How to Use Advanced Algebra II

How to Use Advanced Algebra II
How to Use Advanced Algebra II
Over a period of time, I have developed a set of in-class assignments,
homeworks, and lesson plans, that work for me and for other people who have
tried them. If I give you the in-class assignments and the homeworks, but not
the lesson plans, you only have ⅔ of the story; and it may not make sense
without the other third. So instead, I am giving you everything: the in-class
assignments and the homeworks (the Homework and Activities book), the detailed
explanations of all the concepts (the Conceptual Explanations book), and the lesson
plans (the Teacher's Guide). Once you read them over, you will know exactly
what I have done.
Homework and Activities
The Homework and Activities book is the main text of Advanced Algebra II. It
consists of a series of worksheets, some of which are intended to be used in class
as group activities, and some intended to be used as homework assignments.
Conceptual Explanations
The Conceptual Explanations book serves as a complement to the activities portion
of the course. It is intended for students to read on their own to refresh or clarify
what they learned in class.
Teacher's Guide
The Teacher’s Guide is not an answer key for the homework problems: rather, it is
a day-by-day guide to help the teacher understand how the author envisions the
materials being used.
Instructors should note that this book probably contains more information than
you will be able to cover in a single school year. I myself do not teach from every
chapter in my own classes, but have chosen to include these additional materials
to assist you in meeting your own needs. As you will likely need to cut some
sections from the book, I strongly recommend that you spend time early on to
determine which modules are most important for your state requirements and
personal teaching style.
Please also note that these materials are all available at no cost on the Connexions
website (http://cnx.org/). Instructors wishing to modify or customize these
texts to meet their needs are free to do so under the terms of the Creative
Commons Attribution license, and can take advantage of the Connexions
platform to remix and publish these derivative works. You can also use the site
to rate these books and/or provide feedback.
I hope you enjoy using Advanced Algebra II.
Advanced Algebra II: Activities and
Homework
By:
Kenny Felder
Advanced Algebra II: Activities and
Homework
By:
Kenny Felder
Online:
<http://cnx.org/content/col10686/1.3/ >
CONNEXIONS
Rice University, Houston, Texas
©2008 Kenny Felder
This selection and arrangement of content is licensed under the Creative Commons Attribution License:
http://creativecommons.org/licenses/by/2.0/
Table of Contents
The Philosophical Introduction No One Reads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Functions
1.1 The Function Game: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Function Game: Leader's Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The Function Game: Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Homework: The Function Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Homework: Functions in the Real World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Algebraic Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Homework: Algebraic Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Homework: Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9 Horizontal and Vertical Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 Homework: Horizontal and Vertical Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 Sample Test: Function I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.12 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.13 Homework: Graphing Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.14 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.15 Homework: Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.16 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.17 Homework: Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.18 TAPPS Exercise: How Do I Solve That For y? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.19 Sample Test: Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
2 Inequalities and Absolute Values
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Homework: Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Inequality Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Homework: Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Homework: Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Graphing Inequalities and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Homework: Graphing Inequalities and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Sample Test: Inequalities and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
3 Simultaneous Equations
3.1
3.2
3.3
3.4
3.5
3.6
Distance, Rate, and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Homework: Simultaneous Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Homework: Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
The Generic Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Sample Test: 2 Equations and 2 Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
4 Quadratics
4.1
4.2
4.3
4.4
Multiplying Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Homework: Multiplying Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Homework: Factoring Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
iv
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
Introduction to Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Homework: Introduction to Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Homework: Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
The Generic Quadratic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Homework: Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Sample Test: Quadratic Equations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Graphing Quadratic Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Homework: Graphing Quadratic Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Solving Problems by Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Homework: Solving Problems by Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Homework: Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Sample Test: Quadratics II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
5 Exponents
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Homework: Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Extending the Idea of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Homework: Extending the Idea of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Homework: Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Real Life Exponential Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Homework: Real life exponential curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Sample Test: Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
6 Logarithms
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Introduction to Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Homework: Logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Homework: Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Using the Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
So What Are Logarithms Good For, Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Homework: What Are Logarithms Good For, Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Sample Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
7 Rational Expressions
7.1
7.2
7.3
7.4
7.5
7.6
Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Homework: Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Homework: Rational Expressions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Dividing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Sample Test: Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
8 Radicals
8.1
8.2
8.3
8.4
Radicals (aka* Roots) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Radicals and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Some Very Important Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Simplifying Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
v
8.5
8.6
8.7
8.8
8.9
8.10
Homework: Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A Bunch of Other Stu About Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Homework: A Bunch of Other Stu About Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Homework: Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Sample Test: Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 127
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
9 Imaginary Numbers
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Homework: Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Homework: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Me, Myself, and the Square Root of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
The Many Merry Cube Roots of -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Homework: Quadratic Equations and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 139
Sample Test: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
10 Matrices
10.1 Introduction to Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
10.2 Homework: Introduction to Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
10.3 Multiplying Matrices I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.4 Homework: Multiplying Matrices I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.5 Multiplying Matrices II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.6 Homework: Multiplying Matrices II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
10.7 The Identity and Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.8 Homework: The Identity and Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.9 The Inverse of the Generic 2x2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.10 Using Matrices for Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.11 Homework: Using Matrices for Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.12 Sample Test : Matrices I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.13 Homework: Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.14 Homework: Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10.15 Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.16 Homework: Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.17 Sample Test: Matrices II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
11 Modeling Data with Functions
11.1
11.2
11.3
11.4
11.5
11.6
11.7
Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Homework: Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Homework: Direct and Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
From Data Points to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Homework: From Data Points to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 175
Homework: Calculator Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Sample Test: Modeling Data with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Conics
12.1
12.2
12.3
12.4
??
Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Homework: Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
All the Points Equidistant from a Given Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Homework: Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
vi
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
12.14
All the Points Equidistant from a Point and a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Homework: Vertical and Horizontal Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Parabolas: From Denition to Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Sample Test: Distance, Circles, and Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Distance to this point plus distance to that point is constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Homework: Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
The Ellipse: From Denition to Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Distance to this point minus distance to that point is constant . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Homework: Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Sample Test: Conics 2 (Ellipses and Hyperbolas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
13 Sequences and Series
13.1
13.2
13.3
13.4
13.5
13.6
Arithmetic and Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Homework: Arithmetic and Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Homework: Series and Series Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Homework: Arithmetic and Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Homework: Proof by Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Sample Test: Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
14 Probability
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
How Many Groups? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Homework: Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Introduction to Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Homework: The Multiplication Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Homework: Trickier Probability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Homework: Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Homework: Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Sample Test: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
??
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Attributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
The Philosophical Introduction No One
Reads
1
(*but it's real short so please read it anyway)
Welcome to Advanced Algebra II at Raleigh Charter High School! There are three keys to succeeding in
this math class.
1. Do the homework
2. Ask questions in class if you don't understand anything.
3. Focus on
understanding, not just doing the problem right. (Hint: you understand something
when you say Gosh, that makes sense! I should have thought of that myself !)
Here's how it works. The teacher gets up and explains something, and you listen, and it makes sense, and
you get it. You work a few problems in class. Then you go home, stare at a problem that looks exactly like
the one the teacher put up on the board, and realize you have no idea how to do it. How did that happen?
It looked so simple when the teacher did it! Hmm. . ..
So, you dig through your notes, or the book, or you call your friend, or you just try something, and
you try something else, and eventually. . .ta-da! You get the answer! Hooray! Now, you have learned the
concept. You didn't learn it in class, you learned it when you gured out how to do it.
Or, let's rewind time a bit. You dig through your notes, you just try something, and eventually. . .nothing.
You still can't get it.
That's OK! Come in the next day and say I couldn't get it. This time, when the
experience: So that's why I couldn't get it to
teacher explains how to do it, you will have that Aha!
work!
Either way, you win. But if you don't do the homework, then even if the teacher explains the exact same
thing in class the next day, it won't help. . .any more than it helped the previous day.
The materials in this course-pack were originally developed for Mr. Felder's Advanced Algebra II classes
in the 2001-2002 school year. Every single student in those classes got an A or a B on the North Carolina
End of Course test at the end of the year. You can too! Do your homework, ask questions in class, and
always keep your focus on real understanding. The rest will take care of itself.
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content is available online at <http://cnx.org/content/m19111/1.2/>.
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2
Chapter 1
Functions
1.1 The Function Game: Introduction
1
Each group has three people. Designate one person as the Leader and one person as the Recorder. (These
roles will rotate through all three people.) At any given time, the Leader is looking at a sheet with a list of
functions, or formulas; the Recorder is looking at the answer sheet. Here's how it works.
•
•
•
•
•
•
One of the two players who is
not the Leader says a number.
The Leader does the formula (silently), comes up with another number, and says it.
The Recorder writes down both numbers, in parentheses, separated by a comma. (Like a point.)
Keep doing this until someone guesses the formula. (If someone guesses incorrectly, just keep going.)
The Recorder now writes down the formulanot in words, but as an algebraic function.
Then, move on to the next function.
Sound confusing? It's actually pretty easy. Suppose the rst formula was Add ve. One player says 4
and the Leader says 9. One player says -2 and the Leader says 3. One player says 0 and the Leader
says 5. One player says You're adding ve and the Leader says Correct. At this point, the Recorder
has written down the following:
1.
-
Points:
(4, 9) (−2, 3) (0, 5)
x+5
Answer:
Sometimes there is no possible answer for a particular number.
For instance, your function is take the
square root and someone gives you 4. Well, you can't take the square root of a negative number: 4 is
not in your domain, meaning the set of numbers you are allowed to work on. So you respond that 4 is not
in my domain.
Leader,
do not ever give away the answer!!! But everyone, feel free to ask the teacher if you need
help.
2
1.2 The Function Game: Leader's Sheet
Only the leader should look at this sheet. Leader, use a separate sheet to cover up all the
functions below the one you are doing right now. That way, when the roles rotate, you will
only have seen the ones you've done.
1. Double the number, then add six.
1 This
2 This
content is available online at <http://cnx.org/content/m19125/1.1/>.
content is available online at <http://cnx.org/content/m19126/1.1/>.
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CHAPTER 1. FUNCTIONS
4
2. Add three to the number, then double.
3. Multiply the number by 1, then add three.
4. Subtract one from the number. Then, compute one
divided by your answer.
5. Divide the number by two.
6. No matter what number you are given, always answer 3.
7. Square the number, then subtract four.
8. Cube the number.
9. Add two to the number. Also, subtract two from the original number.
Multiply these two answers.
10. Take the square root of the number. Round up to the nearest integer.
11. Add one to the number, then square.
12. Square the number, then add 1.
13. Give back the same number you were given.
14. Cube the number. Then subtract the original number from that answer.
15. Give back the
lowest prime number that is greater than or equal to the number.
16. If you are given an odd number, respond 1. If you are given an even number, respond 2. (Fractions
are not in the domain of this function.)
1.3 The Function Game: Answer Sheet
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
3 This
3
Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer Points Answer content is available online at <http://cnx.org/content/m19124/1.1/>.
5
16.
Points Answer -
4
1.4 Homework: The Function Game
Exercise 1.1
Describe in words what a
variable is, and what a function is.
There are seven functions below (numbered #2-8). For each function,
•
•
Write the same function in algebraic notation.
Generate three points from that function.
For instance, if the function were Add ve the algebraic notation would be x + 5. The three points might
be
(2, 7), (3, 8),
and
(−5, 0).
Exercise 1.2
Triple the number, then subtract six.
a. Algebraic notation:____________________
b. Three points:____________________
Exercise 1.3
Return 4, no matter what.
a. Algebraic notation:____________________
b. Three points:__________________________
Exercise 1.4
Add one. Then take the square root of the result. Then, divide
that result into two.
a. Algebraic notation:____________________
b. Three points:__________________________
Exercise 1.5
Add two to the original number. Subtract two from the original number. Then, multiply those
two answers together.
a. Algebraic notation:____________________
b. Three points:__________________________
Exercise 1.6
Subtract two, then triple.
a. Algebraic notation:____________________
b. Three points:__________________________
Exercise 1.7
Square, then subtract four.
a. Algebraic notation:____________________
b. Three points:__________________________
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content is available online at <http://cnx.org/content/m19121/1.2/>.
CHAPTER 1. FUNCTIONS
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Exercise 1.8
Add three. Then, multiply by four. Then, subtract twelve. Then, divide by the original number.
a. Algebraic notation:____________________
b. Three points:__________________________
Exercise 1.9
In some of the above cases, two functions always give the same answer, even though they are
dierent functions.
We say that these functions are equal to each other.
For instance, the
function add three and then subtract ve is equal to the function subtract two because they
always give the same answer. (Try it, if you don't believe me!) We can write this as:
x+3−5=x−2
Note that this is not an equation you can solve for
all
x
x
it is a generalization which is true for
values. It is a way of indicating that if you do the calculation on the left, and the calculation
on the right, they will always give you the same answer.
In the functions #2-8 above, there are three such pairs of equal functions. Which ones are
they? Write the algebraic equations that state their equalities (like my
x+3−5 = x−2
equation).
Exercise 1.10
Of the following sets of numbers, there is one that could not possibly have been generated by any
function whatsoever. Which set it is, and why? (No credit unless you explain why!)
a.
b.
c.
d.
e.
(3, 6) (4, 8) (−2, −4)
(6, 9) (2, 9) (−3, 9)
(1, 112) (2, −4) (3, 3)
(3, 4) (3, 9) (4, 10)
(−2, 4) (−1, 1) (0, 0) (1, 1) (2, 4)
1.5 Homework: Functions in the Real World
Exercise 1.11
¢ each.
Laura is selling doughnuts for 35
5
Each customer lls a box with however many doughnuts he
wants, and then brings the box to Laura to pay for them. Let n represent the number of doughnuts
in a box, and let
a.
b.
c.
d.
c
represent the cost of the box (in cents).
If the box has 3 doughnuts, how much does the box cost?
If
c = 245
, how much does the box cost? How many doughnuts does it have?
If a box has n doughnuts, how much does it cost?
Write a function
c (n) that gives the cost of a box, as a function of the number of doughnuts
in the box.
Exercise 1.12
Worth is doing a scientic study of grati in the downstairs boy's room. On the rst day of school,
there is no grati. On the second day, there are two drawings. On the third day, there are four
drawings. He forgets to check on the fourth day, but on the fth day, there are eight drawings. Let
d represent the day, and g represent the number of grati marks that day.
a. Fill in the following table, showing Worth's four data points.
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content is available online at <http://cnx.org/content/m19115/1.2/>.
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d (day)
g (number of grati marks)
Table 1.1
b. If this pattern keeps up, how many grati marks will there be on day 10?
c. If this pattern keeps up, on what day will there be 40 grati marks?
d. Write a function g (d)) that gives the number of grati marks as a function of the day.
Exercise 1.13
Each of the following is a set of points. Next to each one, write yes if that set of points could
have been generated by a function, and no if it could not have been generated by a
function. (You do not have to gure out what the function is. But you may want to try for funI
didn't just make up numbers randomly. . .)
a.
b.
c.
d.
e.
(1, −1) (3, −3) (−1, −1) (−3, −3) ________
(1, π) (3, π) (9, π) (π, π) ________
(1, 1) (−1, 1) (2, 4) (−2, 4) (3, 9) (−3, 9) ________
(1, 1) (1, −1) (4, 2) (4, −2) (9, 3) (9, −3) ________
(1, 1) (2, 3) (3, 6) (4, 10) ________
Exercise 1.14
f (x) = x2 + 2x + 1
a. f (2) =
b. f (−1)
=
c. f 32 =
d. f (y) =
e. f (spaghetti)
√
f. f ( x)
g. f (f (x))
=
Exercise 1.15
Make up a function that has something to do with
movies.
a. Think of a scenario where there are two numbers, one of which depends on the other. Describe
the scenario, clearly identifying the independent variable and the dependent variable.
b. Write the function that shows how the dependent variable depends on the independent variable.
c. Now, plug in an example number to show how it works.
1.6 Algebraic Generalizations
6
Exercise 1.16
a. Pick a number:_____
b. Add three:_____
c. Subtract three from your answer in part (b):_____
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CHAPTER 1. FUNCTIONS
8
d. What happened?_________________________________
e. Write an algebraic generalization to represent this rule._____
f. Is there any number for which this rule will not work?_____
Exercise 1.17
a. Pick a number:_____
b. Subtract ve:_____
c. Double your answer in part (b):_____
d. Add ten to your answer in part (c):_____
e. Divide your answer in part (d) by your original number (a):_____
f. Now, repeat that process for three dierent numbers. Record the number you started with
(a) and the number you ended up with (e).
Started With:_____ Ended With:_____ Started With:_____ Ended With:_____ Started With:_____ Ended With:_____ g. What happened?
h. Write an algebraic generalization to represent this rule.
i. Is there any number for which this rule will not work?
Exercise 1.18
Here are the rst six powers of two.
•
•
•
•
•
•
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
a. If I asked you for 27 (without a calculator), how would you get it? More generally, how do
you always get from one term in this list to the next term?________________
b. Write an algebraic generalization to represent this rule.________________
Exercise 1.19
Look at the following pairs of statements.
• 8 × 8 = 64
• 7 × 9 = 63
• 5 × 5 = 25
• 4 × 6 = 24
9
• 10 × 10 = 100
• 9 × 11 = 99
• 3×3=9
• 2×4=8
a. Based on these pairs, if I told you that
without a calculator) what
29 × 31
30 × 30 = 900,
could you tell me (immediately,
is?________________
b. Express this rulethe pattern in these numbersin words.
c. Whew! That was ugly, wasn't it? Good thing we have math. Write the algebraic generalization for this rule.________________
d. Try out this generalization with negative numbers, with zero, and with fractions.
(Show your work below, trying all three of these cases separately.) Does it always
work, or are there cases where it doesn't?
7
1.7 Homework: Algebraic Generalizations
Exercise 1.20
26 by 2, you get 27 .
x
x+1
that (2 ) (2) = 2
.
In class, we found that if you multiply
We expressed this as a general rule
If you multiply
210
by 2, you get
Now, we're going to make that rule even more general. Suppose I want to multiply
. Well,
2
5
means
2∗2∗2∗2∗2
, and
3
2
means
2∗2∗2
Figure 1.1
23 = 28
a. Using a similar drawing, demonstrate what 103 104 must be.
b. Now, write an algebraic generalization for this rule.________________
Exercise 1.21
The following statements are true.
• 3×4=4×3
• 7 × −3 = −3 × 7
• 1/2 × 8 = 8 × 1/2
7 This
times
23
. So we can write the whole thing out like
this.
25
25
211 .
content is available online at <http://cnx.org/content/m19114/1.2/>.
CHAPTER 1. FUNCTIONS
10
Write an algebraic generalization for this rule.________________
Exercise 1.22
In class, we talked about the following four pairs of statements.
• 8 × 8 = 64
• 7 × 9 = 63
• 5 × 5 = 25
• 4 × 6 = 24
• 10 × 10 = 100
• 9 × 11 = 99
• 3×3=9
• 2×4=8
a. You made an algebraic generalization about these statements: write that generalization again
below. Now, we are going to generalize it further. Let's focus on the
10 × 10
thing.
10 × 10 =
one away from 10; these numbers are, of course, 9 and
11. As we saw, 9 × 11 is 99. It is one less than 100.
b. Now, suppose we look at the two numbers that are two away from 10? Or three away? Or
four away? We get a sequence like this (ll in all the missing numbers):
100
There are two numbers that are
10 × 10 = 100
9 × 11 = 99
1 away from 10, the product is 1 less than 100
8 × 12 = ____
2 away from 10, the product is ____ less than 100
7 × 13 = ____
3 away from 10, the product is ____ less than 100
__×__=___
__ away from 10, the product is ____ less than 100
__×__=___
__ away from 10, the product is ____ less than 100
Table 1.2
c. Do you see the pattern? What would you expect to be the next sentence in this sequence?
d. Write the algebraic generalization for this rule.
e. Does that generalization work when the ___away from 10 is 0? Is a fraction? Is a negative
number? Test all three cases. (Show your work!)
1.8 Homework: Graphing
8
The following graph shows the temperature throughout the month of March.
Actually, I just made this
graph upthe numbers do not actually reect the temperature throughout the month of March. We're just
pretending, OK?
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