on a
parabola. From there, know that the focus and the directrix are
away from it. One
will be above and one will be below, depending on the value of a. If a < 0, the parabola
opens down and the focus moves down while the directrix moves up. If a > 0, the
parabola opens up, the focus moves up, and the directrix moves down.
Q.
Graph the equation of the parabola
y = –2(x – 1)2 + 5.
A.
See Figure 12-3. Because this equation is
already in the proper parabola form, you
should be able to go right to graphing.
The vertex is at (1, 5). The vertical transformation is 2, and the graph is turned
upside down. If you don’t know where this
information comes from, we strongly recommend you go back and read Chapter 3
now, which is chock-full of information on
transforming any function. We’ll be waiting
right here for you when you get back. In
the meantime, the final graph is shown in
Figure 12-3.
Vertex (1, 5)
y = –2(x – 1)2 + 5
Figure 12-3:
Graphing a
vertical
parabola.
Q.
State the vertex, axis of symmetry, focus,
and directrix of y = 3x2 – 4x + 1.
A.
Vertex: (2⁄3, –1⁄3); axis of symmetry: x = 2⁄3;
focus: (2⁄3, –1⁄4); directrix: y = –1⁄12. That’s an
awful lot of fractions, ain’t it? But the
process doesn’t change. Start by subtracting 1 from both sides: y – 1 = 3x2 – 4x.
Then factor out the three: y – 1 = 3(x2 – 4⁄3x).
Now complete the square and be sure to
keep the equation balanced: y – 1 + 4⁄3 =
3(x2 – 4⁄3x + 4⁄9). Simplify and factor: y + 1⁄3 =
3(x – 2⁄3)2. Then solve for y to put the
parabola in its proper form: y = 3(x – 2⁄3)2 – 1⁄3.
Next, use the formulas to figure out all the
parts — the vertex is (2⁄3, –1⁄3), the axis of
symmetry is x = 2⁄3, the focus is (2⁄3, –1⁄4), and
the directrix is y = –1⁄12. Sheesh!
216
Part IV: And the Rest . . .
3.
What’s the vertex of the parabola y = –x2 +
4x – 6? Sketch the graph of this parabola.
Solve It
4.
Find the focus and the directrix of the
parabola y = 4x2.
Solve It
Lying down on the job: Horizontal parabolas
The equation of a horizontal parabola is very similar to the vertical one we discuss
in the last section. Here it is:
x = a(y – v)2 + h
This is a horizontal parabola because y is squared but x is not. Notice, also, that h
and v are still there for the horizontal and vertical shifts, respectively, but that
they’ve switched places. Because this parabola is horizontal, a also switches to
become the horizontal transformation. This is the first time we’ve talked about a
horizontal transformation, so we’ll take a few moments to explain the idea. A horizontal transformation does the same thing as a vertical transformation, but it affects
what the function does from left to right. A horizontal transformation where a is a
fraction in between 0 and 1 is a horizontal shrink, and a horizontal transformation
where a > 1 is called a horizontal stretch. All of these are positive, so the parabola
opens to the right. When a is negative the parabola does the same thing, but the
graph is reflected in the opposite direction (so the parabola opens to the left).
Here are the parts of a horizontal parabola:
⻬ Vertex: (h, v)
⻬ Axis of symmetry: y = v
⻬ Focus: (h +
, v)
⻬ Directrix: x = h –
Chapter 12: Conquering Conic Sections
Q.
Graph the parabola if its equation is
x = y2 – 6y.
217
A.
See Figure 12-4. Complete the square to
get this horizontal parabola in its form:
x = (y – 3)2 – 9. This means that the
parabola’s vertex is located at the point
(–9, 3). There’s no horizontal transformation (a = 1), so from the vertex the graph
moves up 1, over 1; up 1, over 4; up 1, over
9. This gives you the graph in Figure 12-4.
A.
x=
y = y2 – 6y
V (–9, 3)
Figure 12-4:
Graphing a
horizontal
parabola.
Q.
Write the equation of the parabola whose
vertex is (–2, 1) if the focus is at the point
(–4, 1).
. This one takes some
brainpower to work out. Because the focus
is to the left of the vertex, you know that
the parabola is a horizontal one (forgetting,
of course, that you’re in the section of the
book on horizontal parabolas). Start with
the equation of any horizontal parabola:
x = a(y – v)2 + h. Then plug in the vertex
values (–2, 1): x = a(y – 1)2 – 2. Now, all you
have to do is figure out what that pesky
value of a is. Have no fear, pre-calc is here!
You know that the equation to find the
, v). This tells you that
focus is (h +
h+
= –4. You also know that h is –2, so
substitute and get –2 +
= –4. Solve to
get a = –1⁄8. Finally, write the equation of the
parabola: x =
.
218
Part IV: And the Rest . . .
5.
Sketch the graph of x = 2(y – 4)2.
Solve It
6.
Determine whether this parabola opens left
or right: x = –y2 – 7y + 3.
Solve It
Graphing Ellipses: The Fat and the Skinny
An ellipse is defined as the set of all points on a plane, such that the sum of the distances from any point on the curve to two fixed points, the foci, is a constant. Think
of an ellipse as a circle that’s gone flat, like a soda left out overnight. All the fizz has
left. That doesn’t mean that the ellipse has no flavor! It has its own unique parts and
equations, depending on whether (you guessed it) it’s horizontal or vertical. Here’s
what you need to know about any ellipse, whether it’s horizontal or vertical:
⻬ The center is at the point (h, v).
⻬ The longer axis of symmetry is called the major axis, and the distance from the
center to a point on the ellipse along the major axis is represented by a. The
points where the ellipse intersects this axis are called the vertices.
⻬ The shorter axis of symmetry is called the minor axis, and the distance from
the center to a point on the ellipse along the minor axis is represented by b.
The points where the ellipse intersects this axis are called the co-vertices.
⻬ This means that a is always greater than b in an ellipse.
⻬ You can find the foci of the ellipse along its major axis by using the equation
f 2 = a2 – b2.
Figure 12-5 shows how all these pieces fall into place for a horizontal and a vertical
ellipse.
Chapter 12: Conquering Conic Sections
219
vertex
major axis
f
f
a
b
a
c
vertex
f
co-vertex
b.
co-vertex
minor axis
c
major axis
Figure 12-5:
The labels
of a horizontal and a
a.
vertical
ellipse.
minor axis
co-vertex
b
co-vertex
f
vertex
Short and fat: The horizontal ellipse
The equation of a horizontal ellipse is:
Notice that all the variables (except for f, the focus) we mention make their appearance in
the equation. Notice also that because a > b, for any ellipse, a2 > b2. The fact that the bigger
number, a2, is in the denominator of the x fraction tells you that the ellipse is horizontal.
To graph any horizontal ellipse after it’s written in this form, mark the center first. Then
count out a units to the left and right and b units up and down. These four points determine the ellipse’s shape. The vertices are points found at (h ± a, v). The co-vertices are
(h, v ± b). The two foci are f units in the same direction as a. As points, the foci are (h ± f, v).
Q.
State the center, vertices, and foci of the
ellipse 3x2 – 18x + 5y2 = 3.
A.
Center: (3, 0); vertices: (3 ±
, 0); foci:
(5, 0) and (1, 0). As usual, you need to
complete the square to write this ellipse
in its proper form. The constant is already
on the opposite side, so begin by factoring:
3(x2 – 6x) + 5y2 = 3. Next, complete the
square and balance the equation:
3(x2 – 6x + 9) + 5y2 = 3 + 27. Factor the
perfect square: 3(x – 3)2 + 5y2 = 30. Divide
everything by 30:
. This
tells you the center is (3, 0). Then, if a2 = 10,
a=±
, and if b2 = 6, b = ±
. This
gives you the vertices at (3 ±
, 0). It also
tells you the co-vertices are at (3, ±
).
Lastly, f 2 = 10 – 6, so f 2 = 4, which tells you
that f = ± 2 and the foci are at (5, 0) and (1, 0).
Q.
Sketch the graph of the ellipse
.
A.
See Figure 12-6. This ellipse is written in
the proper form, so to graph it, all you
have to do is identify its parts. The center
is (–2, 1). If a2 = 25, then a = ± 5. This means
your vertices are 5 units to the left and the
right from the center, at (–7, 1) and (3, 1).
If b2 = 16, then b = ± 4. This means your
co-vertices are 4 units above and below
the center at (–2, 5) and (–2, –3). This
graph is shown in Figure 12-6.
220
Part IV: And the Rest . . .
V (–7, 1)
Figure 12-6:
Graphing a
horizontal
ellipse.
7.
C (–2, 1)
V (3, 1)
(x + 2)2 + (y – 1)2 = 1
25
16
Sketch the graph of the ellipse 4x2 + 12y2 –
8x – 24y = 0.
Solve It
8.
Write the equation of the ellipse with
vertices at (–1, 1) and (9, 1) and foci at
(4 ±
, 1).
Solve It
Tall and skinny: The vertical ellipse
The equation of a vertical ellipse is:
This equation looks awfully familiar, doesn’t it? The only difference between a horizontal ellipse and a vertical one is the location of a. When the bigger number is under x,
Chapter 12: Conquering Conic Sections
221
it’s a horizontal ellipse. When the bigger number is under y, it’s a vertical ellipse. You
graph this ellipse by marking the center, counting up and down a units to find the vertices, and then counting left and right b units to find the co-vertices. This means your
vertices are at (h, v ± a) and your co-vertices are at (h ± b, v). Your foci move in the
same direction as your vertices, so they’re at (h, v ± f).
Q.
Sketch the graph of the ellipse 81x2 + 4y2 =
324. State the foci of this ellipse.
A.
See Figure 12-7; the foci are (0, ±
).
We wanted to include an example like this
because most of our students freeze when
presented with a question like this one.
Just remember that your only goal is to
write the equation in the proper form. The
equation has no x variable or y variable to
the first degree. That means you don’t have
to complete the square! Say what? All you
need to do is get 1 on the right side of the
equation by dividing everything by 324.
When you do, it reduces to
,
which also conveniently puts it in the form
you want. This ellipse has its center at the
origin (0, 0). Then, a2 = 81, so a moves up
and down 9 units, while b2 = 4, so b moves
left and right 2 units. This gives you the
graph in Figure 12-7. Lastly, f 2 = 81 – 4 = 77,
so f =
. This means your foci are
at (0,
).
81x2 + 4y2 = 324
Figure 12-7:
Graphing a
vertical
ellipse.
Q.
Write the equation of the vertical ellipse
with its center at (–4, 1) if its major axis
has a length of 10 and its minor axis has a
length of 8.
A.
. You have all the
information you need to write the equation. The center is given to you as (–4, 1).
If the major axis has a length of 10, then
2a = 10, or a = 5. Also, the minor axis has
a length of 8, so 2b = 8, or b = 4. Knowing
that the ellipse is vertical tells you to put
a2 under y and b2 under x. This gives you
the equation
.
222
Part IV: And the Rest . . .
9.
Sketch the graph of the ellipse
10.
State the ordered pair for the vertices, the
co-vertices, and the foci of the ellipse
.
.
Solve It
Solve It
Graphing Hyperbolas: No Caffeine Required
A hyperbola is the set of all points where the difference in the distance from two fixed
points (the foci) to any point is constant. Hyperbolas always come in pairs. Each one
is a perfect mirror reflection of the other. Maybe they’re the narcissists of the math
world, always checking themselves out in the mirror that is their axis of symmetry.
There are horizontal and vertical hyperbolas. Regardless of how the hyperbola opens,
you always find the following parts:
⻬ The center is at the point (h, v).
⻬ The graph on both sides gets closer and closer to two diagonal lines known as
asymptotes. The equation of the hyperbola, regardless of whether it’s horizontal
or vertical, gives you two values: a and b. These help you draw a box, and when
you draw the diagonals of this box, you find the asymptotes.
⻬ There are two axes of symmetry:
• The one passing through the vertices is called the transverse axis. The distance from the center along the transverse axis to the vertex is represented
by a.
• The one perpendicular to the transverse axis through the center is called
the conjugate axis. The distance along the conjugate axis from the center
to the edge of the box that determines the asymptotes is represented by b.
• a and b have no relationship; a can be less than, greater than, or equal to b.
⻬ You can find the foci by using the equation f 2 = a2 + b2.
Figure 12-8 shows the parts of a hyperbola — one horizontal and one vertical.
Chapter 12: Conquering Conic Sections
b
a
Focus
Center
(h, v)
Horizontal
Vertex
hyperbola
Focus
Vertex
F
F
V
a
b
Vertical
Figure 12-8: hyperbola
Horizontal
and vertical
hyperbolas
and their
parts.
Center
(h, v)
Focus
Vertex
Hip horizontal hyperbolas (alliteration!)
The horizontal hyperbola is so hip (how hip is it?) that it doesn’t even have to stand
up — it just lies around all day. The equation of a horizontal hyperbola is:
This one looks really similar to the equation of the horizontal ellipse. But if you look
closely, you notice the subtraction sign in between the two fractions. To begin graphing, identify a, which helps determine one edge of a box that you can use to find the
hyperbola’s asymptotes. The corners of this imaginary box are two points of the
asymptotes, so they can be used to draw those lines. The value of a is in the denominator of the x fraction, so it will be left and right from the center. The vertices are at
(h ± a, v). The other edge of the box is found from b, under the y fraction. It moves up
and down. The foci move in the same direction as a and can be found at (h ± f, v). The
.
equation of the asymptotes of a horizontal hyperbola is given by y =
223
224
Part IV: And the Rest . . .
Q.
Sketch the graph of 2x2 – 3y2 + 10x + 6y = 41⁄2.
A.
See Figure 12-9. Put it in its form by completing the square. (Did we mention you’d
be experts at this by the end of this chapter?) Write the equation in order first:
2x2 + 10x – 3y2 + 6y = 41⁄2. Then factor out
the coefficients: 2(x2 + 5x) – 3(y2 – 2y) = 41⁄2.
Watch out for the negative sign there when
factoring. Complete the square and keep
the equation balanced: 2(x2 + 5x + 25⁄4) –
3(y2 – 2y + 1) = 41⁄2 + 25⁄2 – 3. Factor the difference of squares and simplify: 2(x + 5⁄2)2 –
3(y – 1)2 = 30. Last, divide everything by 30:
. From the equation
you know that the center is (–5⁄2, 1), a =
. This gives you the graph in
and b =
Figure 12-9.
2x2 – 3y2 + 10x + 6y = 41
2
Figure 12-9:
Graphing a
horizontal
hyperbola.
Q.
Find the equation of the asymptotes for the
hyperbola
.
A.
y=
. Because this
equation is in its form, the information is
easy to find. h = 0, v = 6, a = 5, and b = 3.
Put those into the equation to find the
asymptotes for a horizontal hyperbola and
get y =
.
,
Chapter 12: Conquering Conic Sections
11.
Find the equation of the hyperbola that has
its center at (4, 1), its vertex at (7, 1), and
one of its asymptotes is 3y = 2x – 5.
12.
Sketch the graph of the equation
(2x – y)(x + 5y) – 9xy = 10.
Solve It
Solve It
Vexing vertical vyperbolas (er, hyperbolas)
The equation of a vertical hyperbola is:
Do you see the differences between the horizontal and vertical hyperbolas? The x and
y switch places (along with the h and v). The a stays on the left, and the b stays on the
right. When you write a hyperbola in its form, you need to make sure that the positive
squared term is always first. The vertices are at (h, v ± a) and the foci are at (h, v ± f).
, sort of like the horiYou can find the asymptotes using the equation y =
zontal hyperbolas.
225
226
Part IV: And the Rest . . .
Q.
Sketch the graph of the hyperbola
16y2 – 25x2 + 64y – 50x = 361.
A.
See Figure 12-10. Start by completing the
square. Rewrite the equation: 16y2 + 64y –
25x2 – 50x = 361. Factor out the coefficients:
16(y2 + 4y) – 25(x2 + 2x) = 361. Now, complete
the square (keep the equation balanced,
too): 16(y2 + 4y + 4) – 25(x2 + 2x + 1) =
361 + 64 – 25. Factor, factor: 16(y + 2)2 –
25(x + 1)2 = 400. Divide the 400:
. The center of this
hyperbola is (–1,–2), where a = 5 (and goes
up and down) and b = 4 (left/right). This
finally gives you Figure 12-10.
Figure 12-10:
Graphing a
vertical
hyperbola.
Q.
Two science stations are two miles apart. They record an explosion, one station two seconds after
the other. Write the equation of the hyperbola that describes the situation by placing both stations on the y-axis with the origin at the center. (Note: Sound travels at 1100 feet per second.)
A.
. If the two stations are two miles apart, that means one is a mile below the
origin (0, –1) and the other is a mile above it (0, 1). Right away, you should notice that all other
measurements in this problem are given in feet, so you have to convert 1 mile to 5,280 feet. This
means the microphones of the two stations are at (0, –5,280) and (0, 5,280). This gives you the foci
of the hyperbola: f = 5,280. To find the vertices, you can set up two different equations based on
the distance (d = rt) the explosion travels to get to the two stations: d1 = 1100t and d2 = 1100(t + 2).
The absolute value of the difference of the distances gives you the vertices of the hyperbola:
. a for your vertex is half of this:
⁄2 = 1100, so a2 = 1,210,000. Now that you know f and a, you can find b2 from the equation
f = a2 + b2: b2 = 26,668,400. Lastly, because the hyperbola has its center at the origin, you can write
2200
2
the equation of the hyperbola as
.
Chapter 12: Conquering Conic Sections
13.
Sketch the graph of the equation x2 + 2x –
4y2 + 32y = 59.
14.
Solve It
Write the equation of the hyperbola that
has its center at (–3, 5), one vertex at
(–3, 1), and passes through the point
(1, 5 –
).
Solve It
Identifying Conic Sections
Often, you’ll be presented with an equation and asked to graph it, but you won’t be
told what type of conic section it is. You have to be able to identify what type it is
before doing any work, but that’s easier than it sounds, because there are only the four
conics, and they have distinct differences:
⻬ Circles have x2 and y2 with equal coefficients on both.
⻬ Parabolas have x2 or y2, but not both.
⻬ Ellipses have x2 and y2 with different (not equal) coefficients on each.
⻬ Hyperbolas have x2 and y2 where exactly one coefficient is negative.
Table 12-1 has all the information you need to know about the four conics in one
handy-dandy chart.
227
228
Part IV: And the Rest . . .
Table 12-1
Types of Conic Sections and Their Parts
Type of Conic
Parts of the Conic
Circle
Center (h, v)
Radius r
Horizontal parabola
Vertex (h, v)
Focus (h + 1 , v)
4a
Directrix x = h – 1
4a
Axis of symmetry x = h
Vertical parabola
Vertex (h, v)
Focus (h, v + 1 )
4a
Directrix y = v – 1
4a
Axis of symmetry y = v
Horizontal ellipse
Center (h, v)
Vertices (h ± a, v)
Co-vertices (h, v ± b)
Foci (h ± f, v)
Vertical ellipse
Center (h, v)
Vertices (h, v ± a)
Co-vertices (h ± b, v)
Foci (h, v ± f )
Horizontal hyperbola
Center (h, v)
Vertices (h ± a, v )
Foci (h ± f, v)
Asymptotes y =
Vertical hyperbola
Center (h, v)
Vertices (h, v ± a)
Foci (h, v ± f )
Asymptotes y =
Chapter 12: Conquering Conic Sections
15.
Sketch the graph of 3x2 + 4y2 – 6x + 16y – 5 = 0.
Solve It
17.
Sketch the graph of 4(x – 2) = 2y2 + 6y.
Solve It
16.
Sketch the graph of 4x2 – 8x – 1 = 4y2 – 4y.
Solve It
18.
Sketch the graph of 2y2 – 4x2 + 8x – 8 = 0.
Solve It
229
230
Part IV: And the Rest . . .
19.
Sketch the graph of 4x2 + 4y2 – 8y + 16x – 4 = 0.
Solve It
20.
Sketch the graph of 3x2 – 4y2 + 3x – 2y – 25⁄2 = 0.
Solve It
Converting from Parametric Form
to Polar Coordinates and Back
So far, you’ve graphed all the conics in rectangular form (x, y). However, you can graph
a conic section in two other ways:
⻬ Parametric form: This form is for conics that can’t be easily written as a function y = f(x). Both x and y are written in two different equations as being dependent on one other variable (usually t).
⻬ Polar form: You recognize this from Chapter 11, where every point is expressed
as (r, θ).
We show you how to deal with both of these forms in the following sections.
Parametric form for conic sections
Parametric form defines both x and y in terms of another arbitrary value called the
parameter. Most often, this is represented by t, as real-world applications set the definitions based on time. You can find x and y by picking values for t. Why change? In parametric form you can find how far an object has moved over time (the x equation) and
the object’s height over time (the y equation).
Chapter 12: Conquering Conic Sections
Q.
Sketch the curve given by the parametric
equations x = 2t + 1, y = t2 – 3t + 1, and
1 < t ≤ 5.
A.
See Figure 12-11. Create a table for t, x,
and y. Pick values of t between the interval
values given to you, and then figure out
what the x and y values are for each t value.
Table 12-2 shows these values, and
Figure 12-11 shows the graph of this
parametric function.
Even though t > 1 in the given interval, you need to start your table off with this value
to see what the function would have been. Your graph has an open circle on it at this
point to indicate that the value isn’t included in the graph or the interval.
Table 12-2
t value
Figure 12-11:
Graphing a
function
written in
parametric
form.
Plug and Chug a Parametric Equation
x value
231
y value
1
3
–1
2
5
–1
3
7
1
4
9
5
5
11
11
232
Part IV: And the Rest . . .
21.
Sketch the graph of the parametric equations x = t 2 – 1, y = 2t, and –2 ≤ t ≤ 3.
Solve It
22.
Sketch the graph of the parametric
equations x =
,y=
, and t > 1.
Solve It
Changing from parametric form
to rectangular form
The only other way to graph a parametric curve is to write it in rectangular form. To
do this, you must solve one equation for the parameter and then substitute that value
into the other equation. It’s easiest if you pick the equation you can solve for the
parameter (choose the equation that’s linear). To show you how it works, we use the
example from the last section.
Q.
Write the parametric equations x = 2t + 1,
y = t2 – 3t + 1, and 1 < t ≤ 5 in rectangular
form.
A.
y=
. First, solve the equation
that’s linear for t:
. Then substitute
this value into the other equation for t:
y=
equation to get y =
. Simplify this
.
Chapter 12: Conquering Conic Sections
23.
Eliminate the parameter and find an equation in x and y whose graph contains the
curve of the parametric equations x = t2,
y = 1 – t, and t ≥ 0.
24.
Eliminate the parameter of the parametric
equations x = t – 5 and y =
.
Solve It
Solve It
Conic sections on the polar coordinate plane
Conic sections on the polar coordinate plane are all based on a special value known as
eccentricity, or e. This value describes what kind of conic section it is, as well as the
conic’s shape. It’s difficult to know what kind of conic section you’re dealing with until
you know what the eccentricity is:
⻬ If e = 0, the conic is a circle.
⻬ If 0 < e < 1, the conic is an ellipse.
⻬ If e = 1, the conic is a parabola.
⻬ If e > 1, the conic is a hyperbola.
When you know e, all conics are expressed in polar form based on (r, θ), where r is the
radius and θ is the angle. See Chapter 11 for more information on polar equations.
All conics in polar form are written based on four different equations:
r=
or
r=
or
where e is eccentricity and k is a constant value. To graph any conic section in polar
form, substitute values of θ and plug and chug away until you get a picture!
233
234
Part IV: And the Rest . . .
Q.
Graph the equation r =
A.
See Figure 12-12. First, notice that the
equation as shown doesn’t fit exactly into
any of the equations we just gave you. All
those denominators begin with 1, and this
equation begins with 4! To deal with this,
factor out the 4 from the denominator to
. Notice that this makes e the
get
same in the numerator and denominator
(1⁄4) and that k is 2. Now that you know e is
1
⁄4, that tells you the equation is an ellipse.
Plugging in values gives you points θ = 0,
r = 2⁄3; θ = π⁄2, r = 1⁄2; θ = π, r = 2⁄5; θ = 3π⁄2, r = 1⁄2.
This gives you the ellipse in Figure 12-12.
, which is the same as
0.5
Figure 12-12:
Graphing
a conic
section
written in
polar form.
25.
0.25
–0.5 –0.25
–0.25
0.25 0.5 0.75
–0.5
Graph the equation of r =
any vertices.
Solve It
1
and label
26.
Identify the conic section whose equation
is r =
Solve It
by stating its eccentricity.
Chapter 12: Conquering Conic Sections
Answers to Problems on Conic Sections
a
Find the center and radius of the circle 2x2 + 2y2 – 4x = 15. Then graph the circle. The center is
(1, 0) and the radius is ≈ 2.92.
Rewrite the equation so the x and y variables are together to get 2x2 – 4x + 2y2 = 15. Factor out
the coefficient: 2(x2 – 2x) + 2y2 = 15. Complete the square: 2(x2 – 2x + 1) + 2y2 = 15 + 2. Factor and
get 2(x – 1)2 + 2y2 = 17. Divide everything by 2 to write the circle in its form: (x – 1)2 + y2 = 8.5.
This means the center is (1, 0) and the radius is
, or about 2.92.
b
Write the equation of the circle with the center (–1, 4) if the circle passes through the point
(3, 1). The answer is (x + 1)2 + (y – 4)2 = 25.
If you’re given the center and a point, you can find the distance between the two points using
the distance formula from Chapter 1:
d=
.
This tells you the radius of the circle. Now that you know both the radius and the center, you
can write the equation: (x + 1)2 + (y – 4)2 = 25.
c
What’s the vertex of the parabola y = –x2 + 4x – 6? Sketch the graph of this parabola. The
answer is (2, –2); see the following graph.
Add the 6 to both sides: y + 6 = –x2 + 4x. Now, factor out the coefficient: y + 6 = –1(x2 – 4x).
Complete the square and balance the equation: y + 6 – 4 = –1(x2 – 4x + 4). Simplify and factor:
y + 2 = –1(x – 2)2. Lastly, subtract 2 from both sides to write the equation in its proper form:
y = –1(x – 2)2 – 2. This means the vertex is located at the point (2, –2).
d
Find the focus and the directrix of the parabola y = 4x2. The focus is (0, 1⁄16) and the directrix is
y = –1⁄16.
There’s no square to complete, so if it helps you to fill in the missing information with zeros,
then rewrite the equation as y = 4(x – 0)2 + 0. This puts the vertex at the origin (0, 0). Because
a = 4, the focus is
units above this point at (0, 1⁄16) and the directrix is the line that runs
units below the vertex, perpendicular to the axis of symmetry at y = –1⁄16.
e
Sketch the graph of x = 2(y – 4)2. See the graph for the answer.
This equation is written in the proper form, unless you’d like to rewrite it as x = 2(y – 4)2 + 0
because the h is missing. This is a horizontal parabola with its vertex at (0, 4). It opens to the
right, with a horizontal transformation of 2.
235
236
Part IV: And the Rest . . .
f
Determine whether this parabola opens left or right: x = –y2 – 7y + 3. The answer is the
parabola opens to the left.
Did we get you on this one? Did you start completing the square? We know you’re used to doing
it by now, but make sure you look at the directions of the problem. This one just asks you
whether the parabola opens to the left or right, and you can tell from the leading coefficient
of –1 (without doing any work at all) that the parabola opens to the left. If you did actually
want to graph this equation, the standard form is y = –1(x + 7⁄2)2 + 61⁄4.
g
Sketch the graph of the ellipse 4x2 + 12y2 – 8x – 24y = 0. See the graph for the answer.
You have to complete the square twice for this one, so maybe we made up for that last question
here. There’s no constant to move, so rewrite the equation and factor out the coefficients:
4(x2 + 2x) + 12(y2 – 2y) = 0. Now complete the square and balance the equation: 4(x2 + 2x + 1) +
12(y2 – 2y + 1) = 0 + 4 + 12. Factor: 4(x + 1)2 + 12(y – 1)2 = 16. Divide everything by 16:
. But don’t start to graph it yet, as each ellipse has a coefficient of 1 on
the numerator, and you have to divide that 3 in the second fraction to get
.
Now that the equation is written in standard form, you can graph it. The center is (1, 1), the
vertices are (3, 1) and (–1, 1), and the co-vertices are (1, 2.2) and (1, –0.2).
C
h
Write the equation of the ellipse with vertices at (–1, 1) and (9, 1) and foci at (4 ±
answer is
, 1). The
.
Knowing the vertices tells you the center, because it’s halfway between them (the midpoint of
the segment connecting them — see Chapter 1 for a refresher). This means the center is at (4, 1)
and that each vertex is 5 units away from the center, so a = 5 and a2 = 25. The foci are ±
237
Chapter 12: Conquering Conic Sections
units away from the center, which tells you that f =
. Now that you know a and f, you can
find b2 using f 2 = a2 – b2. In this case, b2 = 4. Now you can write the equation:
i
Sketch the graph of the ellipse
.
. See the graph for the answer.
How convenient! This equation is written in the proper form, so you don’t have to complete the
square. Did we make up for our sneaky question in number 6, yet? The center is (1, –2), a ≈ 2.83,
and b ≈ 2.45. That gives you the following ellipse:
j
State the ordered pair for the vertices, the co-vertices, and the foci of the ellipse
. The vertices are (–1, 8) and (–1, 0); the co-vertices are (–2, 4) and (0, 4);
and the foci are (–1, 4 ±
).
The sneakiest part of this problem is that the x half of the equation isn’t written as a fraction,
but that’s easy to remedy by writing the denominator of 1:
. This tells you
that the center is (–1, 4), a = 4, and b = 1. The vertices are (–1, 4 ± 4) = (–1, 8) and (–1, 0). The
co-vertices are (–1 ± 1, 4) = (–2, 4) and (0, 4). Lastly, f 2 = a2 – b2, so f = ±
the foci at (–1, 4 ±
k
, which gives you
).
Find the equation of the hyperbola that has its center at (4, 1), its vertex at (7, 1), and one of its
asymptotes is 3y = 2x – 5. The answer is
.
You’re given the center (4, 1) and the equation of the asymptote, which you can rewrite in
slope-intercept form by dividing by 3 to get y =
. Because the vertex is 3 units to the
right, this is a horizontal hyperbola. The slope of the asymptote, , is the value of . If we lost
you there, you have to write the equation of the asymptote in point-slope form because you know
the point is the center (4, 1) and the slope is m = 2⁄3. This gives you the equation
Adding 1 to both sides makes the equation look like the equation of the asymptote for a horizontal hyperbola and helps you identify
and b, you can write the equation:
at the same time. Now that you know the center, a,
.
.
238
Part IV: And the Rest . . .
l
Sketch the graph of the equation (2x – y)(x + 5y) – 9xy = 10. See the graph for the answer.
Does the term –9xy make you a little nervous? Don’t be. Just FOIL out the binomials and get
2x2 + 10xy – xy – 5y2 – 9xy = 10. Notice that all the xy terms cancel to give you 2x2 – 5y2 = 10. You
can divide everything by 10 to write this equation in its form and get
, or ≈ 2.24; and b =
, or ≈ 1.41.
this hyperbola is at (0, 0); a =
m
. The center of
Sketch the graph of the equation x2 + 2x – 4y2 + 32y = 59. See the graph for the answer.
You just knew we were gonna say, “Complete the square,” didn’t you? Okay, let’s get going.
Factor out the coefficients, including the 1 in front of the x2: 1(x2 + 2x) – 4(y2 – 8y) = 59.
Complete the square and balance away: 1(x2 + 2x + 1) – 4(y2 – 8y + 16) = 59 + 1 – 64. Factor and
simplify: 1(x + 1)2 – 4(y – 4)2 = –4. Divide everything by –4:
. Did you notice
how it suddenly became a vertical hyperbola because the y fraction is positive? You have to
rewrite it to put it in its correct form:
n
. That gives you the following figure.
Write the equation of the hyperbola that has its center at (–3, 5), one vertex at (–3, 1), and
passes through the point (1, 5 –
). The answer is
.
This one requires a thinking cap, that’s for sure. We suggest drawing it out on a sheet of graph
paper and marking the center, the vertex, and the point, but that’s it. Doing so shows you that
it’s a vertical hyperbola, so start with the form for any vertical hyperbola:
.
Based on what you’re given, you know that h = –3, v = 5, and the vertex is 4 units below the
center, so a = 4. What do you do with the point the question gives you? Remember that all
points are (x, y), so x = 1 and y = 5 –
. Plug all these values into the equation that you started
Chapter 12: Conquering Conic Sections
with and get
b2. This equation simplifies to
write the equation:
o
, which means you only have one variable to solve for,
. Solving it gets you b2 = 16. This means you can finally
.
Sketch the graph of 3x2 + 4y2 – 6x + 16y – 5 = 0. See the graph for the answer.
This is an ellipse because x and y are both squared but have different coefficients. Add 5 to
both sides: 3x2 + 4y2 – 6x + 16y = 5. Rewrite the equation with the x and the y together: 3x2 – 6x +
4y2 + 16y = 5. Factor the coefficients: 3(x2 – 2x) + 4(y2 + 4y) = 5. Complete the square and balance
the equation: 3(x2 – 2x + 1) + 4(y2 + 4y + 4) = 5 + 3 + 16. Factor the perfect square trinomials and
simplify: 3(x – 1)2 + 4(y + 2)2 = 24. Divide everything by 24:
. That gives you
the following graph.
p
Sketch the graph of 4x2 – 8x – 1 = 4y2 – 4y. See the graph for the answer.
Rewrite the equation first: 4x2 – 8x – 4y2 + 4y = 1. You should recognize that you have a hyperbola on your hands because there are an x2 and a y2 where exactly one has a negative coefficient. Factor out the coefficients: 4(x2 – 2x) – 4(y2 – y) = 1. Complete the square and balance the
equation: 4(x2 – 2x + 1) – 4(y2 – y +1⁄4) = 1 + 4 – 1. Factor the perfect squares: 4(x – 1)2 – 4(y – 1⁄2)2 =
4. Divide everything by 4:
. In this particular hyperbola, the values of a and
b are both 1. Knowing this and the center gives you the following graph.
q
Sketch the graph of 4(x – 2) = 2y2 + 6y. See the graph for the answer.
Notice right away that the equation doesn’t have an x2, so this is a horizontal parabola. Go
ahead and distribute the 4 first: 4x – 8 = 2y2 + 6y. Now, factor the coefficient on the y2 variable:
4x – 8 = 2(y2 + 3y). Completing the square for this one gets you fractions; half of 3 is 3⁄2 and that
value squared is 9⁄4. Add this inside the parentheses, and don’t forget to add 2 · 9⁄4 to the other
side to keep the equation balanced: 4x – 8 + 9⁄2 = 2(y2 + 3y + 9⁄4). Simplify and factor: 4x – 7⁄2 =
239
240
Part IV: And the Rest . . .
2(y + 3⁄2)2. Begin to solve for x by adding over the 7⁄2: 4x = 2(y + 3⁄2)2. Now divide each term by 4:
x=
r
. This gives you a horizontal parabola with vertex at (7⁄8, –3⁄2).
Sketch the graph of 2y2 – 4x2 + 8x – 8 = 0. See the graph for the answer.
You have a hyperbola to graph this time. We’ll skip the narrative on how to complete the
square and show the steps only (we’re confident you’re a pro at it by now): 2y2 – 4x2 + 8x = 8;
2y2 – 4(x2 – 2x) = 8; 2y2 – 4(x2 – 2x + 1) = 8 – 4; 2y2 – 4(x – 1)2 = 4;
. This gives you
the following hyperbola.
s
Sketch the graph of the equation 4x2 + 4y2 – 8y + 16x – 4 = 0. See the graph for the answer.
You should recognize that this is a circle because of the x2 and y2 with equal coefficients on
both. Here are the steps to completing the square: 4x2 + 4y2 – 8y + 16x = 4; 4x2 + 16x + 4y2 – 8y = 4;
4(x2 + 4x) + 4(y2 – 2y) = 4; 4(x2 + 4x + 4) + 4(y2 – 2y + 1) = 4 + 16 + 4; 4(x + 2)2 + 4(y – 1)2 = 24.
Because this is a circle, you need to get coefficients of 1 in front of both sets of parentheses by
.
dividing by 4: (x + 2)2 + (y – 1)2 = 6. This circle has its center at (–2, 1) and its radius is
t
Sketch the graph of 3x2 – 4y2 + 3x – 2y – 25⁄2 = 0. See the graph for the answer.
This is another hyperbola because the coefficient on y2 is negative while the coefficient on
x2 is positive. Here are the usual steps: 3x2 + 3x – 4y2 – 2y = 25⁄2; 3(x2 + x) – 4(y2 + 1⁄2y) = 25⁄2;
3(x2 + x + 1⁄4) – 4(y2 + 1⁄2y + 1⁄16) = 25⁄2 + 3⁄4 – 1⁄4; 3(x + 1⁄2)2 – 4(y + 1⁄4)2 = 12;
.
Chapter 12: Conquering Conic Sections
This gives the following hyperbola.
u
Sketch the graph of the parametric equations x = t2 – 1, y = 2t, and –2 ≤ t ≤ 3. See the graph for
the answer.
Set up a table of t, x, and y where you pick the t and find x and y. Be sure to stay within the interval defined by the problem. Here’s our chart:
t
x
y
–2
3
–4
–1
0
–2
0
–1
0
1
0
2
2
3
4
3
8
6
These (x, y) points give you the following graph:
v
Sketch the graph of the parametric equations x =
the answer.
Another table comes in handy here:
t
x
y
1
undefined
undefined
2
1
1
3
1
4
1
5
⁄2
⁄3
⁄2
1
⁄4
1
,y=
, and t > 1. See the graph for
241
242
Part IV: And the Rest . . .
These points give you the following graph:
w
Eliminate the parameter and find an equation in x and y whose graph contains the curve of the
parametric equations x = t2, y = 1 – t, and t ≥ 0. The answer is y = 1 ±
.
Solve for t in the first equation by taking the square root of both sides: ±
= t. Substitute this
value into the other equation: y = 1 ±
.
Some teachers may require that you don’t leave the square root in this equation and instead
write it in a form that looks more like a conic section. Subtract 1 from both sides: y – 1 = ±
.
2
Then square both sides to get (y – 1) = x. That looks more like a parabola now, doesn’t it?
x
Eliminate the parameter of the parametric equations x = t – 5 and y =
y=
.
. The answer is
The first equation is easy to solve for t by adding 5 to both sides: x + 5 = t. Substitute this value
into the other equation and get y =
.
If you’re required by your teacher to write this equation without the square root, square both
sides to get y2 = x + 5. This, too, looks like a parabola.
y
Graph the equation of r =
and label any vertices. See the graph for the answer.
Notice that the given equation is the same thing as
e = 1. This makes it a parabola.
A
Identify the conic section whose equation is r =
is this conic is a hyperbola because e = 4⁄3.
You have to factor the 3 out of the denominator first:
thing as
, which tells you that k = 8 and
by stating its eccentricity. The answer
. This is the same
, and that tells you that k = 3 and e = 4⁄3, which is why this one is a hyperbola.
Chapter 13
Finding Solutions for Systems
of Equations
In This Chapter
䊳 Using the substitution and elimination methods to solve linear equations
䊳 Solving larger systems of equations
䊳 Graphing inequalities
䊳 Discovering decomposing partial fractions
䊳 Mastering matrices to solve systems of equations
N
o, a system of equations is not how to organize, arrange, or classify them. A system of
equations is a collection of linear equations involving the same set of variables. The
point is to find one solution, if there is one, that works in all the equations. Solving one
equation for one variable is almost always possible and usually pretty easy to do. Two variables require at least two equations, and three variables require at least three. You see, you
need a unique equation for every variable present if you have to solve the system.
It goes without saying that the bigger the system of equations becomes, the longer and
harder it may be to solve. Solving a system involves several techniques, and sometimes it
may be easier to solve certain systems certain ways. That’s why math textbooks show all the
techniques — so you know when to use each technique.
Of course, you can choose to always solve all systems using one technique, but another technique may require fewer steps, which will save you time, not to mention money (money for
aspirin, that is, for all the headaches you’d get if you solved all equations one specific way!).
A Quick-and-Dirty Technique Overview
Here’s a handy guide to all the techniques we cover in this chapter and when it’s best to
use them:
⻬ If a system has two or three variables, you can use substitution or elimination to solve.
⻬ If a system has four or more variables, you should use matrices, in which you have the
following choices:
• The Gaussian method
• Inverse matrices
• Cramer’s Rule
244
Part IV: And the Rest . . .
We discuss all these techniques in detail in this chapter. And if that’s not enough information for you, we also discuss systems of inequalities, which can only be solved by
graphing.
Solving Two Linear Equations
with Two Variables
When you’re presented with a system of linear equations in two variables, the best
methods to solve them are known as substitution and elimination. As we mention earlier in this chapter, you can use either method to solve any system of this type, but
textbooks usually show you both methods because each one has its unique advantages. In keeping with this spirit, we show you both methods, but we also explain when
to use both as well.
Just remember that with each system of equations in two variables, you have to find
the solution to both variables, usually x and y. Don’t stop until you have both, or else
you may only get half the credit on your next test for doing half the work.
Also remember that sometimes, systems of equations don’t have a solution.
Remember when you used to graph two straight lines on one graph to determine the
point of intersection? Well, because that’s the least accurate of all the methods to
solve, mathematicians came up with the other methods that we talk about in this
chapter. However, if you recall, sometimes the two lines were parallel to each other —
without an intersection! This meant that there was no solution. The fact that there may
be no solution may pop up from time to time using these other methods as well.
So how will you recognize a system of equations with no solution without using a
graph? That’s easy — you end up with an equation that just doesn’t make sense. It may
say 2 = 7 or –1 = 10, but you’ll know right away that it has no solution. It’s also possible
that you’re given the same line (in disguise) twice. If you were to graph that system,
you’d end up with one line on top of another.
These two lines share infinite points, so you say that the system has infinite solutions.
These equations boil down at some point to an identity — the left and right sides of the
equation are exactly the same (such as 2 = 2, or 10x = 10x, or 4y – 3 = 4y – 3), and these,
too, are easy to recognize.
The substitution method
In the substitution method, you solve one equation for one variable and then substitute
this expression for that variable in the other equation. If one of the two equations
you’re given has already been solved for one variable, huge bells and whistles should
go off inside your head. You know you’ve got a winner for the substitution method. Of
course, if one equation can be easily solved for one variable (one variable has a coefficient of 1), you also know that substitution is a good bet.
Chapter 13: Finding Solutions for Systems of Equations
Q.
that you know half of your answer (y in this
case), you can substitute that value into
one of the original equations to get the
other half (x).
Solve the system of equations:
.
A.
x = 3, y = 1. Notice how the first equation
says “x = . . . ”? This tells you to use substitution. You can take this expression and
substitute it into the other equation where
it says x. This gets you 2(4y – 1) + 5y = 11.
The substitution method makes your job
easier because you end up with one equation in one variable — and this one is easy
to solve! When you do, you get y = 1. Now
1.
Use substitution to solve the system
245
Save yourself some time and steps
by substituting the first answer you
get into the equation that has
already been solved for a variable.
For this example, because you know
that x = 4y – 1 and you figure out that
y = 1, it takes very few steps to figure
out that x = 3.
2.
The sum of two numbers is 14 and their difference is 2. Find the numbers.
Solve It
Solve It
The elimination method
Elimination is the method of choice when both of the linear equations given to you are
written in standard form:
Ax + By = C
Dx + Ey = F
where A, B, C, D, E, and F are all real numbers. It’s called standard form because it’s
supposed to be the standard way that textbooks depict linear equations. But the truth
is that textbook authors like to keep you on your toes so they write linear equations in
all kinds of forms.
In the two equations, if the x and y terms are opposite of each other, then you should
choose elimination. In the elimination method, you add the two equations together so that
one of the variables disappears (is eliminated). Sometimes, however, you must multiply
one or both equations by a constant in order for the terms to have opposite signs. This
way, when you add the two equations together, one of the variables will be eliminated.
246
Part IV: And the Rest . . .
Q.
Solve the system
A.
x = 2, y = –2. Notice that you could solve
this system using substitution, because in
the second equation, you can solve for y by
subtracting 3x from both sides. But notice
that the y terms are exact opposites of
each other. If you add the two equations
you get 5x = 10. This means that you can
divide both sides by 5 to easily solve for x
and get x = 2. Substituting this back into
either equation, you find that y = –2.
.
Q.
Solve the system
A.
x = 1, y = –1. To solve this system by substitution is possible, but it would eventually mean dividing one of the coefficients
and creating those ugly fractions we all
hate. Instead, you can avoid the fractions
by using the elimination method. The fact
that both equations are written in standard
form is another vote in favor of the elimination method. Notice that the y terms have
opposite signs, so you can eliminate them
(you can eliminate any variable you
choose, but it’s all about the least amount
of steps). It’s a little like you’re finding the
least common multiple of both coefficients,
in this case the 3 and the 5. The smallest
number that both of those go into is 15, so
you have to multiply the top equation by 5
and the bottom equation by 3. This gives
you
.
. Adding these two
equations together gives you 22x = 22,
which gives you the solution x = 1. You
then have to substitute this value back into
one of the two original equations to solve
for y. In this example, y = –1.
3.
4.
Solve the system
Solve It
Solve the system
.
Solve It
.
247
Chapter 13: Finding Solutions for Systems of Equations
Not-So-Straight: Solving Nonlinear Systems
The substitution and elimination methods are common tools for systems of equations
that include nonlinear equations. Yes, now at least one of your two given equations will
often be a quadratic equation (it could also be a rational function or some other type).
The method you choose to use for these types of systems depends on the types of
equations that you’re given. We break the following sections into those types and
show you how to best solve each one.
One linear equation and one not
When one equation is linear and the other equation isn’t linear, it’s best to use the substitution method. That’s because the linear equation can be easily solved for one variable. You can then substitute this value into the other equation to solve. Most often,
that means solving a quadratic equation at some point, so if you need to brush up on
those techniques, see Chapter 4.
Q.
Solve the system of equations
A.
x = –3, y = –9; x = 1, y = –1. As we mention
earlier, it’s usually easier to solve the linear
equation first. The second given equation
is the linear one, and it’s easier to solve for
y (no pesky coefficients to divide). Doing
so gets you y = 2x – 3. After you substitute
this expression into the first equation for y,
5.
Solve
.
you get x2 + 2x – 3 = 0. This quadratic polynomial factors to (x + 3)(x – 1) = 0. Then,
using the zero product property (for more
information, see Chapter 4), you get two
solutions: x = –3 or x = 1. Uh-oh. Now what?
No stopping you! Two solutions for x
means twice the substitution and twice the
y answers. If x = –3, then y = –9, and if x = 1,
then y = –1.
.
6.
Solve the system of equations
Solve It
Solve It
.
248
Part IV: And the Rest . . .
Two equations that are nonlinear
In this section, we explore what happens when both of the given equations are nonlinear. These steps will also, most likely, create one final quadratic equation to solve.
Solving these systems requires the elimination method because one of the quadratic
terms must cancel to solve for the other variable. This is probably easiest to see with
an example, so read on.
Q.
Solve the system
A.
(x, y) = (0, –5), (± 3, 4). Notice right away
that the x2 terms in both equations have
the same coefficient. If you multiply the
second equation by –1 and then add the
two equations together, you get y2 + y = 20.
Subtracting the 20 from both sides gets you
7.
Solve
Solve It
the quadratic equation, y2 + y – 20 = 0, that
you have to factor: (y + 5)(y – 4) = 0. Solve
and get y = –5 and y = 4. Substituting y = –5
into the second equation gets you x2 + 5 =
5, or x2 = 0, which means that x = 0.
Substituting y = 4 into the same equation
gets you x2 – 4 = 5 or x2 = 9, which gives you
x = ± 3. Both of these solutions work.
8.
Solve
Solve It
Systems of equations disguised
as rational equations
Sometimes you’ll see two equations that look like they’re rational equations. That
means that the variable is in the denominator of the equation. As we discuss in
Chapter 3, sometimes rational functions have undefined values. Keep this in mind with
the final solutions you find to the equation — they may not really work! Always check
the solutions to these types of equations because you never know which ones are
actually solutions and which ones aren’t until you double-check. We recommend that
you always start off by substituting the rational expressions into the other equation so
you can deal with a more normal-looking system (as if there is such a thing).
Chapter 13: Finding Solutions for Systems of Equations
and v =
Q.
, you can conveniently rewrite the
Solve the system
entire system as
A.
Solve
Solve It
. You can then
use any of the methods you’re already
familiar with to solve the system. For this
example, if you multiply the second equation by 2, you get 2u – 2v = –22. You can
then add the two equations to eliminate v
and get 5u = –11. This means that u = –11⁄5.
Then you can work your way backwards to
get x = –5⁄11. We’d then suggest substituting
your u value to find your v, which gets you
on your way to finding that y = 5⁄44.
x = –5⁄11, y = 5⁄44. Okay, so you see all those
fractions right off the bat and you decide to
throw in the towel and walk away, right?
Wrong! Make the fractions go away by
starting off with a substitution. Notice, first
of all, that you can rewrite the given
system as
9.
249
. By letting u =
10.
Solve
Solve It
Solving More than Two (Equations
and/or Variables)
Now that you have a grip on solving a system of equations in two variables, we’ll
change gears to systems with three variables. Remember that for each variable you
need at least one unique equation. In other words, two variables need at least two
equations, three variables need at least three equations, four variables need four, and
so on. Why change to three variables, though? Well, we live in a three-dimensional
world, so we need three variables to represent it.
Most of the time when you’re given systems larger than
, you want to use elimination. You have to take two equations at a time and eliminate one variable. Then, you
have to take another two equations and eliminate the same variable. If you start with a
system, this will knock you down to a
system, which you then must solve. If
you start with a
system, you work down to a
system, which you must then
work down to
and solve. Sounds fun, doesn’t it?
250
Part IV: And the Rest . . .
Q.
Solve the system of equations
.
A.
x = 1, y = –2, z = 3. To make things easier
in this section in general, we’ll always label
the given equations from top to bottom
with capital letters. For this example, the
top equation x – y + z = 6 is equation A, the
middle equation is equation B and the
bottom equation is equation C. Notice that
equation C is already missing the z variable. Also notice that equations A and B
have z variables that are exact opposites of
each other.
If you add those two together (A + B),
you’d get a brand new equation (we’ll call
it D) which is 2x – y = 4.
If you then take equation D and add C to it
(D + C), you’d get one more equation, E,
which is 3x = 3, which is solved easily to
get x = 1. You can then substitute this value
into equation C and get 1 + y = –1, which
means that y = –2. You can also use the fact
that x = 1 in equation B to get 1 – z = –2, or
–z = –3, or finally z = 3.
Q.
Solve the system
A.
x = 2, y = –1, z = –3, and w = 1. Most of our
students panic with systems that are bigger
than
, so we decided to show one example
of a
so you can see how it works. A: x + y
+ z + w = –1; B: 2x + y + z = 0; C: 2y + z – w = –6;
D: x – z + 2w = 7. Notice that B is missing its w
variable, so if you can use the others to eliminate w as well, that’ll be the first success.
.
A + C gives you a new equation, E: x + 3y +
2z = –7.
–2A + D gives a new equation, F: –x –2y – 3z = 9.
So now you have three equations with three
variables: B, D, and F. Next, you have to pick
another variable to eliminate — it doesn’t
matter which one; we eliminate x.
E + F gives you equation G: y – z = 2.
Meanwhile, B + 2F also eliminates x with equation H: –3y – 5z = 18.
Finally, 3G + H eliminates y with one last equation, J: –8z = 24; z = –3.
Using this solution in equation H gives you –3y
– 5(–3) = 18; –3y + 15 = 18; –3y = 3; y = –1. Now
that you know z and y, use them in equation E:
x + 3(–1) + 2(–3) = –7; x – 3 – 6 = –7; x = 2. Then
use all those answers in equation A: 2 – 1 – 3 +
w = –1; w = 1.
11.
Solve the system
Solve It
12.
Solve
Solve It
Chapter 13: Finding Solutions for Systems of Equations
13.
14.
Solve
Solve It
Solve
Solve It
Graphing Systems of Inequalities
A system of inequalities is more than one inequality in more than one variable. Up until
now in your math career, you’ve probably seen systems of inequalities that have all
been linear. In pre-calc, you continue with those types of problems but then move up
to nonlinear systems of inequalities. That’s right, you’ll be seeing quadratics and
conics, too. For a review of how to graph one inequality, see Chapter 1. If you’re
quacky on quadratics, see Chapter 3. And if conics sound crazy, see Chapter 12. If, on
the other hand, you’re raring to go, read on.
The only way to solve a system of inequalities is to graph it. You end up with (hopefully) two overlapping shaded regions — the overlap is the solution. Every single point
in the overlap is a solution to the system. What happens if there’s no overlap? Well,
there’s no solution!
When you multiply or divide an inequality by a negative number, the inequality sign
changes: < becomes >, ≤ becomes ≥, and vice versa. This is a pretty important fact to
remember because it affects your shading in the end.
Q.
Sketch the graph of the inequality
them. The top equation in slope-intercept
form is y ≤ –3x + 5, and the bottom
equation becomes
A.
See Figure 13-1. Because both of these
inequalities are linear, you have to put
them in slope-intercept form to graph
. Graphing
both on the same coordinate plane gives
you Figure 13-1.
251
252
Part IV: And the Rest . . .
y < –1x + 2
2
y < –3x + 5
Figure 13-1:
Graphing a
system of
linear
inequalities.
Q.
A.
Sketch the graph of
Solve It
. For these types of
problems, we recommend that you pick
test points to see where to shade. For
example, the origin (0, 0) is a great point to
try in the original equations to see whether
it works. Is 02 + 02 < 16? Yes, so you shade
inside the circle. In the second equation, is
0 – 2(0) > –4? Yes, so you also shade below
the line. See the graph in Figure 13-2.
See Figure 13-2. This time, the top equation is a circle and the bottom equation is a
line. The circle is in the proper form to
graph, so you don’t have to do any work
there (other than to graph, that is), while
the bottom equation in slope-intercept
Figure 13-2:
Another
graph of a
system of
inequalities;
this time,
one equation isn’t
linear.
15.
form is
Sketch the graph of the inequality
x 2 + y 2 < 16
y < 1x + 2
2
.
16.
Sketch the graph of
Solve It
Chapter 13: Finding Solutions for Systems of Equations
17.
Sketch the graph of
.
18.
Sketch the graph of
.
Solve It
Solve It
We’re Partial to . . . Decomposing
Partial Fractions!
Call us crazy, but we’re partial to partial fractions! (We know, but we really couldn’t
resist.) The process known as decomposing partial fractions takes one fraction and
breaks it down as the sum or difference of two (or more) other fractions. This requires
being an expert at factoring, so if you need a review, turn to Chapter 4 and read up on
how to do it. If you’re a pro by now, then you know to always follow these general steps:
1. Factor the denominator.
2. Write separate fractions, one for each factor of the denominator based on
these rules:
a. If the factor is linear, it has some constant in the numerator.
b. If the factor is quadratic, it has a linear expression in the numerator.
Note: If any factor has a power on it, you have to create one fraction for each power,
from 1 on up to the highest degree. This is probably best shown with an example.
Suppose that you’re able to factor the polynomial in a particularly long denominator into
(x – 4)(x + 1)3(2x – 1)(3x2 – 6x + 2)2. You would have to create the sum of seven different
fractions:
.
The first denominator’s factor is linear, so its numerator is a constant. The second
denominator’s factor is linear with a degree of 3, so you need to create three different
constant numerators: one for the first degree, the second for the second degree, and
the third for the third degree. The third factor is also linear, so it got one fraction with
a constant on the top. Lastly, the final term is quadratic and second degree, so it got
two linear terms in the numerator: one for the first degree and the second for the
second degree.
253
254
Part IV: And the Rest . . .
Q.
Find the partial fraction decomposition of
A.
.
. Begin by factoring the denominator of the given fraction into
.
Because both factors in the denominator are linear, you break them into two different
fractions with constant numerators:
. Now, multiply every
fraction by the factored denominator, which results in some big-time cancelling:
. You now have the
simplified equation 7x + 5 = A(x + 2) + B(x – 1). When you multiply this out, you get the equation
7x + 5 = Ax + 2A + Bx – B. Now, gather like terms: 7x + 5 = Ax + Bx + 2A – B. Factor the x out on
the right side: 7x + 5 = (A + B)x + 2A – B. Notice how both sides match up, which means that
7x = (A + B)x, or 7 = A + B and 5 = 2A – B. This gives you a system of equations to solve:
.
Now you know why the textbooks usually include this material in the chapter with systems of
equations (as well as why we include it here — aren’t we clever?). Add these two equations to
eliminate B and get 12 = 3A, or 4 = A. Substituting this into the top equation gets you 7 = 4 + B,
or 3 = B. You can now use these values to write the sum of two fractions:
19.
Find the constants A and B:
20.
.
Solve It
21.
Find the form of the partial fraction
decomposition for
the constants.
, but don’t find
Solve It
Find the partial fraction decomposition for
22.
Find the partial fraction decomposition for
.
Solve It
.
.
Solve It
Chapter 13: Finding Solutions for Systems of Equations
There Is No Spoon: Working with a Matrix
A matrix is a collection of numbers arranged in rows and columns. Each number inside
the matrix is called an element. A matrix comes in handy when you have a bunch of
data that you need to keep track of. Usually, a matrix is named by its dimensions, or
how big it is. This is also sometimes known as the order of the matrix and is always the
number of rows by the number of columns. For example, if matrix M is
, it has four
rows and three columns. After the data is organized in this fashion, you can add, subtract, and even multiply matrices.
There’s no such thing as matrix division, so don’t worry about it. There’s also an operation known as scalar multiplication, which means you multiply the entire matrix by a
constant.
To add or subtract matrices, you have to operate on their corresponding elements. In
other words, you add or subtract the first row/first column in one matrix to or from the
exact same element in another matrix. The two matrices must have the same dimensions;
otherwise, an element in one matrix won’t have a corresponding element in the other.
Figure 13-3 shows two matrices and what their sum and differences are. Figure 13-4
shows the scalar multiplication 3A.
–5
6
2
1
0
6
–3
2
1
2
B = –8
–2
4
10
–3
5
3
–9
Figure 13-3:
Addition and
–3
subtraction
of matrices. A + B = –2
0
5
10
3
2
5
–8
–7 –3
A − B = 14 –10
4
9
–8
–1
10
A=
Figure 13-4:
–5
Multiplying
3A = 3 6
matrix A
2
by 3.
1
0
6
–3
–15
2 = 18
1
6
3
0
18
–9
6
3
Multiplying matrices is another can of worms. First of all, to multiply two matrices AB
(the matrices are written right next to each other, with no symbol in between), the
number of columns in matrix A must match the number of rows in matrix B. If matrix A
is
and matrix B is
, the product AB has dimensions
. And remember,
when it comes to matrix multiplication, AB doesn’t equal BA; in fact, just because AB
exists doesn’t even mean that BA does as well.
For all problems in this section, M =
P=
.
,N=
, and
255
256
Part IV: And the Rest . . .
Q.
Q.
Find 3M – 2N.
A.
Find MP.
A.
.
First, substitute in each given equation:
You need to multiply each element of each
row of the left matrix by each element of
each column of the right matrix.
.
The sum of the first row times the first
column: –5(–1) – 1(4) + 3(2) + 6(–5) = –23.
This is the first row, first column answer.
Follow the order of operations and
multiply in the scalars:
.
Then subtract the two matrices, watching
the negative signs:
.
The sum of the first row times the second
column: –5(2) – 1(4) + 3(3) + 6(2) = 7. This
is the first row, second column answer.
The sum of the first row times the third
column: –5(–1) – 1(0) + 3(1) + 6(–1) = 2.
This is the first row, third column answer.
The sum of the second row times the
first column: 0(–1) + 2(4) – 2(2) + 6(–5) =
–26. This is the second row, first column
answer.
The sum of the second row times the
second column: 0(2) + 2(4) – 2(3) + 6(2) =
14. This is the second row, second
column answer.
The sum of the second row times the
third column: 0(–1) + 2(0) – 2(1) + 6(–1)
= –8. This is the second row, third
column answer.
Putting these all into a matrix gives you
the answer
.
Chapter 13: Finding Solutions for Systems of Equations
23.
24.
Find 4N.
Find 4N + 5M.
Solve It
Solve It
25.
26.
Find 3M – P.
Solve It
Find NP.
Solve It
Getting It in the Right Form:
Simplifying Matrices
You can write any system of equations in matrix form. To do so, follow these steps:
1. Write all the coefficients in one matrix, called the coefficient matrix. Each
equation gets its own row in the matrix, and each variable gets its own column,
written in the same order as the equations.
2. Multiply this times another column matrix with all the variables in it, called
the variable matrix, in order from top to bottom.
3. Set this product equal to a column matrix with the answers in it, sometimes
called the answer matrix.
257
258
Part IV: And the Rest . . .
Row echelon form is similar to matrix form; however, you only work with the coefficient matrix. Across any row, the first number element (besides 0) that you run into
is called the leading coefficient. For a coefficient matrix to be in row echelon form,
⻬ Any row with all 0s in it must be the bottom row.
⻬ The leading coefficient in any row must be to the right of the leading coefficient in the row above it.
Reduced row echelon form takes row echelon form and makes all the leading coefficients the number 1. Also, each element above or below a leading coefficient must
be 0. Figure 13-5 shows a matrix (a) in reduced row echelon form and a matrix (b)
not in reduced row echelon form.
Figure 13-5:
A matrix (a)
in reduced
row echelon
form and
(b) not in
reduced
row echelon a.
form.
0
1
0
0
0
1
2
7
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
b.
Finally, augmented form takes the coefficient matrix and tacks on an extra column —
a column with the answers in it so that you can look at the entire system in one convenient package.
These ways of writing systems of equations in matrices come in handy when dealing with systems that are
or larger. Your goal is to get the matrix into row echelon form using elementary row operations. These operations are different from the
operations we discuss in the previous section because they’re done on only one
row at a time. Here are three row operations you can perform:
⻬ Multiply each element of a row by a constant
⻬ Interchange any two rows
⻬ Add two rows together
We stay consistent with Pre-Calculus For Dummies and use the same notation that we
use there to represent these elementary row operations. This means that 4r2 → r2
multiplies the second row by 4 to change the second row.
swaps row one with
row three. r3 + r1 → r1 adds row three to row one and changes row one. 4r2 + r1 → r1
first multiplies row two by 4 and then adds that to row one to change row one.
You can use any combination of these row operations to get the given matrix into
row echelon form. Use reduced row echelon form only if you’re specifically told to
do so by your teacher or textbook, as it takes more steps.
We only focus on the forms and the row operations in this section. To really dig in
deep and discover how to get a matrix in row echelon form, read on to the next
section.
Chapter 13: Finding Solutions for Systems of Equations
Q.
Write the system of equations
as a matrix system.
A.
. The matrix on the left is
Q.
Using the augmented matrix from the last
example, use elementary row operations
to find –3r2 → r2.
Write the system from the previous question as an augmented matrix.
A.
. Just take the coefficient matrix
and add on the answer matrix. Voilà!
You’ve got yourself an augmented matrix.
Notice that the vertical line separates
the two and lets you know that this is an
augmented matrix and not a normal
matrix.
the coefficient matrix, containing all the
coefficients from the system. The second
matrix is the variable matrix, and the third
one, on the right, is the answer matrix.
This completes the job for this question.
27.
28.
Now, using your answer from 27, find
Solve It
Solve It
29.
Now, keep going and find r1 + r2 → r2.
Solve It
259
30.
Lastly, find 3r2 + r1 → r1.
Solve It
Solving Systems of Equations Using Matrices
You can solve a system of equations using matrices in three ways. Putting a matrix in
row echelon form using the techniques described in the last section is called Gaussian
elimination. The second way uses a method called inverse matrices, and the third
method is called Cramer’s Rule. Your book may cover all these techniques, or it may
cover only one, or it may not even cover this material at all. But we’re here for you if
you need us! We dedicate one section to each of these ways, to keep it nice and simple.
.
260
Part IV: And the Rest . . .
Gaussian elimination
The process of putting a matrix in row echelon form is called Gaussian elimination. We
focus in this section on matrices in augmented form because that’s most commonly
what you’ll be asked to do, but know that the rules don’t change if you’re asked to do
this with some other form of matrix equation. The goals of using the elementary row
operations are simple: get a 1 in the upper-left corner of the matrix, get 0s in all positions underneath this 1, get 1s for all leading coefficients diagonally from the upper-left
to the lower-right corners, and then get 0s below each of them. When you get to that
point, you use a process called back substitution to solve for all the variables in the
system.
Q.
1 for this reason. Follow the elementary
Put the system of equations
row operation
in augmented form and then write the
equation in row echelon form.
A.
to get
.
Now, to make the first element of row two
0, you need to add –2. So take –2r1 + r2 → r2
, x = 21⁄11, y = –3⁄11. You wrote
to get
this system as an augmented matrix
. This gives you an
equation in the second row that’s easy
to solve for y:
in the last section. Now you
, or y = –3⁄11. Now
you can work backwards using back
need to get it into row echelon form. First,
you need to get a 0 below the element in
the upper-left corner. This is easiest if you
get a 1 in the upper-left corner first. In
fact, some textbooks may say that row
echelon form has leading coefficients of
31.
Solve the system of equations
substitution in the top equation:
You know the value of y so substitute that
32.
Use Gaussian elimination to solve
.
by writing it in augmented form and then
putting the matrix in row echelon form.
Solve It
, or x = 21⁄11.
in and get
Solve It
.
Chapter 13: Finding Solutions for Systems of Equations
Inverse matrices
Another way to solve a system is by using an inverse matrix. This process is based on
the idea that if you write a system in matrix form, you’ll have the coefficient matrix
multiplying the variable matrix on the left side — if only you could divide a matrix
you’d have it made in the shade! Well, if you look at the simple equation 3x = 12, you
can solve it by dividing both sides by 3,
, to get x = 4. This is the same thing as
multiplying by its multiplicative inverse on both sides, which turns out to be the same
in matrices! You have to use an inverse matrix. Remember from Chapter 3 that if f(x) is
a function, its inverse is denoted by f –1(x). This is true for matrices as well: If A is the
matrix, A–1 is its inverse. If you have three matrices (A, B, and C) and you know that
AB = C, then you can solve for B by multiplying the inverse matrix A–1 on both sides:
A–1[AB] = A–1C, which simplifies to B = A–1C.
Finding a matrix’s inverse
But how do you find a matrix’s inverse? Realize first that only square matrices have
inverses. The number of rows must be equal to the number of columns. Even then, not
every square matrix has an inverse. If the determinant (which we talk more about in
the next section) of a matrix is 0, it doesn’t have an inverse. The definition of a determinant involves a lot of math mumbo jumbo which, in our humble opinion, won’t help
you much with finding the answer. We’d rather cut to the chase and simply show you
how to find the determinant.
When a matrix does have an inverse, you can use several ways to find it depending on
how big the matrix is. If it’s a
, you can find it by hand using a simple formula. If it’s
or bigger, you can find it by hand, but that doesn’t mean that you should. Most
textbooks give you the inverse matrix for these bigger matrices. You can also use a
graphing calculator or the Internet to find an inverse matrix.
In the meantime, if matrix A is the
matrix
, its inverse is found using:
Using an inverse matrix to solve a system
Now that you can find the inverse matrix, all you have to do to solve is:
1. Write the system as a matrix equation.
2. Create the inverse matrix.
3. Multiply this inverse in front of both sides of the equation.
4. Cancel on the left side; multiply the matrices on the right.
5. Multiply the scalar.
261
262
Part IV: And the Rest . . .
Q.
To multiply a matrix by its inverse
cancels everything on the left
except for the variable matrix,
Set up the matrix equation for the system
by using inverse matrices.
A.
.
x = 1, y = 2. First set up the matrix
equation
That means all you have to do is multiply
the matrices on the right and then multiply
. Now,
find the inverse matrix using the
formula from earlier in this chapter:
the scalar:
. This gives
the solutions from top to bottom as x = 1
and y = 2.
.
Now, multiply this inverse on the
left of both sides of the equation:
.
33.
34.
Solve the system
using inverse matrices.
Solve the system
using inverse matrices.
Solve It
Solve It
Cramer’s Rule
Cramer’s Rule is a method based on determinants of matrices that’s used to solve
systems of equations. The determinant of a
determinant of a
textbooks. If A =
matrix
is ad – bc. The
matrix is found using a process called diagonals in some
, then first rewrite the first two columns immediately
following the third. Draw three diagonal lines from the upper left to the lower right and
three diagonal lines from the lower left to the upper right, as shown in Figure 13-6.
Chapter 13: Finding Solutions for Systems of Equations
Figure 13-6:
How to find
|A|=
a matrix’s
determinant.
a1
a2
a3
b1
b2
b3
c1
c2
c3
a1
a2
a3
263
b1
b2
b3
Then multiply down the three diagonals from left to right and up the other three. Find
the sum of the numbers of the products on the top and the sum of the products on the
bottom. Finally, find the difference of the top and bottom. This is the same thing as:
For a
x=
Q.
system
:
y=
Solve the system of equations
. Do the same thing
using Cramer’s Rule.
A.
for y =
x = –4, y = 3. Using Cramer’s Rule by substituting the coefficients and the constants,
x=
=
. This is
why if you’re looking for a pain-free way of
solving a system, we recommend Cramer’s
Rule. See how easy that was?
. Find the determinants in
the numerator and the denominator:
35.
36.
Find the determinant of
Use Cramer’s Rule to solve
.
.
Solve It
Solve It
264
Part IV: And the Rest . . .
Answers to Problems on
Systems of Equations
a
Use substitution to solve the system
. The answer is r = 7 and s = –1.
Substituting the fact that s = 13 – 2r, change the first equation to, say, r + 13 – 2r = 6. This simplifies to –r + 13 = 6, or –r = –7, which means that r = 7. Now that you know this value, you can substitute it into the second equation: s = 13 – 2(7) = 13 – 14 = –1. The final answer: r = 7, s = –1.
b
The sum of two numbers is 14 and their difference is 2. Find the numbers. The answer is x = 8
and y = 6.
First, you need to change the given words into a system of equations using variables. The sum
of two numbers being 14 becomes x + y = 14, and their difference of 2 becomes x – y = 2. The
first equation has an x variable with a coefficient of 1, so you can solve for it easily by subtracting y from both sides: x = 14 – y. Now, substitute this expression for x in the other equation:
14 – y – y = 2. Combine like terms: 14 – 2y = 2. Solve for y: –2y = –12; y = 6. Now that you’ve got
that on lockdown, substitute it into the other equation to solve for x: x = 14 – 6; x = 8.
c
Solve the system
. The answer is (x, y) =
.
We didn’t do it on purpose, but sometimes the answers to these questions just aren’t pretty, so
don’t expect ‘em to be. Even though you know that the answer is going to fractionville, we still
recommend that the first thing you do in any equation of this type is to get rid of the fractions
by multiplying every term by the LCD. The LCD for both equations in this problem turns out to
be 6, so get multiplying:
=
. Now that this looks more like all
the other systems you’ve been dealing with, which variable would you like to eliminate? y?
Excellent choice. Multiply the top equation by 3 and the bottom by 2:
these two equations eliminates y: 17x = –78, which means that x =
to solve for y:
d
Solve the system
or
. Adding
. Substitute this value in
.
. The answer is x = k, y =
.
You’ve never seen this type of answer before, so stick with us and we’ll explain what happened.
Notice first of all that all you have to do is multiply the top equation by –2 to get –6x + 4y = –8,
which is the exact opposite of the bottom equation, 6x – 4y = 8. If you add these two you get
0 = 0, which is always true. Therefore, this system has infinite solutions. Lots of answers will work
in this system (actually, an infinite number of them). If you graph this system on a coordinate
plane, you get two lines that lie on top of each other. How many points do those two lines share
in common? All of them. Some books ask you to write this out using variables to represent constants. For example, if you arbitrarily pick that x = k, you can plug that into the top equation to
get 3k – 2y = 4, which means that –2y = 4 – 3k, or y =
.
Chapter 13: Finding Solutions for Systems of Equations
e
Solve
. The answer is x = 2 and y = 3 or x = –3 and y = 8.
As we explain in the section this problem was presented in, sometimes you end up with
quadratics that, when solved, have two solutions. First, solve the linear equation for a variable, like x in the second equation: x = 5 – y. Now substitute this into the first equation:
(5 – y)2 – y = 1. FOIL out the binomial to get 25 – 10y + y2 – y = 1. Combine like terms:
25 – 11y + y2 = 1. Now get 0 on one side of the equation: 24 – 11y + y2 = 0. This factors to
(3 – y)(8 – y) = 0, which, when you use the zero product property, gets you two solutions
for y: y = 3 and y = 8. Accept that both of these are true and substitute them, one at a time,
into the original quadratic equation to get the most possible solutions for x. First: If y = 3,
then x2 – 3 = 1; x2 = 4; x = ± 2. 2 + 3 = 5 works in the second equation, but notice that –2 + 3 = 5
doesn’t. That means that when y is 3, x only equals 3. Now do the same thing for y = 8:
x2 – 8 = 1; x2 = 9; x = ± 3. In the second equation, 3 + 8 = 5 is false, but –3 + 8 = 5 is true,
so the other solution is x = –3, y = 8.
f
Solve the system of equations
. The answer is x = 4 and y = 5 or x = 5 and y = 4.
First, solve the linear equation for x: x = 9 – y. Plug this into the second given equation:
(9 – y)y = 20. Distribute to get 9y – y2 = 20 and get a quadratic to solve. Get 0 on one side:
0 = y2 – 9y + 20, which factors to 0 = (y – 5)(y – 4). This means that y is 5 or 4. Plug them
both into either original equation: y = 5: x + 5 = 9, x = 4. y = 4: x + 4 = 9, x = 5.
g
Solve
. The answer is no solution.
This is also a first for you, but don’t be surprised if your textbook or teacher throws these
monkey wrenches at you, too. First, notice that both given equations have y2 in them, with the
same signs. If you multiply the second equation by –1, you get
. Now add both
equations together to get x2 – x = 6. Next, get the equation to equal 0: x2 – x – 6 = 0. This factors to (x – 3)(x + 2) = 0, which does give two solutions, x = 3 or x = –2. However, if you plug
either one of these values into either of the original equations, you discover that y has no
solution. This means that the system has no solution, either.
h
Solve
. The answer is
.
The y terms have opposite signs, so it’s easier to eliminate them after you multiply the
second equation by 4:
. Adding these two equations gets you that
–9x2 = –256, or x2 = 256⁄9, which finally means that x = ± 16⁄3. Now notice that both of the original
equations have x2 in them, but no x term. If you square 16⁄3 or –16⁄3, you get the same result:
256
⁄9. This means the posi-tive and negative signs don’t really matter when it comes to solving
for y:
.
Next, simplify: 768 – 16y2 = –400. Subtract 768 from both sides: –16y2 = –1168. Now, by
. Wow, that made even our heads hurt!
dividing –16, you get y2 = 73, or y = ±
265
266
Part IV: And the Rest . . .
i
Solve
u=
. The solution is x = 4 and y = 3. First, rewrite the system by letting
and v =
and getting
. Now, multiply the first equation by 3 and the
. Adding these two equations gets you 189u = 27, which
second equation by 7:
means that u = 1⁄7. Here’s where you have to pay attention though! If this were a multiple choice
test, most people would pick 1⁄7 for the solution to this system, and they’d be wrong. That’s why
you have us, to help remind you that you’re not done yet! That’s because you found u but not
x or y. Work your way backwards:
. If the numerators are equal, the denominators have
to be as well: 7 = x + 3 means that 4 = x. And you would have gotten the multiple choice correct!
Now you can use that to get that y = 3 from the following steps:
Because this is a rational expression, also be sure to always check your solution to see if it’s
extraneous. In other words, if x = 4 or y = 3, do you get 0 in the denominator of either given
equation? In this case, the answer is no — so these answers are legit!
j
Solve
. The answer is x = 5 and y = –1.
If you let u =
and v =
second equation by 2 and get
backwards from there:
.
, you can rewrite the system as
. Now, multiply the
. Add them to get 24u = 4, or u = 1⁄6. Work your way
; 6 = x + 1; x = 5. The following steps get you to the solution for y:
Chapter 13: Finding Solutions for Systems of Equations
k
Solve the system
. The answer is (x, y, z) = (5, –1, 2).
A is the top equation: 3x – 2y = 17. B is the middle equation: x – 2z = 1. C is 3y + 2z = 1. C has no
x variable so you use the other two to eliminate x. A – 3 × B will do just that.
becomes
. Add these two equations to get –2y + 6z = 14, which you can call
equation D. 2C + 3D eliminates y:
becomes
. Add these two to
get 22z = 44, or z = 2. Plug this into equation C: 3y + 2(2) = 1; 3y + 4 = 1; 3y = –3; y = –1. Lastly,
plug z = 2 into equation B: x – 2(2) = 1; x – 4 = 1; x = 5.
l
Solve
. The answer is no solution.
Right away, notice that all the coefficients on the middle and bottom equations are exact
opposites of each other. When you add these two equations you get 0 = 4. Because this
equation is false, there’s no solution.
m
Solve
. The answer is x = 4, y = 3, and z = 5.
This system has three variables, so you only need three unique equations. Because you
have four, you just have an additional one to play with. A: 2x + 3y + 4z = 37. B: 4x – 3y + 2z =
17. C: x + 2y – 3z = –5. D: 3x – 2y + z = 11.
A + B eliminates y: 6x + 6z = 54. Divide by 6 to get equation E: x + z = 9.
C + D also eliminates y: 4x – 2z = 6. Divide by 2 to get equation F: 2x – z = 3.
E + F eliminates z: 3x = 12; x = 4.
Work backwards by plugging x = 4 into equation F: 2(4) – z = 3; 8 – z = 3; –z = –5; z = 5.
Now x = 4 and z = 5 go into equation A: 2(4) + 3y + 4(5) = 37; 8 + 3y + 20 = 37; 3y + 28 = 37; 3y =
9; y = 3.
n
Solve
. The answer is a = 1, b = 3, c = –2, and d = –4.
To avoid confusion between A and a, we’re naming the equations in this system a little
differently:
T: 3a + b + c + d = 0; S: 4a + 5b + 2c = 15; N: 4a + 2b + 5d = –10; L: –5a + 3b – d = 8. Now start
eliminating!
2T + S:
gives you equation M: –2a + 3b – 2d =
15. That way, you now have three equations with a, b, and d in them: N, L, and the new one,
M. Start over and eliminate another variable.
267
268
Part IV: And the Rest . . .
5M + 2N:
gives you equation R: –2a + 19b = 55.
M – 2L:
gives you equation P: 8a – 3b = –1. You’re
down to two equations and two variables (happy dance)!
4R + P:
. Add these two equations to get 73b = 219. This
gives you your first solution, b = 3. Work your way backwards to get the other solutions.
P: 8a – 3(3) = –1; 8a – 9 = –1; 8a = 8; a = 1.
M: –2(1) + 3(3) – 2d = 15; –2 + 9 – 2d = 15; 7 – 2d = 15; –2d = 8; d = –4.
T: 3(1) + 3 + c – 4 = 0; 3 + 3 + c – 4 = 0; 2 + c = 0; c = –2.
o
Sketch the graph of
. See the graph for the answer.
Put the first two equations in slope-intercept form first. The top equation is y ≥ –2x + 9; the
second equation is y ≤ 2x – 1. Put them all on the same graph.
y
x
p
Sketch the graph of
. See the graph for the answer.
Both of these equations are circles. If you don’t recognize them as such, you should turn to
Chapter 12 to read up on conic sections. Graph them both on the same graph.
Chapter 13: Finding Solutions for Systems of Equations
q
Sketch the graph of
269
. See the graph for the answer.
The first equation is a parabola, x2 – 2 > y. The second equation is the line x – 4 < y. Graph them
both on the same graph.
r
Sketch the graph of
. See the graph for the answer.
The third equation is a square root function. If you don’t remember how to graph it, turn to
Chapter 3 and get a refresher.
s
Find the constants A and B:
. The answer is A = 6 and B = –5.
We’ve just started the decomposition process for you. The first
thing you should do is multiply everything by the factored denominator
; in other words,
. Multiply everything out to get x – 38 = Ax – 3A + Bx + 4B. Collect the
like terms on the right: x – 38 = Ax + Bx – 3A + 4B. Now factor out the x: x – 38 (A + B)x – 3A + 4B.
The coefficients of the x terms are equal, which gives you one equation: 1 = A + B. The constants are also equal, which gives you a second equation: –38 = –3A + 4B. This system of equations is solved as follows:
gives you –35 = 7B, or –5 = B. Now, 1 = A – 5 tells you that 6 = A.
. Adding these two equations
270
Part IV: And the Rest . . .
t
Find the form of the partial fraction decomposition for
The answer is
, but don’t find the constants.
.
Note the directions here. We just ask you to set up the problem, not to perform the decomposition. The denominator is already factored for you. The hardest part about this problem is to
remember that you need to use one term for every power of this binomial, up to its degree of 2.
In other words, you need one fraction with (x – 1)1 in it and you need another with (x – 1)2 in it.
This is why the answer is
u
.
Find the partial fraction decomposition for
. The answer is
.
First, factor the given denominator to become
. Because each factor is linear, set
up three different fractions with constants on the top: one for the (x – 1) factor, one for the
(x – 2) factor, and one for the (x – 2)2 factor.
.
Multiply every term by the factored denominator, cancel, and get 2x2 – 21x + 18 = A(x – 2)2 +
B(x – 1)(x – 2) + C(x – 1). Multiply this all out to get 2x2 – 21x + 18 = Ax2 – 4Ax + 4A + Bx2 – 3Bx +
2B + Cx – C. Collect the like terms: 2x2 – 21x + 18 = Ax2 + Bx2 – 4Ax – 3Bx + Cx + 4A + 2B – C. Factor
out the x2 and the x on the right side: 2x2 – 21x + 18 = (A + B)x2 + (–4A – 3B + C)x + 4A + 2B – C.
Set the coefficients of x2 equal to each other: 2 = A + B. Set the coefficients of x equal to each
other: –21 = –4A – 3B + C. Lastly, set the constants equal to each other: 18 = 4A + 2B – C. Solve
this system of equations using elimination to get A = –1, B = 3, and C = –16. This is where the
answer
v
comes from.
Find the partial fraction decomposition for
. The answer is
.
Factor the denominator by using grouping: 2x2(x – 2) + 3(x – 2) becomes (x – 2)(2x2 + 3). Use
this to make two different fractions with a constant on top of the linear factor and a linear
expression on top of the quadratic factor:
. Multiply every
term by the factored denominator, cancel, and get 11x2 – 7x + 14 = (Ax + B)(x – 2) + C(2x2 + 3).
Multiply it all out to get 11x2 – 7x + 14 = Ax2 – 2Ax + Bx – 2B + 2Cx2 + 3C. Collect the like terms
and factor to get 11x2 – 7x + 14 = (A + 2C)x2 + (–2A + B)x – 2B + 3C. This gives you a system with
three equations:
which gives you the answer
. Solving this system tells you that A = 3, B = –1, and C = 4,
.
271
Chapter 13: Finding Solutions for Systems of Equations
w
Find 4N. The answer is
.
First, write out the problem by substituting the given matrix N in
. Distribute
the 4 to every element inside the matrix to get the answer:
x
Find 4N + 5M. The answer is
.
.
First, substitute the given matrices into the expression:
.
Distribute both scalars to every element of their matrix:
.
Add these two matrices by adding their corresponding elements to get
y
.
Find 3M – P. The answer is no solution.
These matrices aren’t the same dimensions, so you can’t add them. There’s no solution.
A
Find NP. The answer is
.
Substitute the given matrices into the expression
. Check to see if
you can even multiply them. The matrix on the left is
and the one on the right is
, so
you can multiply them. Multiply every row from the left matrix by every column from the right
matrix.
The sum of the first row times the first column: 2(–1) + 4(4) + 5(2) – 8(–5) = 64.
The sum of the first row times the second column: 2(2) + 4(4) + 5(3) – 8(2) = 19.
The sum of the first row times the third column: 2(–1) + 4(0) + 5(1) – 8(–1) = 11.
The sum of the second row times the first column: 10(–1) + 3(4) – 2(2) – 3(–5) = 13.
The sum of the second row times the second column: 10(2) + 3(4) – 2(3) – 3(2) = 20.
The sum of the second row times the third column: 10(–1) + 3(0) – 2(1) – 3(–1) = –9.
Putting these all into a matrix gives you the answer:
B
.
Using the augmented matrix from the last example, use elementary row operations to find
–3r2 → r2. The answer is
.
Just multiply the second row of the given equation by –3 to get
.
272
Part IV: And the Rest . . .
C
Now, using your answer from problem 27, find
. The answer is
Swap the first row with the second row and get
D
.
.
Now, keep going and find r1 + r2 → r2. The answer is
.
Add each element from row one to its corresponding element in row two to change row two
and get
E
.
Lastly, find 3r2 + r1 → r1. The answer is
.
Temporarily multiply the second row by 3 to get [–9 –30 | –9]. Add these to the corresponding
elements in row one to change row one and get
F
Solve the system of equations
.
by writing it in augmented form and then putting the
matrix in row echelon form. The answer is x = 9⁄5 and y = 17⁄25.
The matrix in augmented form is
left corner:
. You need to add a –3 to the second row to get a 0 under the one:
–3r1 + r2 → r2 gives you
the second row:
substitute
G
. Multiply the top row by 1⁄2 to get a 1 in the upper
; this is in row echelon form. Set up an equation from
. Solve this equation to get y = 17⁄25. Use that answer and back
to get that x = 9⁄5.
Use Gaussian elimination to solve
. The answer is x = 1, y = 1, and z = 1.
Set up the system as an augmented matrix:
gets a 1 in the upper-left corner:
–3r1 + r2 → r2 gets a 0 under the 1 in the second row:
.
.
.
Chapter 13: Finding Solutions for Systems of Equations
3r1 + r3 → r3 gets a 0 under the 1 in the third row:
.
gets a 1 in the next position along the diagonal:
.
–4r2 + r3 → r3 gets a 0 under the 1 you just created:
or z = 1. Back substitute
273
. This means that –z = –1,
to get that y = 1. Back substitute again: x – 2(1) – 1(1) = –2,
or x = 1.
H
Solve the system
using inverse matrices. The answer is x = –1 and y = 6.
First, write the system as a matrix equation:
. Now, find the inverse matrix
using the handy formula we show you in the section in which this question appears:
. Multiply this inverse on both sides of the equation:
. Multiply the two matrices:
the scalar:
I
. Multiply
. Your solutions from top to bottom are x = –1 and y = 6.
Solve the system
using inverse matrices. The answer is x = 5 and y = –1.
Write the system as a matrix equation:
. Find the inverse:
. Multiply this on both sides:
. Multiply the matrices:
scalar:
J
. Multiply the
.
Find the determinant of
Because this is a
after the third one:
. The answer is 93.
matrix, you have to use diagonals. First, rewrite the first two columns
. The sum of the diagonals from bottom-left to
274
Part IV: And the Rest . . .
top-right is: (–2)(4)(4) + (–1)(6)(2) + (5)(–3)(–1) = –32 – 12 + 15 = –29. The sum of the diagonals
from top-left to bottom-right is: (2)(4)(5) + (–1)(6)(–2) + 4(–3)(–1) = 40 + 12 + 12 = 64. The difference of the bottom sum minus the top sum is 64 – (–29) = 93.
K
Use Cramer’s Rule to solve
. The answer is x = 1, y = –1, and z = 1.
Set up the quotient to find x first:
. We use an online matrix calculator to
find the determinants here. Out of all the tips we could give you, this is probably the best one.
Learn how to use your calculator or the Internet to calculate matrix operations like a determinant. If your teacher doesn’t allow you to use a calculator on a test, you have to use the
process of diagonals to find the determinant here.
Solve for y:
Solve for z:
. Ditto.
. Ditto squared.
Chapter 14
Sequences, Series, and
Binomials — Oh My!
In This Chapter
䊳 Finding general sequences and series
䊳 Dealing with arithmetic sequences and series
䊳 Solving geometric sequences and series
䊳 Using the binomial theorem
T
his chapter is all about patterns. No, we’re not making quilts, although we could . . . nah!
We’ll stick with patterns of numbers, not cloth. Namely, we explore sequences, series,
and the binomial theorem.
A sequence is an ordered list of numbers that follow a pattern. A series is the sum of the
terms in a sequence, so it adds a pattern. The binomial theorem is the result of discovering
the pattern of an expanded binomial.
Hmm . . . we think we’re sensing a pattern of patterns here.
One mathematical term that comes up in this chapter is factorial, which you may remember
from your previous math classes. A factorial, n!, read “n factorial,” is defined as 1 · 2 · 3 · . . . ·
(n – 1) · n.
Major General Sequences and
Series: Calculating Terms
Mathematically, a sequence is usually written in the following form: {an} = a1, a2, a3, . . . an.
Here, n is the number of terms, an is the term of the sequence, and a1 is the first term, a2 is the
second term, and so on. Similarly, a series can be written as the sum of the terms: a1 + a2 +
a3 + . . . + an. The pattern of sequences and series can usually be described by a general
expression or rule. Because sequences and series can be infinite, this expression allows you
to find any number in the list without having to find all the numbers. If you’re not given the
general expression, you can find it if you’re given the first few terms of a sequence or series.
Sometimes a term in a sequence depends on the term(s) before it. These are called recursive
sequences. A famous example of a recursive sequence is the Fibonacci Sequence: 1, 1, 2, 3, 5,
8, 13, . . . where each term is the sum of the two before it.
276
Part IV: And the Rest . . .
Q.
Write the first five terms of the sequence
an = n2 + 3.
Q.
Write a general expression for the sequence
to find the nth term: –1⁄2, 2⁄3, –3⁄4, 4⁄5.
A.
4, 7, 12, 19, 28. To find each term, you just
plug the number of the term (n) into the
formula: a1 = (1)2 + 3 = 4; a2 = (2)2 + 3 = 7;
a3 = (3)2 + 3 = 12; a4 = (4)2 + 3 = 19;
a5 = (5)2 + 3 = 28.
A.
an =
Write the first five terms of the sequence:
2.
1.
an =
3.
Write a general expression for the sequence
to find the nth term: 2, 2, 8⁄3, 4, 32⁄5, . . .
Solve It
alternates between negative and positive.
To deal with this, multiply by powers of –1:
(–1)n. Next, notice that the sequence’s
numerator is the same as the term number
(n), so n becomes your numerator. Finally,
you can see that the denominator is simply
one number larger than the numerator
(and term number), so it can be written as
n + 1. Putting these pieces together, you
.
get: an =
.
Solve It
. First, notice that the sign
Write a general expression for the sequence
to find the nth term: 2, 4, 10, 28, 82, . . .
Solve It
4.
Write the next two terms of the sequence:
1, 1, 2, 3, 7, 16.
Solve It
Chapter 14: Sequences, Series, and Binomials — Oh My!
5.
Find the sum of the first five terms of the
series: 2n – 1 + 1.
Solve It
6.
277
Find the sum of the first five terms of the
series: 3n + 2n.
Solve It
Working Out the Common Difference:
Arithmetic Sequences and Series
One special type of sequence is called an arithmetic sequence. In these sequences, each
term differs from the one before it by a common difference, d. As a result, you have a
formula for finding the nth term of an arithmetic sequence:
an = a1 + (n – 1)d
where a1 is the first term, n is the number of terms, and d is the common difference.
To find the sum of an arithmetic sequence, also called an arithmetic series, you have to
add a given number of terms together. This you can write in summation notation:
This is read as “the kth partial sum of an” where n = 1 is the lower limit, k is the sum’s
upper limit, a1 is the first term, and ak is the last term. To find a sum, simply plug the
lower and upper limits into the formula for an to find a1 and ak and then simplify.
Q.
Find the 60th term of the arithmetic
sequence: 4, 7, 10, 13, . . .
A.
a60 = 181. The easiest way to begin this
problem is to find the formula for the nth
term. To do so, you need a1, which is 4. You
also need the common difference, d, which
can be found by subtracting two sequential
terms; for example: a2 – a1 = 7 – 4 = 3. Plugging these into the general formula and
simplifying, you get: an = a1 + (n – 1)d =
4 + (n – 1)3 = 4 + 3n – 3 = 1 + 3n. Now you
can find the 60th term by plugging in 60 for
n: a60 = 1 + 3(60) = 1 + 180 = 181.
Q.
Find
A.
–68. To find the sum, you just have to use
the arithmetic series formula. For this, you
need k (which is 8), a1, and ak. Start by finding a1: –3(1) + 5 = 2. Then find a8: –3(8) + 5 =
–19. Finally, plug these into the formula:
Sn = (8⁄2)(2 + –19) = (4)(–17) = –68.
.
278
Part IV: And the Rest . . .
7.
Find the 50th term of the arithmetic
sequence: –6, –1, 4, 9, . . .
Solve It
9.
Find the general formula of an arithmetic
sequence where a5 = –5 and a20 = –35.
Find
Solve It
Find the general formula of an arithmetic
sequence where a1 = –3 and a15 = 53.
Solve It
10.
Find
.
Solve It
Solve It
11.
8.
.
12.
Write the arithmetic series 2 + 7⁄3 + 8⁄3 + 3 + 10⁄3
in summation notation and find the result.
Solve It
Chapter 14: Sequences, Series, and Binomials — Oh My!
279
Be Fruitful and Multiply: Simplifying
Geometric Sequences and Series
When consecutive terms in a sequence have a common ratio, the sequence is called a
geometric sequence. To find that ratio, r, you divide each term by the term before it,
and the quotient should be the same. Just like the other sequences, a1 denotes the first
term. To find the next term, multiply by the common ratio, r. Another pattern! The formula for the nth term of a geometric sequence is:
As with other sequences, you can find the sum of geometric sequences, called geometric
series. To find a partial sum of a geometric sequence you can use the following formula:
Here, n = 1 is the lower limit, k is the sum’s upper limit, r is the common ratio, and a1 is
the first term. To find a sum, simply plug the lower and upper limits into the formula
for an to find a1 and ak and then simplify.
Unlike any other sequence, you can actually find the value of an infinite sum of many
geometric sequences. As long as r lies within the range –1 < r < 1, you can find the infinite
sum. If r lies outside that range, an will grow infinitely, so the sum won’t have a limit. To
find the infinite sum of a geometric series where r is within the range –1 < r < 1, use the
following formula:
Because we just plug and chug, geometric sequences and series are pretty easy to deal
with. You just need to remember your rules for simplifying fractions. You can do it!
Q.
Find the 10th term of the geometric
sequence: 3, –6, 12, –24, . . .
A.
a10 = –1,536. For the formula for the nth term
of a geometric sequence, you need a1 and r.
a1 is given in the problem: 3. To find r, all you
need to do is divide a2 by a1: –6⁄3 = –2. Now
you can simply plug these values into the
formula:
= –1,536.
Q.
A.
Find the sum:
.
242
⁄27. To use the partial sum formula,
you need to know a1, r, and k. From the
problem, you can identify r as 1⁄3 and k
as 5. To find a1, simply plug in 1 for n:
a1 = 6(1⁄3)1–1 = 6 · (1⁄3)0 = 6 · 1 = 6. Now
all you have to do is plug and chug:
= 242⁄27.
280
Part IV: And the Rest . . .
13.
Find the 16th term of a geometric sequence
given a1 = 5 and a2 = –15.
Solve It
15.
Find the sum
.
Find the sum of the infinite geometric
series: 1⁄6 + 1⁄3 + 2⁄3 +. . . .
Solve It
Find the 8th term of a geometric sequence
given a2 = 6 and a6 = 486.
Solve It
Solve It
17.
14.
16.
Find the partial sum of the geometric
series: 1⁄6 + 1⁄3 + 2⁄3 + . . . 32⁄3.
Solve It
18.
Find the sum:
Solve It
.
Chapter 14: Sequences, Series, and Binomials — Oh My!
281
Expanding Polynomials Using
the Binomial Theorem
Binomials are polynomials with exactly two terms. Often, binomials are raised to powers
to complete computations, and when you multiply out a binomial so that it doesn’t
have any parentheses, it’s called a binomial expansion. One way to complete binomial
expansions is to distribute terms, but if the power is high, this method can be tedious.
An easier way to expand binomials is to use the binomial theorem:
Here, a is the first term, b is the second term, and
is the combinations
formula. For example, to find the binomial coefficient given by
, plug the values into
the formula and simplify:
.
This seems like a lot of work, but trust us — if you just take it one step at a time, this
method will save you an immense amount of time!
Q.
Write the expansion of (3x – 2)4.
A.
81x4 – 216x3 + 216x2 – 96x + 16. To expand, simply replace a with 3x, b with –2, and n with 4 to get
. Now, to simplify
this mess, start with the combinations formula for each term:
.
Then, raise the monomials to the specified powers:
.
Finally, combine like terms and simplify: 81x4 – 216x3 + 216x2 – 96x + 16.
282
Part IV: And the Rest . . .
19.
Find the 5th term of (x + y)12.
Solve It
21.
Expand (k – 4)5.
Solve It
20.
Find the 8th term of (2x – 3y)10.
Solve It
22.
Expand (y + 4z)6.
Solve It
Chapter 14: Sequences, Series, and Binomials — Oh My!
Answers to Problems on Sequences,
Series, and Binomials
a
Write the first five terms of the sequence: an =
.
The answer is –1, – ⁄4, 0, ⁄8, ⁄5. To find each term, you plug the number of the term (n) into the formula:
1
1
1
= –1, –1⁄4, 0, 1⁄8, 1⁄5.
b
Write a general expression for the sequence to find the nth term: 2, 4, 10, 28, 82, . . .
The answer is an = 3n – 1 + 1. At first, this one seems tricky. But if you subtract 1 from each term,
the pattern becomes apparent: 1, 3, 9, 27, 81 . . . these are multiples of 3! In fact, they can be
found by taking 3 to sequential powers: 30, 31, 32, 33, 34, where the power is 1 less than the term,
or n – 1. So, putting that all together, we get an = 3n – 1 + 1.
c
Write a general expression for the sequence to find the nth term: 2, 2, 8⁄3, 4, 32⁄5, . . .
The answer is
. Here’s the hint for this one: The third term has a 3 in the denominator, and
the fifth term has a 5 in the denominator. If you write each as an unreduced fraction over the
term, n, the pattern for the denominator reveals itself: 2⁄1, 4⁄2, 8⁄3, 16⁄4, 32⁄5. The denominator is n. Now
you just have to figure out the pattern for the numerator. Easy! They’re all multiples of 2. The
first term is 21, the second is 22, and so on. So, your general expression is simply
d
.
Write the next two terms of the sequence: 1, 1, 2, 3, 7, 16, . . .
The answer is 65, 321. Ah! This is one of those recursive sequences we were mentioning. This
pattern is found by adding the first term squared to the second term, and so on: 12 + 1 = 2, 12 + 2
= 3, 22 + 3 = 7. . . . So, to find the next two terms, you just need to continue the pattern: 72 + 16 =
65 and 162 + 65 = 321.
e
Find the sum of the first five terms of the series: 2n–1 + 1.
The answer is 36. This one is an easy plug and chug. Simply plug in values 1 through 5 for n
and simplify: 21 – 1 + 1 = 2, 22 – 1 + 1 = 3, 23 – 1 + 1 = 5, 24 – 1 + 1 = 9, 25 – 1 + 1 = 17. Now you just need to
add the terms: 2 + 3 + 5 + 9 +17 = 36.
f
Find the sum of the first five terms of the series: 3n + 2n.
The answer is 393. Just like the last one, plug and chug away: 31 + 2(1) = 5, 32 + 2(2) = 13, 33 +
2(3) = 33, 34 + 2(4) = 89, 35 + 2(5) = 253. Now just add ‘em up: 5 + 13 + 33 + 89 + 253 = 393.
g
Find the 50th term of the arithmetic sequence: –6, –1, 4, 9, . . .
The answer is a50 = 239. To start, find the formula for the nth term. For this, you need a1, which
is –6. Next, you need the common difference, d, found by subtracting two sequential terms:
a2 – a1 = –1 – (–6) = 5. From here, simply plug these into the general formula and simplify: an =
–6 + (n – 1)5 = –6 + 5n – 5 = 5n – 11. Now that you have the general formula, you can find the
50th term by plugging in 50 for n: a50 = 5(50) – 11 = 250 – 11 = 239.
h
Find the general formula of an arithmetic sequence where a1 = –3 and a15 = 53.
The answer is an = 4n – 7. Here, you have the first term, a1, so you just need to find d. You can
use a15 to find it. Simply plug in a1, a15, and n = 15 into the general formula and solve using algebra: 53 = –3 + (15 – 1)d; 53 = –3 + 14d; 56 = 14d; 4 = d. Now you can plug it in to find the general
formula: an = –3 + (n – 1)4 = –3 + 4n – 4 = 4n – 7.
283
284
Part IV: And the Rest . . .
i
Find the general formula of an arithmetic sequence where a5 = –5 and a20 = –35.
The answer is an = –2n + 5. This one is a bit more complicated but completely doable! To start,
recognize that you have two terms, which means you can create two equations. Then you have
a system of equations (check out Chapter 13 for a refresher). Solve these to find your missing
variables and you can write your general formula. Start by writing your two equations with
your given values and simplify: –5 = a1 + (5 – 1)d = –5 = a1 + 4d and –35 = a1 + (20 – 1)d = –35 =
a1 + 19d. Next, use elimination to solve the system by multiplying the first equation by –1 and
adding the two together to get –30 = 15d. Therefore, d = –2. Now, simply substitute this back
into either equation to find a1: –5 = a1 + 4(–2); –5 = a1 – 8; 3 = a1. Finally, plug in a1 and d to find
the general formula: an = 3 + (n – 1)(–2); an = 3 + –2n + 2; an = –2n + 5.
j
Find
.
The answer is 35⁄2. To find the sum, you just have to use the arithmetic series formula. For this,
you need k (which is 5), a1, and ak. Start by finding a1: 1⁄2 · (1) + 2 = 5⁄2. Then find a5: 1⁄2 · (5) + 2 = 9⁄2.
Finally, plug these into the formula: Sn = (5⁄2)(5⁄2 + 9⁄2) = (5⁄2)(14⁄2) = (5⁄2)(7) = 35⁄2.
k
Find
.
The answer is 77. Follow the same steps as the previous problem. However, notice that the
lower limit is 4, so you need to start by finding a4: 2(4) – 3 = 5, and this is like your a1. The
number of terms from 4 to 10 is 7, so k = 7. You also need a10 = 2(10) – 3 = 17. Finally, just plug in
the values: Sn = ( 7⁄2)(5 + 17) = ( 7⁄2)(22) = 7(11) = 77.
l
Write the arithmetic series: 2 + 7⁄3 + 8⁄3 + 3 + 10⁄3 in summation notation, and find the result.
The answer is
. For summation notation, you need to find the general formula
and know how many terms you’re dealing with. In this case, you have five terms. Therefore, you
know your upper limit, k, is 5. For the general formula, you need the first term a1 = 2 and the
common difference, d, which is found by subtracting two sequential terms: a2 – a1 = 7⁄3 – 2 = 1⁄3.
Plug these in and simplify to find your general formula: an = 2 + (n – 1)1⁄3 = 2 + 1⁄3 · n – 1⁄3 = 5⁄3 + 1⁄3 · n.
Then, plug the general formula into the summation notation and add the values given in
the original problem to find the result:
m
.
Find the 16th term of a geometric sequence given a1 = 5 and a2 = –15.
The answer is a16 = –14,348,907. To find the 16th term, you need to find the general formula.
For that, you need a1 and r. a1 is given in the problem: 3. To find r, all you need to do is divide a2
by a1: –15⁄5 = –3. Now you can simply plug these values into the formula:
–14,348,907.
n
=
Find the 8th term of a geometric sequence given a2 = 6 and a6 = 486.
The answer is 4,374. This time you don’t have the first term, so you have to set up a system of
equations and simplify: 6 = a1 · r2 – 1; 6 = a1r1 and 486 = a1 · r6 – 1; 486 = a1r5. Isolate a1 and use substitution to solve for r: a1 = 6⁄r; 486 = (6⁄r)r5; 486 = 6r4; 81 = r4; 3 = r. Then substitute r back into either
equation to find a1: 6 = a1(3); 2 = a1. Now you can set up the general formula: an = 2 · 3n – 1. Finally,
to find the 8th term, plug in n = 8: a8 = 2 · 38 – 1 = 2 · 37 = 2 · 2,187 = 4,374.
o
Find the sum
.
The answer is 21⁄8. Start by finding a1 = 4(–1⁄2)1 – 1 = 4(–1⁄2)0 = 4 · 1 = 4.
Because r = –1⁄2, you have everything you need to plug into partial sum formula:
= 21⁄8.
Chapter 14: Sequences, Series, and Binomials — Oh My!
p
Find the partial sum of the geometric series: 1⁄6 + 1⁄3 + 2⁄3 + . . . 32⁄3.
The answer is 127⁄6. Here you have a1 and you can find r by dividing a2 by a1: 1⁄3 ÷ 1⁄6 = 2. The trick
here is that you need to find n. To do so, plug the last term into the general formula and use properties of exponents to solve for n: 32⁄3 = 1⁄6 · 2n – 1; 64 = 2n – 1; 26 = 2n – 1; 6 = n – 1; 7 = n. Now that you
= 127⁄6.
have all the variables, plug them in to find the partial sum:
q
Find the sum of the infinite geometric series: 2⁄3 + 1⁄3 + 1⁄6 +. . . .
The answer is 4⁄3. You need to start by finding r, which is a2 divided by a1: 1⁄3 ÷ 2⁄3 = 1⁄2. a1 is 2⁄3. So all
you need to do is plug these values into the appropriate formula and use what you know about
= 4⁄3.
fractions to simplify:
r
Find the sum:
.
The answer is 9⁄5. Start by finding a1 by plugging 1 into the general formula: 3(–2⁄3)1 – 1 = 3(–2⁄3)0
= 3. Next, notice that you have r, –2⁄3. From here, plug in these values to find the infinite sum:
= 9⁄5.
s
Find the 5th term of (x + y)12.
The answer is 495x8y4. All you have to do to find the 5th term is use the handy binomial
theorem. In this case, n is 5, a = x, and b = y:
. Because the original binomial
doesn’t have any coefficients, the coefficient just comes from the combinations formula:
. Multiply the other terms by this and you get 495x8y4.
t
Find the 8th term of (2x – 3y)10.
The answer is –2,099,520x3y7. In this case, a = 2x, b = –3y, and n is 8. Plug these into the
binomial theorem and simplify:
= –2,099,520x3y7.
u
Expand (k – 4)5.
The answer is k5 – 20k4 + 160k3 – 640k2 + 1280k +1024. To expand, simply replace a with k,
b with –4, and n with 5 to get:
.
To simplify, start with the combinations formula for each term:
.
Next, raise the monomials to the specified powers:
.
5
4
3
2
Last, combine like terms and simplify: k – 20k + 160k – 640k + 1280k +1024.
285
286
Part IV: And the Rest . . .
v
Expand (y + 4z)6.
The answer is y6 +24y5z + 240y4z2 + 1280y3z3 + 3,840y2z4 + 6,144yz5 + 4,096z6. Here, replace a
with y, b with 4z, and n with 6 to get:
.
Then, to simplify this mess, start with the combinations formula for each term:
.
Raise the monomials to the specified powers:
.
6
5
4 2
3 3
2 4
5
Finally, combine like terms and simplify: y +24y z + 240y z + 1280y z + 3,840y z + 6,144yz +
4,096z6.
Chapter 15
The Next Step Is Calculus
In This Chapter
䊳 Using different techniques to find limits of functions
䊳 Applying limit laws to find limits of combined functions
䊳 Discovering continuity and discontinuity in functions
Y
our high school math career probably began with Algebra I, and then you repeated
some of the information in Algebra II, and then again the same material presented itself
in Pre-calculus (albeit a bit morphed and complicated since Algebra 1). The end of pre-calc
is the beginning of calculus. Calculus teachers assume you’ve been paying attention during
your math years, or they assume that you bought this book and are now a math genius, and
that most of the material you’ve learned has stuck with you. Because of this, calc teachers
move pretty quickly into new material.
Calculus is the study of change. Until now, all the information you use to solve problems
has been constant, so your answers have always been constant. For example, up until calc,
in a distance problem the rate at which a car is moving remains constant. The slope of a
straight line is always a constant. The volume of a shape is always a constant. But in calculus, all these can move and grow and change. For example, the car can accelerate, decelerate, and accelerate again, all within the same problem, which changes the whole outcome.
The line can now be a curve so that its slope changes over time. The shape that you’re trying
to find the volume of can get bigger or smaller, so that the volume changes over time.
All this change may send you screaming in terror to the nearest antianxiety salt lick, but
chances are, if you’ve made it this far in your math career, you can handle it. So just in case
your Pre-calc course ends with a preview of calc, we include the first couple of topics here.
Finding Limits: Graphically, Analytically,
and Algebraically
If you haven’t noticed by now, graphing functions has slowly become more complex and
intricate. The more complicated the function is, the more complicated the graph tends to be.
By now you’ve seen functions that are undefined at certain values; the graph has either a
hole or a vertical asymptote, which affects your domain. The end of pre-calc (and the beginning of calc) looks at the limit of a function — what the function would do if it could.
In symbols, a limit is written as
, which is read as “the limit of f(x) as x approaches
n is L.” L is the limit that you’re looking for. For the limit of a function to exist, the left limit
and the right limit must be equal.
288
Part IV: And the Rest . . .
⻬ A left limit starts at an x value that’s less than the number n that x is approaching and gets closer and closer from the left. This is written as
.
⻬ A right limit starts at an x value that’s greater than the number n that x is
approaching and gets closer and closer from the right. This is written as
.
When the left limit and right limit are the same, only then does the function have a
= L.
limit. When
=
= L, then
You can find a limit in three different ways: graphically, analytically, and algebraically.
We take a look at each one so you know how to handle them.
Before you try any of the techniques for finding a limit, always try plugging in the value
that x is approaching into the function. If you get an answer, then that’s also your limit.
We also recommend only using the graphing method when you’ve been given the graph
and asked to find the limit. The analytical method works for any function, and sometimes it’s the only method you have in your pocket that will work. However, if the algebraic method works, then go with it — the analytical method is just too long and tedious.
Graphically
When you’re given the graph of a function and asked to find the limit, just read the
graph as x approaches the given value and see what the y value would have been (or
was if the function is defined).
Q.
In the given graph for f(x), find
, and
.
f(x)
,
A.
= 1,
= –2, and
=
doesn’t exist. The function is defined at x =
–3 because you can see a dot there, so the
limit of the function is the same as the
= 1.
function’s value: f(x) = 1, so
The function isn’t defined at x = 4 because
the graph has a hole, but if you move along
the graph from the left as x approaches 4,
y also approaches a value of –2. Because
this is the same from the right side, then
= –2. The graph has an asymptote
at x = 6, but the left limit is completely different from the right limit, so the limit as x
approaches 6 doesn’t exist. Some teachers
and books write this as “DNE” (does not
exist) because we see it frequently (and
we’re lazy as mathematicians).
Chapter 15: The Next Step Is Calculus
1.
In the given graph for g(x), find
, and
.
2.
,
Solve It
In the given graph for h(x), find
, and
.
289
,
Solve It
g(x)
Analytically
“Analytically” means “systematically,” and that’s exactly how you find the limit using
this method. You set up a chart, and the value that x is approaching goes in the middle
of the top row. On the left you put values that get closer to the value that x is
approaching, and you do the same thing on the right. The second row should be the y
values when you plug the top row into the function. Hopefully, the bottom row values
approach the same number from the left and right, and voilà — you have your limit!
Q.
Find the limit of
A.
chart. Here’s the table that we set up. Note:
Yours doesn’t have to look like this; there’s
no one way that works all the time in finding the limit analytically, but most of the
tables we’ve seen are set up similarly. Try
to get those x values really close to the
value you’re approaching — that’s usually
the best way to find the limit.
.
= 6. Notice that if you plug 2
into the function for x, you get 0 in the
denominator. That means the function is
undefined there. If the directions ask you
to find this limit analytically, you’d set up a
x
1
1.9
1.99
1.999
2
2.001
2.01
2.1
3
y
5
5.9
5.99
5.999
???
6.001
6.01
6.1
7
Notice that the y values from both the left and the right seem to be approaching 6; that’s your limit.
290
Part IV: And the Rest . . .
3.
Find the limit of
Solve It
.
4.
Find the limit of
.
Solve It
Algebraically
To find a limit algebraically, you can use four different techniques: plugging in, factoring, rationalizing the numerator, and finding the lowest common denominator. Always
start by plugging the given number into the function just to see whether it works. If the
answer is undefined, then move on to one of the other three techniques — each one
depending on the given function.
Plugging in
This first technique asks you to substitute the given value into the function. If you get
an undefined value, like 0 in the denominator of the fraction, try something else. But
when it works, this technique is the quickest way to find a limit. We like short methods, and we hope you do, too!
Factoring
When the function is a rational function with polynomials in the numerator and the
denominator, try factoring them. If you’ve forgotten how to factor a polynomial and
need a refresher, see Chapter 4. Chances are, some factors will cancel from both the
numerator and the denominator. You can then substitute the given value into the cancelled version and usually get an answer that’s also your limit.
When you follow the steps of factoring the rational function on the top and bottom,
canceling, and plugging in the given value, if you still get an undefined function, then
the limit does not exist (DNE).
Chapter 15: The Next Step Is Calculus
291
Rationalizing the numerator
When you see square roots in the rational function in the numerator and plugging in
doesn’t work, always try to rationalize the numerator. That’s right — you multiply by
the conjugate of the numerator on both the top and bottom of the fraction. When you
do, you usually see a bunch of terms cancel, and the function will simplify down to a
point where you can plug in the given value and find the limit.
Finding the lowest common denominator
When the rational function is a complex rational function (fractions inside of the fraction), find the common denominators and add or subtract terms, then cancel and simplify. You can then plug in the given value to find the limit.
Q.
Find the limit of
A.
=
.
.
Find the common denominator of the fractions on the top first:
. Subtract and then simplify the top now that you have
a common denominator:
. Notice that
you can now plug 0 into this function and get the limit:
5.
Find the limit of
algebraically.
Solve It
6.
.
Find the limit of
Solve It
.
292
Part IV: And the Rest . . .
Knowing Your Limit Laws
Calculus also provides you with a few limit laws that help you find the limits of combined functions: added, subtracted, multiplied, divided, and even raised to powers. If
you can find the limit of each individual function, you can find the limit of the combined function as well.
If
and
, the limit laws are:
⻬ Addition law:
⻬ Subtraction law:
⻬ Multiplication law:
⻬ Division law:
⻬ Power law:
For this whole section, use the following to answer the questions:
Q.
Find
.
A.
= 7. The limit of g(x)
is 2 and the limit of f(x) is –5. To find
, use the subtraction law:
2 – (–5) = 7. It really is that easy!
7.
Find
Solve It
.
8.
Find
Solve It
.
Chapter 15: The Next Step Is Calculus
9.
Find
.
10.
Find
.
Solve It
Solve It
Determining Continuity
The word “continuity” in math means the same thing as it does in your everyday life.
Something that’s continuous has a stability or a permanence to it . . . it just doesn’t
ever stop. In pre-calc, you’ve seen functions that have holes in their graph, jumps in
their graph, or asymptotes — just to name a few. A graph that doesn’t have holes,
jumps, or asymptotes keeps going forever, and we call that function continuous.
Polynomial functions, exponential functions, and logarithmic functions are continuous
at every point. If you’re ever asked to determine the continuity of one of these types of
functions, don’t bother — the answer is that it’s always continuous.
We usually look at specific values in the domain to determine continuity instead of looking at the entire function. Even discontinuous functions are discontinuous only at certain
places. The discontinuity at a certain x value in any function is either always removable
(a hole in the graph) or nonremovable (an asymptote). It all depends on the factored version of the polynomial in the denominator. If a factor cancels, the discontinuity there is
removable. If the factor doesn’t cancel, the discontinuity is nonremovable.
Three things must be true for a function to be continuous at x = c:
⻬ f(c) must be defined. When you plug c into the function, you must get a value
out again. For example, getting 0 in the denominator is unacceptable and therefore a discontinuity.
⻬ The limit of the function as x approaches c must exist. The left and right limits
must be the same. If they aren’t, the function is discontinuous there.
⻬ The function’s value and the limit must be the same. f(c) =
. If the value
of the function is one thing and the limit is something different, that’s not good;
the function is discontinuous there.
293
294
Part IV: And the Rest . . .
Here’s the graph of a function where each one of the above situations fails:
At f(4) the graph has a hole. The function is undefined there, and therefore discontinuous at x = 4.
At f(–3) the function jumps. The limit as x approaches –3 from the left is 4,
and from the right the limit is 1, so the limit doesn’t exist, and the function is
discontinuous.
At f(0) the function is defined at one point: f(0) = –4, but the limit as x approaches
0 from the left and from the right is –2. These two values must be the same for the
function to be continuous.
One point where the function is continuous is at x = 2. f(2) =
Q.
Is f(x) =
A.
f(x) is continuous at x = 3. First, check to
make sure the function is defined at x = 3.
When you plug 3 in for x you get f(3) = 3⁄5.
Next, make sure the limit is defined.
Because you can plug in the value on the
last step and get a value, you can also plug
3 in and get 3⁄5 for the limit. Because these
two values are the same, the function is
continuous at x = 3.
continuous at x = 3?
Q.
Explain why f(x) =
= 2.
is not continuous
at x = –2. Is this discontinuity removable or
nonremovable?
A.
The function is not defined at f(–2). This
rational function doesn’t factor, and therefore, nothing cancels and the discontinuity
that exists is nonremovable.
Chapter 15: The Next Step Is Calculus
11.
Determine whether g(x) =
continuous at x = 5.
is
Is g(x) =
Is g(x) =
continuous at x = 0?
Solve It
Solve It
13.
12.
continuous at x = –3?
14.
Solve It
Determine whether h(x) =
is continuous at x = –2.
Solve It
15.
Determine whether p(x) =
16.
Determine all numbers at which q(x) =
is continuous.
is continuous at x = 1.
Solve It
Solve It
295
296
Part IV: And the Rest . . .
Answers to Problems on Calculus
a
In the given graph for g(x), find
,
, and
.
= 3,
DNE, and
≈ –2.2.
Looking at the graph for g(x), when x = –5, g(x) = 3. When x approaches –2 from the left the limit
is 4, but when x approaches –2 from the right the limit is –2. Because these two values aren’t
the same,
does not exist (DNE). And when x approaches 1, the graph has a y value
somewhere between –2 and –3. Because this isn’t an exact science, just do your best to approximate the limit. It looks to us like it’s about –2.2.
b
In the given graph for h(x), find
,
, and
=⬁,
.
= 2, and
≈ –2.5.
This answer also comes directly from the given graph. Trace your fingers along the graph as x
gets closer and closer to –2 from the left and notice that the line keeps going up. Also notice
that as x approaches this same value from the right, the graph is also going up. Like, forever
and ever up. That’s why
= ⬁. Meanwhile, as x approaches 5, h(x) approaches 2, and as
x approaches 0, it looks like f(x) is awful close to –2.5.
c
Find the limit
. The answer is 11.
See our chart for the analytical evaluation of this limit:
x
–2
–1.1
–1.01
–1.001
–1
–0.999
–0.99
–0.9
0
y
10
10.9
10.99
10.999
???
11.001
11.01
11.1
12
By looking at the y values in the second row, it looks like from both the left and the right that y
is approaching 11.
d
Find the limit
. The answer is 7.
Here’s the chart we used for this limit:
x
1
1.9
1.99
1.999
2
2.001
2.01
2.1
3
y
4
6.7
6.97
6.997
???
7.003
7.03
7.3
10
Looking at this one, we call it 7. What do you think?
e
Find the limit of
algebraically. The answer is 7.
Plugging 2 into the function gives you a 0 in the denominator, so you must try another technique. This rational function has a numerator that factors. (You did the limit analytically in
. This reduces to 3x + 1, which
problem 4.) When you factor it, you should get
gives you a function that you can plug 2 into: 3(2) + 1 = 6 + 1 = 7.
f
Find the limit of
. The answer is
.
Substitute 5 into this equation and you also get 0 in the denominator. Noticing that the numerator has a square root, you should find yourself thinking something along the lines of, “Perhaps
I should multiply by the conjugate to rationalize the numerator.” If you did think that or something close to it, give yourself a huge pat on the back.
297
Chapter 15: The Next Step Is Calculus
Here’s how to multiply by the conjugate:
. FOIL out the numerators and
watch the square roots disappear. However, don’t multiply out the denominators — it’ll cancel
easier if you don’t:
.
The numerator and the factor on the left on the bottom both cancel and give you
. Now when you plug 2 in, you find that the limit is
g
Find
. The answer is
.
.
Because you know both limits, to find the limit of their quotient, divide their limits as well:
=
h
.
Find
. The answer is
.
= (–5)2
Plug in the information that you know based on the given limits:
.
i
Find
. The answer is
Plug and chug away:
j
Find
.
.
=
=
.
is undefined.
This time, putting the limit of h(x) in the denominator also puts 0 in the denominator. That’s an
undefined limit.
k
Determine whether g(x) =
is continuous at x = 5. x = 5 is a removable discontinuity.
Factor the given equation first:
=
. Cancel to get
when you plug 5 into this simplified expression, you do get an answer of
the original, given equation. The graph is going to look and act like
nal equation is
or
. Notice that
, or
. But this isn’t
, but because the origi-
, there is still going to be a hole in the graph (try
plugging 5 into either of them and see what happens). This is why 5 is a removable discontinuity.
l
Is g(x) =
continuous at x = 0? The answer is yes, x = 0 is continuous.
You can simply plug 0 into this function and get a value out:
makes the function continuous.
. This
298
Part IV: And the Rest . . .
m
Is g(x) =
continuous at x = –3? x = –3 is a nonremovable discontinuity.
Plugging –3 into the original function
gives you 0 in the denominator, so you know
right away that it’s discontinuous. When you factor and simplify to
denominator, so the discontinuity is nonremovable.
n
Determine whether h(x) =
, you still get 0 in the
is continuous at x = –2. The function is continuous
at x = –2.
If you don’t know how to deal with piece-wise functions like this, get a refresher from Chapter 3.
First, look at f(–2) = 3(–2) – 1 = –6 – 1 = –7. The function exists.
Now, look at
= (–2)2 – 11 = 4 – 11 = –7.
= 3(–2) – 1 = –6 – 1 = –7. Next, look at
Because the left limit matches the right limit, the function has a limit as x approaches –2.
Lastly, because the function value matches the limit value, the function is continuous at x = –2.
o
Determine whether p(x) =
is continuous at x = 1. The function isn’t continuous
at x = 1.
p(1) =
. The function exists.
=
, but
= 4(1) + 3 = 4 + 3 = 7. These two values aren’t equiva-
lent, so there is no limit and the function is discontinuous at x = 1.
p
Determine all numbers at which q(x) =
is continuous. The function is contin-
uous everywhere except x = 1. In interval notation, this is written as
.
The only places that piece-wise functions have potential discontinuities are where the function
may break into pieces — where the interval begins or ends.
= (–2)2 – 3 = 4 – 3 = 1, and
= –2 + 3 = 1.
=
, so the graph is
continuous there.
= 1 + 3 = 4, but
graph is discontinuous there.
= 12 – 3(1) = 1 – 3 = –2.
doesn’t equal
, so the
Part V
The Part of Tens
T
In this part . . .
his part has a summary of the parent graphs we cover
in Chapter 3, including how to transform them. Think
of it as a quick guide to all the topics we cover regarding
graphing and transforming parent functions. This part also
includes a chapter on the mistakes we commonly see in
pre-calc and how to avoid them (please avoid them!).
Chapter 16
Ten Uses for Parent Graphs
In This Chapter
䊳 Graphing polynomial functions
䊳 Picturing absolute value and rational functions
䊳 Visualizing exponential and logarithmic functions
䊳 Sketching trig functions
A
picture is worth a thousand words, and graphing is just math in pictures! These pictures can give you important information about the characteristics of a function. The
most common graphs are called parent graphs. These graphs are in their original, unshifted
form. Any parent graph can be stretched, shrunk, shifted, or flipped. They’re extremely
useful because you can use them to graph a more complicated version of the same function
using transformations (see Chapter 3). That way if you’re given a complex function (say a
crazy-looking quadratic), you automatically have a basic idea of what the graph will look
like without having to plug in a whole bunch of numbers first. Essentially, by knowing what
the parent looks like, you get a good idea about the kids . . . the apple doesn’t fall far from
the tree, right? Now, let’s check out those family pictures!
Squaring Up with Quadratics
The basic quadratic is simplicity itself: y = x2. Its graph is a parabola with a vertex at the
origin, reflected over the y-axis (see Figure 16-1). You can find out more about graphing quadratics in Chapters 3 and 12.
Figure 16-1:
The graph of
the function
y = x2
passes
through the
origin and is
symmetric
across the
y-axis.
302
Part V: The Part of Tens
Cueing Up for Cubics
The parent graph of the cubic function, y = x3, also passes through the origin. This
graph is symmetric over the origin (see Figure 16-2). We cover cubics in Chapter 3.
Figure 16-2:
The graph of
the parent
function
y = x3.
Rooting for Square Roots and Cube Roots
A square root graph looks like a parabola that has been rotated clockwise 90° and cut
in half. It’s cut in half (only positive) because you can’t take the square root of a negative number. The parent graph is pictured in Figure 16-3.
Figure 16-3:
The parent
graph of
y=
starts at the
origin and
curves up to
the right.
Chapter 16: Ten Uses for Parent Graphs
Cube root functions are the inverse of cubic functions, so their graphs are going to
reflect that. The parent graph of a cube root function passes through the origin and is
symmetric over it, as shown in Figure 16-4.
Figure 16-4:
The parent
graph of
has a point
of inflection
(bend) at the
origin.
Graphing Absolutely Fabulous
Absolute Value Functions
Because the absolute value function turns all inputs into non-negative values (0 or positive), the parent graph is only above the x-axis. Figure 16-5 shows the parent graph in
its characteristic V shape.
Figure 16-5:
The graph of
the function
y = |x | has a
vertex at the
origin and is
symmetric
over the
y-axis.
303
304
Part V: The Part of Tens
Flipping over Rational Functions
In Chapter 3, we take you through the steps for graphing rational functions. These
involve finding asymptotes, intercepts, and key points. Because these functions don’t
really have parent graphs per se, we thought we’d show you an example of the most
basic rational function: y = 1⁄x (see Figure 16-6). To see more, flip back to Chapter 3.
Figure 16-6:
A graph of
the rational
function
y = 1⁄x
with its
asymptotes.
Exploring Exponential Graphs
and Logarithmic Graphs
The parent graph of an exponential function is y = bx where b is the base. Because b
has to be some number to graph, we thought we’d show you the graph of y = ex. This
graph passes through the point (0, 1) and a horizontal asymptote of the x-axis, as
shown in Figure 16-7. We cover exponential graphs in Chapter 5.
Figure 16-7:
The exponential
graph of
y = ex
increases
without
bound to
the right.
1
Chapter 16: Ten Uses for Parent Graphs
The inverse (see Chapter 3) of an exponential function is a logarithmic function. So
we’re going to show you the inverse of the graph of y = ex, which is the graph of y =
logex, which is known as the natural log, or y = ln x (see Chapter 5). This graph passes
through the point (1, 0) and a vertical asymptote of the y-axis, as shown in Figure 16-8.
We also cover logarithmic graphs in Chapter 5.
Figure 16-8:
The logarithmic
graph of
y = logex
increases
slowly to the
right.
1
Seeing the Sine and Cosine
A sine graph looks like a wave. The parent graph passes through the origin and has an
amplitude of 1. The period is 2π, which means that the wave repeats itself every 2π.
Figure 16-9 shows one full period of the parent sine graph.
Figure 16-9: 1
The parent
graph of 0
sine:
y = sinθ. –1
π
2
π
3π
2
2π
Like sine, the graph of cosine is a wave. This parent graph passes through the point
(0, 1) and also has an amplitude of 1 and a period of 2π. You can see the parent graph in
Figure 16-10. For more information about graphing sine and cosine, turn to Chapter 7.
1
Figure 16-10:
The parent 0
graph of
cosine: –1
y = cosθ.
π
2
π
3π
2
2π
305
306
Part V: The Part of Tens
Covering Cosecant and Secant
Remember reciprocals? Well, cosecant is the reciprocal of sine, so the graph of cosecant reflects that. We’ve lightly sketched the parent sine graph along with the parent
cosecant graph in Figure 16-11 so you can see the relationship. Specific graphing information can be found in Chapter 7.
Figure 16-11:
The graph of
the parent 1
cosecant
function: 0
y = cscθ
with a light –1
sketch of
y = sinθ.
π
2
π
3π
2
2π
Again, like cosecant, secant is the reciprocal of cosine, so the graph of secant is similar
to the graph of cosine. To picture this, we’ve lightly drawn the graph of cosine along
with the parent graph of secant in Figure 16-12.
Figure 16-12:
The graph of
the parent 1
secant
function: 0
y = secθ
with a light –1
sketch of
y = cosθ.
π
2
π
3π
2
2π
Tripping over Tangent and Cotangent
One repeating pattern of the graph of tangent is its asymptotes, where the function is
undefined. Like other trig graphs, a tangent graph has a period where it repeats itself.
In this case, it’s π. In Figure 16-13, we show you one period of the parent tangent graph.
For more information about graphing tangents, turn to Chapter 7.
Chapter 16: Ten Uses for Parent Graphs
Figure 16-13:
One period
of the
1
parent
−π 0 π π 3π π 5π 3π
tangent −π
2 4–1 4 2 4
4 2
function:
y = tanθ.
Like tangents, the parent graph of cotangent has asymptotes at regular intervals. Also
like tangents, the period of cotangent is π. In Figure 16-14, we show you one period of
the parent cotangent graph. You can get more information about graphing cotangents
in Chapter 7.
Figure 16-14:
The graph of
the parent
function
y = cotθ.
π
4
π
2
3π
4
π
307
308
Part V: The Part of Tens
Chapter 17
Ten Pitfalls to Pass Up in Pre-Calc
In This Chapter
䊳 Avoiding some common mistakes that pre-calc students make
䊳 Reviewing some relevant pre-calc rules
O
ne of our favorite video games from childhood (we know this dates us, but . . .) featured a little guy with a square head running through the square jungle, swinging on
square vines, and jumping over square alligators in square swamps. He was avoiding the pitfalls of the jungle. Consider this chapter the vine you can use to jump over the pitfalls that
normally trip up the pre-calc student. And you don’t have to be a video game geek to understand this chapter!
Going Out of Order (Of Operations)
Operations in an expression or an equation aren’t meant to be done from left to right. For
example, 3 – 7(x – 2) doesn’t equal –4(x – 2) or –4x + 8. You’re supposed to do multiplication
first, which means distributing the –7 first: 3 – 7x + 14. Now combine like terms to get –7x + 17.
Remember your order of operations (PEMDAS) all the time, every time:
Parentheses (and other grouping devices)
Exponents
Multiplication and Division, from left to right in order as you find them
Addition and Subtraction, also from left to right
To further review the order of operations, see Chapter 1.
FOILed Again! FOILing Binomials Incorrectly
When multiplying binomials, always remember to multiply them in the correct order. You
remember FOIL — First, Outside, Inside, Last. This includes squaring any binomial. The
biggest mistake we see in these situations is something like: (x – 4)2 = x2 + 16. That’s forgetting a whole lot of multiplying, though. It should look like this: (x – 4)2 = (x – 4)(x – 4) =
x2 – 4x – 4x + 16 = x2 – 8x + 16.
310
Part V: The Part of Tens
Gettin’ Down with Denominators
Don’t fall for a big trap and break a fraction up incorrectly.
doesn’t equal
.
If you don’t believe us, pick a value for x and plug it into both equations and see if you
get the same answer twice. You won’t, because it doesn’t work. You’re doing the order
of operations wrong if you change
to
(see the section “Going Out of Order
(Of Operations),” earlier in this chapter). The division bar is a grouping symbol, and
you have to simplify the numerator and denominator separately before doing the division.
is simply
, it doesn’t simplify. However, you may come across a
case where a fraction does simplify. For example,
simplifies to
because 3 goes into every term.
Combining Terms That Can’t Be Combined
Yet another mistake we see frequently is students combining terms that aren’t meant
to be combined.
4x – 1 suddenly becomes 3x, which it’s not. 4x – 1 is simplified, meaning that it’s an
equation that doesn’t contain any like terms. 3a4b5 + 2a5b4 is also simplified. Those
exponents are close, but close only counts in horseshoes and hand grenades. When
counting in the real world (as opposed to the algebra one you probably feel stuck in
now), you can’t combine apples and bananas. Four apples plus three bananas is still
four apples and three bananas. It’s the same in algebra: 4a + 3b is simplified.
Forgetting to Flip the Fraction
When dealing with complex fractions, don’t take all the rules that you’ve learned and
throw them out the window.
doesn’t become
. If you think it does,
you’re probably forgetting that a division bar is division.
=
. To divide a fraction, you must multiply by its reciprocal.
=
.
Chapter 17: Ten Pitfalls to Pass Up in Pre-Calc
Losing the Negative (Sign)
We know that in life you’re not supposed to be negative, but in math, don’t throw away
a negative sign — especially when subtracting polynomials.
(4x3 – 6x + 3) – (3x3 – 2x + 4) isn’t the same thing as 4x3 – 6x + 3 – 3x3 – 2x + 4. If you did
it that way, you didn’t subtract the whole second polynomial, only its first term. The
right way to do it is 4x3 – 6x + 3 – 3x3 + 2x – 4, which simplifies to x3 – 4x – 1.
Similarly, when subtracting rational functions, take care of that negative sign.
. What happened? You forgot to subtract the whole
second polynomial on the top. Instead, you should’ve done this:
.
Oversimplifying Roots
When it comes to roots, we’ve seen all kinds of errors. For instance,
suddenly
becomes 3 in a problem and loses the root altogether. Or
becomes
, losing the index on the root.
Don’t add or subtract roots that aren’t like terms, either.
ever. They’re not like terms, so you can’t add them.
isn’t
, now or
is it — it’s done.
Avoiding Exponent Errors
When multiplying monomials, don’t multiply the exponents. We’ve seen students
who’ve been dealing with exponents for a long time suddenly do something like this:
x4 · x3 = x12. Also, when dealing with a power over a product, you must apply the power
to everything. (2x5y)3 isn’t 2x5y3 or 2x15y3. You must raise everything inside the parentheses to the third power, so the answer should be 8x15y3.
Watch out when dealing with negatives and exponents in some calculators. –42 and
(–4)2 represent –16 and 16 respectively, so be sure you know which configuration
you’re looking for when you punch it into your calculator.
311
312
Part V: The Part of Tens
Canceling Too Quickly
You can cancel terms when adding or subtracting if you have two terms that are exact
opposites of each other. In multiplication and division, you can cancel terms if one
common factor divides into all terms.
Here are the most common canceling mistakes we see:
⻬ Canceling constants. If you see the rational expression
, it doesn’t equal
3x – 2. The 5 in the denominator has to divide into both terms on the top.
⻬ Canceling variables. For the same reason as the preceding bullet,
isn’t 4x – 3x + 2 because the denominator doesn’t divide into everything on the
top, only the first term.
⻬ Canceling everything. To keep you aware,
isn’t
, which would be creating a constant out of two
polynomials dividing.
Dealing with Distribution
When a polynomial is multiplied by a monomial, the process is known as distribution.
Think of it like delivering the newspaper to every house on the block.
We’ve also seen students who don’t distribute to every term, especially when the polynomial gets long.
3(2x5 – 6x4 + 3x3 – x2 + 7x – 1) isn’t equal to
6x5 – 6x4 + 3x3 – x2 + 7x – 1,
or 6x5 – 18x4 + 3x3 – x2 + 7x – 1,
or even 6x5 – 18x4 + 9x3 – x2 + 7x – 1,
or anything other than 6x5 – 18x4 + 9x3 – 3x2 + 21x – 3.
Index
•A•
AAS, Law of Sines, 177–178
absolute value
defined, 10, 21
inequalities, 26
parent graphs, 43
absolute value bars, 207
absolute value functions, 44, 303
absolute value marks, 9
Addition law, 292
additive identity, 8
ambiguous case, Law of Sines, 179–180
amplitude, 45, 127
angle of depression, 110
angle of elevation, 110
answer key
advanced identities, 170–176
basic trig identities, 153–159
binomials, 283–286
calculus, 296–298
complex numbers, 207–210
conic sections, 235–242
exponential functions, 100–104
functions, 62–71
fundamentals, 20–24
logarithmic functions, 100–104
oblique triangles, 186–192
polar coordinates, 207–210
real numbers, 36–39
roots, 86–90
sequences, 283–286
series, 283–286
systems of equations, 264–274
transforming trig functions, 137–142
the unit circle, 121–126
answer matrix, 257
Archimedean spiral polar equation, 204
arcs, 119–120
area, 184–185
arithmetic sequences, 277–278
arithmetic series, 277–278
ASA, Law of Sines, 177–178
associative property of addition, 8
associative property of multiplication, 8
asymptote
defined, 222
graphing rational functions, 304
horizontal, 50–51
oblique, 50
vertical, 50–51, 134
augmented form, 258
axis of symmetry, 214
•B•
back substitution, 260
binomial expansion, 281
binomial theorem, 281–282
binomials
answer key, 283–286
FOILing incorrectly, 309
guess and check method, 74
British method, 74
•C•
calculators, 311
calculus
answer key, 296–298
determining continuity, 293–295
finding limits
algebraically, 290–291
analytically, 289–290
graphically, 288–289
overview, 287–288
limit laws, 292–293
canceling too quickly, 312
cardioid polar equation, 204
Cartesian coordinate system, 12
center, 212
circle polar equation, 204
closed interval, 28
coefficient matrix, 257
co-function identity, 149
combinations formula, 281
combinations of transformations, 47–50
combining functions, 54
common difference, 277–278
common ratio, 279
314
Pre-Calculus Workbook For Dummies
commutative property of addition, 8
commutative property of multiplication, 8
completing the square, 63, 76, 211
complex conjugate, 78
complex coordinate plane, 195
complex fraction
flipping, 310
simplifying denominator, 158
complex numbers
answer key, 207–210
performing operations with and graphing,
195–198
complex rational function, 291
compound interest formula, 98
conic sections
answer key, 235–242
changing from parametric form to rectangular
form, 232–233
circles, 212–213
graphing ellipses
horizontal, 219–220
overview, 218–219
vertical, 220–222
graphing hyperbolas
horizontal, 223–225
overview, 222–223
vertical, 225–227
identifying, 227–230
overview, 211
parabolas
horizontal, 216–218
overview, 213–214
vertical, 214–216
parametric form for, 230–232
on polar coordinate plane, 233–234
conjugate
complex, 78
complex number, 196
conjugate axis, 222
constants
canceling, 312
defined, 73
ellipses, 218
irrational, 91
continuity, determining, 293–295
continuous compound interest formula, 98
conventions used in book, 2
coordinate plane
defined, 41
unit circle, 112–114
cosecant
reciprocal of sine, 306
secant, 134–136
symbol, 107
cosine
double angle identities, 163
parent graphs, 128–130, 305
symbol, 107
cotangent
parent graphs, 306–307
symbol, 107
vertical asymptote, 131–133
co-terminal angle, 112
co-vertices, 218
Cramer’s Rule, 262–263
cube root
parent graphs, 302–303
simplifying radicals and exponents, 30
cube root functions, 45
cubic function
overview, 44–45
parent graphs, 302
•D•
decomposing partial fraction, 253–254
degrees, converting to radians, 107
denominator
defined, 30
lowest common, 151, 291
overview, 310
rationalizing, 33
depressed polynomial, 79
Descartes’ Rule of Signs, 78, 80
diagonals, 262
difference identity, 166
directrix, 214
discontinuous piece-wise function, 53
discriminant, 78
disjointed sets, 28
distance, 15
distribution, 312
Division law, 292
does not exist (DNE), 290
domain
functions, 41
graphing equalities and inequalities, 12
range, 58–59
double angle identity, 163–164
Index
•E•
eccentricity, 233
element, 255
elementary row operations, 258
elimination method, 245–246
ellipses
horizontal, 219–220
overview, 218–219
vertical, 220–222
equalities, solving, 10–12
even function, 41
even-odd identity, 148
excluded value, 58
exponent
errors, 311
simplifying, 30–32
exponential functions
answer key, 100–104
continuity, 293
solving, 91–93
solving equations using, 96–97
word problems, 98–99
exponential graphs, 304–305
extraneous solution, 21
•F•
factor theorem, 80
factorable polynomial, 73–75
factorial, 275
factoring
functions, 290
overview, 74
proofs, 151
flatness, figuring, 184–185
focus, 214
FOIL method backwards, 309
FOILing binomials incorrectly, 309
fractions
complex
flipping, 310
simplifying denominator, 158
decomposing partial, 253–254
forgetting to flip, 310
proofs, 151
functions
answer key, 62–71
defined, 41
domain and range, 58–59
evaluating composition of, 56–57
exponential
answer key, 100–104
continuity, 293
solving, 91–93
solving equations using, 96–97
word problems, 98–99
inverse of, 60–61
logarithmic
answer key, 100–104
continuity, 293
solving equations using, 93–97
operating on, 54–55
parent graphs
absolute value functions, 44
combinations of transformations, 47–50
cube root functions, 45
cubic functions, 44–45
horizontal transformations, 46
quadratic functions, 43
reflections, 47
square root functions, 44
translations, 46–47
vertical transformations, 45–46
piece-wise, 53–54
rational, 50–52
symmetry, 41–42
transforming trig
answer key, 137–142
parent graphs, 128–130
period graphs, 127–128
secant and cosecant, 134–136
tangent and cotangent, 131–133
Fundamental Theorem of Algebra, 78
fundamentals
answer key, 20–24
equalities, solving, 10–12
graphs
calculating midpoint using, 16
creating using plug and chug method, 13
creating using slope-intercept form, 13–15
finding distance using, 15
finding slope using, 16–19
overview, 12
inequalities
expressing solutions in interval notations,
28–29
solving, 25–27
order of operations, 7–10
rationalizing, 33–35
simplifying exponents and radicals, 30–32
315
316
Pre-Calculus Workbook For Dummies
•G•
Gaussian elimination, 260
GCF (greatest common factor)
factoring, 151
overview, 74
geometric sequences, 279–280
geometric series, 279–280
graphing
calculating midpoint using, 16
ellipses
horizontal, 219–220
overview, 218–219
vertical, 220–222
finding distance using, 15
finding slope using, 16–19
hyperbolas
horizontal, 223–225
overview, 222–223
vertical, 225–227
overview, 12
parent graphs
absolute value functions, 44, 303
combinations of transformations, 47–50
cosecant and secant, 134, 306
cube root functions, 45
cubic functions, 44–45
cubics, 302
exponential and logarithmic, 304–305
horizontal transformations, 46
quadratic functions, 43
quadratics, 301
rational functions, 304
reflections, 47
sine and cosine, 305
square root functions, 44
square roots and cube roots, 302–303
tangent and cotangent, 306–307
transforming trig functions, 127–142
translations, 46–47
vertical transformations, 45–46
period graphs, 127–128
piece-wise functions, 53–54
plug and chug method, 13
polar coordinates, 199–201
polar equations
Archimedean spiral, 204
cardioid, 204
circle, 204
lemniscate, 204
limaçon, 205–206
rose, 204
polynomials, 82–85
rational functions, 50–52
slope-intercept, 13–15
systems of inequalities, 251–253
greater than or equal to symbol, 25
greater than symbol, 25
greatest common factor (GCF)
factoring, 151
overview, 74
guess and check method, 74
•H•
half-angle identity, 165–166
half-life of radioactive element formula, 98
Heron’s Formula, 185
high order polynomial
Descartes’ Rule of Signs, 78
imaginary roots, 78–79
rational roots, 79
synthetic division, 79–81
horizontal asymptote, 50–51
horizontal ellipse, 219–220
horizontal hyperbolas, 223–225
horizontal parabolas, 216–218
horizontal shrink, 216
horizontal stretch, 216
horizontal transformations
overview, 46
trig graphs, 127–128
hyperbolas
horizontal, 223–225
overview, 222–223
vertical, 225–227
•I•
icons used in book, 3–4
identities
additive, 8
co-function, 149
difference, 166
double angle, 163–164
even-odd, 148
half-angle, 165–166
multiplicative, 8
periodicity, 150
product to sum, 166
Pythagorean, 147
Index
ratio
reciprocal identities and, 145
to show expression in terms of sine and
cosine, 153
reciprocal, 145–146
sum and difference, 161–163
trig
advanced, 161–176
basic, 145–159
imaginary number, 195
imaginary roots, 78–79, 80
inequalities
expressing solutions in interval notations,
28–29
graphing, 14
solving, 25–27
initial side, x-axis, 112
intercepts, 13
interval notations, 28–29
inverse matrices, 261–262
inverse of functions, 60–61
irrational constant, 91
•L•
Law of Cosines, 180–181
Law of Sines
ambiguous case, 179–180
ASA and AAS, 177–178
LCD (lowest common denominator)
finding limits algebraically, 291
trig proofs, 151
leading coefficient
defined, 73
simplifying matrices, 258
leading coefficient test, 83
left limit, 288
lemniscate polar equation, 204
less than or equal to symbol, 25
less than symbol, 25
limaçon polar equation, 205–206
limits, finding
algebraically
factoring, 290
finding lowest common denominator, 291
plugging in, 290
rationalizing numerator, 291
analytically, 289–290
graphically, 288–289
overview, 287–288
linear equations, solving with two variables
elimination method, 245–246
substitution method, 244–245
linear term, 73
logarithmic functions
answer key, 100–104
continuity, 293
solving equations using
overview, 93–96
exponents and logs, 96–97
logarithmic graphs, 304–305
lowest common denominator (LCD)
finding limits algebraically, 291
trig proofs, 151
•M•
major axis, 218
matrices
overview, 255–257
simplifying, 257–259
solving systems of equations using
Cramer’s Rule, 262–263
Gaussian elimination, 260
inverse matrices, 261–262
overview, 259
methods
British, 74
elimination, 245–246
guess and check, 74
plug and chug
linear equations, 12
overview, 13
parametric equations, 231
parent graphs, 43
substitution, 244–245
midpoint, calculating, 16
minor axis, 218
monomials, 33
Multiplication law, 292
multiplicative identity, 8
multiplicity, roots, 79
•N•
negative coefficient, 46
negative measure, 112
negative numbers, 26
negative root, 78
317
318
Pre-Calculus Workbook For Dummies
negative sign, 311
negative slope, 16
nonlinear systems, solving
one linear equation and one not, 247
systems of equations disguised as rational
equations, 248–249
two equations, 248
nonremovable discontinuity, 293
numbers, properties of, 41
numerator, 30
•O•
oblique asymptote, 50
oblique triangles
ambiguous case, SSA, 179–180
answer key, 186–192
figuring flatness, 184–185
Law of Cosines, 180–181
Law of Sines, 177–178
to solve word problems, 182–184
odd function, 41
open interval, 28
order of matrix, 255
order of operations, 7–10, 309
ordered pair, 41
origin symmetry, 41
oversimplifying roots, 311
•P•
parabolas
horizontal, 216–218
overview, 213–214
quadratic functions, 43
vertical, 214–216
parameter value, 230
parametric form
changing to rectangular form, 232–233
for conic sections, 230–232
parent graphs
absolute value functions, 44, 303
combinations of transformations, 47–50
cosecant and secant, 134, 306
cube root functions, 45
cubic functions, 44–45
cubics, 302
exponential and logarithmic, 304–305
horizontal transformations, 46
quadratic functions, 43
quadratics, 301
rational functions, 304
reflections, 47
sine and cosine, 305
square root functions, 44
square roots and cube roots, 302–303
tangent and cotangent, 306–307
transforming trig functions, 128–130
translations, 46–47
vertical transformations, 45–46
parent square root function, 44
partial fractions, decomposing, 253
PEMDAS order of operations, 8, 309
period graph, 127–128
periodicity identity, 150
piece-wise function, 53–54
plug and chug method
linear equations, 12
overview, 13
parametric equation, 231
parent graphs, 43
point-in-plane, 112
polar coordinate plane, 233–234
polar coordinates
answer key, 207–210
changing to and from, 202–203
graphing, 199–201
polar equations, graphing
Archimedean spiral, 204
cardioid, 204
circle, 204
lemniscate, 204
limaçon, 205–206
rose, 204
polar equations
Archimedean spiral, 204
cardioid, 204
circle, 204
lemniscate, 204
limaçon, 205–206
rose, 204
pole, 199
polynomial
depressed, 79
expanding using binomial theorem, 281–282
factorable, 73–75
find equations with, 81–82
graphing, 82–85
Index
high order
Descartes’ Rule of Signs, 78
imaginary roots, 78–79
rational roots, 79
synthetic division, 79–81
negative sign, 311
quadratic
completing square, 76
graphing, 213
quadratic formula, 76–77
polynomial functions, 293
positive measure, 112
positive root, 78
positive slope, 16
Power law, 292
power-reducing formulas, 169
powers, 30. See also exponent
Pre-calc rules icon, 3
product to sum identity, 166
products
changing to sums, 166–167
expressing sums as, 168
logs, 96
properties of equality, 8
properties of numbers, 41
Pythagorean identity, 147
Pythagorean Theorem, 108
•Q•
quadratic equation
parent graphs, 301
standard form, 73
quadratic functions, 43
quadratic polynomial
completing the square, 76
graphing, 213
quadratic formula, 76–77
quadratic term, 73
quotients, 96
•R•
radians, converting to degrees, 107
radicals. See also roots
order of operations, 9
simplifying, 30–32
radicand, 58
radius, 212
range, 41, 58–59
ratio identity
reciprocal identities, 145
to show expression in terms of sine and
cosine, 153
rational exponent, 30
rational functions
cosecant and secant, 134
graphing, 50–52
negative sign, 311
overview, 58
parent graphs, 304
rational root, 79
Rational Root Theorem, 79–80
rationalizing
denominator, 33–35
numerator, 291
ratios
right triangle, 115–117
trigonometric, 107–109
reciprocal identity, 145–146
reciprocals, 306
rectangular coordinate system, 199
rectangular form
changing parametric form to, 232–233
conics, 230
recursive sequences, 275
reduced row echelon form, 258
reference angle, 115
reflections, 47
reflexive property, 8
relation, 12
Remember icon, 4
removable discontinuity, 293
right limit, 288
right triangles
finding ratios on unit circle, 115–117
solving word problems with, 110–112
roots
answer key, 86–90
factorable polynomials, 73–75
finding equations with, 81–82
graphing polynomials, 82–85
high order polynomials
Descartes’ Rule of Signs, 78
imaginary roots, 78–79
rational roots, 79
synthetic division, 79–81
oversimplifying, 311
319
320
Pre-Calculus Workbook For Dummies
roots (continued)
quadratic polynomials
completing the square, 76
quadratic formula, 76–77
simplifying, 30
rose polar equation, 204
row echelon form, 258
•S•
SAS, Law of Cosines, 180–181
scalar multiplication operation, 255
secant
cosecant, 134–136
parent graphs, 306
symbol, 107
second root, 30
sequences
answer key, 283–286
arithmetic, 277–278
calculating terms, 275–277
simplifying geometric sequences, 279–280
series
answer key, 283–286
arithmetic, 277–278
calculating terms, 275–277
simplifying geometric series, 279–280
shift, 46
shrink, 45
simplifying expressions, 8
sine
cosine, 128–130
parent graphs, 305
symbol, 107
sinusoids, 128
slope, 16–19
slope-intercept graph form, 13–15
SOHCAHTOA acronym, 108
solving the triangle, 177
square root
parent graphs, 302–303
proofs and, 151
simplifying, 30
square root function, 44, 58
SSA, ambiguous case, 179–180
SSS, Law of Cosines, 180–181
standard form
defined, 73
elimination method, 245
standard position, 112
stretch, 45
substitution method, 244–245
Subtraction law, 292
sum and difference identities, 161–163
sums
changing products to, 166–167
expressing as products, 168
symmetric property, 8
symmetry, 41–42
synthetic division, 79–81
systems of equations
answer key, 264–274
decomposing partial fractions, 253–254
graphing systems of inequalities, 251–253
matrices
overview, 255–257
simplifying, 257–259
solving more than two, 249–251
solving nonlinear systems
one linear equation and one not, 247
systems of equations disguised as rational
equations, 248–249
two equations that are nonlinear, 248
solving two linear equations with two variables
elimination method, 245–246
substitution method, 244–245
solving using matrices
Cramer’s Rule, 262–263
Gaussian elimination, 260
inverse matrices, 261–262
overview, 259
technique overview, 243–244
systems of inequalities, 251–253
•T•
tangent
parent graphs, 306–307
symbol, 107
vertical asymptote, 131–133
term of the sequence, 275
terminal side, 112
theorems
binomial, 281–282
factor, 80
Fundamental Theorem of Algebra, 78
Pythagorean Theorem, 108
Rational Root Theorem, 79–80
third root, 30, 302–303
Tip icon, 3
transformations
absolute value functions, 44
combinations of, 47–50
cube root functions, 45
cubic functions, 44–45
Index
exponential equation with base, 2, 92
horizontal, 46, 127–128
logarithmic equation with base, 2, 95
quadratic functions, 43
reflections, 47
square root functions, 44
translations, 46–47
vertical, 45–46, 127–128
transforming trig functions
answer key, 137–142
parent graphs, 128–130
period graphs, 127–128
secant and cosecant, 134–136
tangent and cotangent, 131–133
transitive property, 8
translations, 46–47
transverse axis, 222
triangles
oblique
ambiguous case, 179–180
answer key, 186–192
figuring flatness, 184–185
Law of Cosines, 180–181
Law of Sines, 177–178
to solve word problems, 182–184
right
finding ratios on unit circle, 115–117
solving word problems with, 110–112
trig graphs, 127–128
trig identities
advanced
answer key, 170–176
changing products to sums, 166–167
double angle identities, 163–164
expressing sums as products, 168
half-angle identities, 165–166
power-reducing formulas, 169
sum and difference identities, 161–163
basic
answer key, 153–159
co-function identities, 149
even-odd identities, 148
periodicity identities, 150
Pythagorean identities, 147
reciprocal identities, 145–146
trig proofs, 151–152
trigonometry
answer key, 121–126
arcs, 119–120
ratios, 107–109
solving equations, 117–119
solving word problems with right triangles,
110–112
unit circle
coordinate plane and, 112–114
finding right triangle ratios on, 115–117
trinomial, 74
•U•
undefined function, 50
undefined value, 58
union symbol, 28
unit circle
answer key, 121–126
coordinate plane, 112–114
finding right triangle ratios on, 115–117
•V•
variable matrix, 257
variables
canceling, 312
solving two linear equations with two variables
elimination method, 245–246
substitution method, 244–245
vertex, 43, 214, 218
vertical asymptote
graphing rational functions, 50–51
tangent and cotangent, 131
vertical ellipse, 220–222
vertical hyperbolas, 225–227
vertical parabolas, 214–216
vertical transformations
overview, 45–46
trig graphs, 127–128
•W•
Warning icon, 4
word problems
exponential, 98–99
solving with oblique triangles, 182–184
solving with right triangles, 110–112
•X•
X-axis symmetry, 41
•Y•
Y-axis symmetry, 41
321
Notes
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