# COMPASS Placement Test Review Packet

COMPASS Placement Test Review Packet For preparing to take the COMPASS Placement Test East Georgia College Table of Content The COMPASS Test Test-taking Strategies Additional Resources for the Compass Placement Test 3 5 6 Reading Review I. Finding the Main Idea II. Vocabulary: Word Meaning and Context III. Supporting Details IV. Inferences V. Implied Main Ideas and Central Points Online Reading Resources 7 8 9 11 12 14 18 Writing Review I. Punctuation II. Basic Grammar and Usage III. Sentence Structure IV. Rhetorical Skills Online Grammar & Writing Resources 19 20 25 38 38 47 Math Review I. General Overview Topics Covered in EGC Math Classes Are You Prepared? II. Refreshing your Math Additional Math Review Resources 48 50 50 57 88 107 EGC COMPASS Placement Test Review Packet 2 The COMPASS Test The COMPASS placement test is offered in Reading, Writing, and Math. The test helps to determine whether you have the knowledge to succeed in the classes you are planning to take or whether taking some preparatory classes will ensure your success. Taking the three tests separately is usually helpful to ensure best results in all three tests. The COMPASS test is a self-adjusting, multiple choice test that is taken at the computer. The answer to your current question will determine the next question; it will stop once it has determined your level. Consequently the test is untimed and has a different number of questions for each student. It also means that you will see questions that you don’t know, because the test will ask you more and more difficult questions until it has found something that you don’t know. Just do your best you can for each question the test presents to you. You will receive paper, pencil, and a calculator. Since you will work on the computer you will not be allowed to bring food or drink. Once you have completed a test you will receive a printout with your scores and a recommendation of classes to take. You should then make an appointment with an advisor to discuss your course work. EGC COMPASS Placement Test Review Packet 3 The COMPASS Entrance exam can taken at the main campus in Swainsboro. Students are not required to schedule an appointment for the COMPASS Exam and is offered at the main campus in Swainsboro Monday - Thursday 8:00am-4:00pm and on Fridays 8:00am-10:00am except holidays and during Orientations. Students who are unable to test at the Swainsboro office are able to take the exam at a COMPASS Internet Remote Testing Site of their choosing. Please be aware that each institution charges a fee for this service which the student will be responsible for. To find one in your area, please visit: http://www.act.org/COMPASS/sites/index.html. Once you have located a testing site convenient for you, please contact the Office of Admissions to set up the registration. After you have taken the COMPASS Entrance exam in the any or all of the three subjects of Reading, Writing, and Algebra, you will receive a printout of your scores explaining whether or not you are exempt or placed in Learning Support Courses. Here are the locations and hours for each campus: Swainsboro Campus 131 College Circle Swainsboro, GA 30401 (478) 289-2017 Office hours: o Monday-Thursday: 8:00am - 6:00pm o Friday: 8:00am - 12:00pm EGC @ Statesboro 10449 US Highway 301 South Statesboro, GA 30458 (912) 623-2400 Office hours: o Monday to Thursday: 8:00am - 6:00pm o Friday: 8:00am - 12:00pm Testing Center website: http://www.ega.edu/admissions EGC COMPASS Placement Test Review Packet 4 Test-aking Strategies 1. Take the Placement Test Seriously Giving your best during the test can save you several terms of math, reading, and writing classes, and therefore a lot of time and money. What you don’t know, you don’t know. That’s fine. But if you know something, make sure you show it on the test so that you are placed into the appropriate class for your skill level. 2. Prepare For the Test It is important that you review your knowledge before you take the test, particularly if you have not been in school for many years. Go over the following parts in this review packet to refresh your memory about the things you once knew. This packet is not designed to help you learn material that you never knew. For that you should take a class. It is equally important, however, that you are physically prepared for the test. Be sure to get enough sleep the night before, and eat something nutritious before arriving for the test. Don’t consume anything with caffeine or a lot of sugar right before the test. Caffeine might make you feel more jittery and less patient, causing you to skip important steps. Too much sugar will give you a short energy boost followed by a sense of fatigue. Drink water or tea instead. 3. Take ȱTime The Compass Test is not timed which means that you can take as much time as you need. Make use of that! Read the questions carefully, think about them, do your work on paper, and then choose an answer. Your score does not depend on how long you take for each question. Your score only depends on whether you choose the right answer. 4. Read the Questions Carefully Don’t assume anything. Follow the instructions of the question exactly. Read all the details very carefully. A simple 'not' can change everything around. It helps to copy the question onto paper and underline the important information or rewrite it in your own words. EGC COMPASS Placement Test Review Packet 5 5. Work Math Problems out on Paper Since the COMPASS Test is a test that you take on the computer make sure to copy math problems onto paper and work them step by step. It’s worth it! Working a problem out carefully and minding all the details gets you the points to place you in the right class. 6. Take a Break You can take a break whenever you like! Just go to the testing supervisor, and s/he will save your work. You can continue when you come back. You can even come back the next day. This is very important because in order to do well on the test you need to concentrate. So if you need to use the restroom, go. If you are thirsty or hungry, go drink and eat. If you are tired, get up and take a walk or go home and come back the next day. Additional Resources for the COMPASS placement test The testing centers website: http://www.ega.edu/admissions The testing centers website for the COMPASS placement test: http://www.ega.edu/compass The COMPASS website: http://www.act.org/compass/student/index.html Sample test questions on the COMPASS website: http://www.act.org/compass/sample/index.html EGC COMPASS Placement Test Review Packet 6 Math Review – Part I On the previous page you found a flow chart of the math sequence. The placement test will determine where you enter the sequence. Your educational goal will determine where you will exit the sequence. Please see an advisor for that. In the following pages the topics are listed that are covered in each class. TOPICS covered in EGC Math Classes To be successful studying the topics covered in these courses, students should be appropriately prepared by: 1. Taking the prerequisite math course within the last three years with a passing grade of A or B, or within the last one year with a passing grade of C, or 2. Placing into the course by taking the COMPASS placement test. MTH 20 – Basic Math Fractions, Decimals, Integers addition, subtraction, multiplication, division, Order of operations Ratio and Proportion Percent percents decimals fractions Measurements Metric system American system Geometry Statistics Place value, rounding, inequalities, exponents, power of ten Prime numbers, multiples, prime factorization, least common multiples EGC COMPASS Placement Test Review Packet 49 MTH 60 – Introductory Algebra I I. Integer arithmetic a.The four basic operations of addition, subtraction, multiplication, and division b. Absolute value, exponents, order of operations II. One variable linear equations and inequalities III. Application (i.e. word/story) problems with formulas IV. Graphing lines a. Finding and interpreting slope b. Finding and interpreting intercepts c. Interpret relationships between variables d. Modeling with linear equations MTH 65 – Introductory Algebra II 1. Systems of linear equations in two variables a. Graphing method b. Substitution method c. Addition method d. Applications 2. Working with algebraic expressions a. Add, subtract, multiply, and divide by a monomial b. Factoring polynomials 3. Solving quadratic equations a. Square Root Property (includes – simplify and approximate numeric square roots) b. Factoring Property c. Quadratic Formula d. Graphing (includes – interpret vertex, axis of symmetry and vertical/horizontal intercepts) e. Applications 4. Relations and functions a. Function notation b. Evaluate EGC COMPASS Placement Test Review Packet 50 MTH 70 – Review of Introductory Algebra 1. 2. 3. 4. Solving equations A. Linear equations B. Quadratic equations C. Rational equations D. Radical equations Graphing A. Linear functions B. Quadratic functions Simplifying expressions A. Polynomial expressions B. Rational expressions Function concepts A. Domain B. Range C. Function notation D. Graph reading MTH 95 – Intermediate Algebra 1. 2. 3. 4. Applications and Modeling A. Linear functions B. Quadratic functions C. Exponential functions Graphing A. Linear functions B. Quadratic functions C. Exponential functions Solve equations and inequalities A. Symbolically B. Numerically C. Graphically Function concepts A. Domain B. Range C. Inverses D. Compositions E. Transformations EGC COMPASS Placement Test Review Packet 51 MTH 111 – College Algebra (MTH 111B or MTH 111C) 1. Graphing and solving equations and applications involving: A. Polynomial functions B. Rational functions C. Exponential functions D. Logarithmic functions 2. Functions Operations A. Inverses of functions B. Compositions of functions C. Transformations of functions MTH 112 – Elementary Functions (Trigonometry) 1. 2. 3. 4. Right triangle trigonometry Law of Sines and Law of Cosines and their applications Solutions to trigonometric equations Applications A. Vectors B. Parametric equations C. Polar coordinates and graphs D. Complex numbers MTH 211 – Foundations of Elementary Math I 1. Topics for Math 211: A. Problem solving B. Set Theory – union, intersection, complement, Venn Diagrams C. Historic Numeration Systems D. Whole Number Operations – properties, algorithms, models, non-decimal bases. E. Number Theory – divisibility, primes, GCD, LCM, modular arithmetic. EGC COMPASS Placement Test Review Packet 52 ARE YOU PREPARED? The mini quizzes on the following pages are meant to serve only as an indicator of a few of the math skills that you are expected to know at the beginning of each course. Do not use these problems as a study guide thinking that they will adequately prepare you for the course. These example problems are merely representative of some of the most important concepts that are taught in the prerequisite courses. The courses will offer little or no time for any type of review; they assume that you are prepared to do the work the first day of class. Below is a sample of some skills you should have BEFORE entering MTH 20 – Basic Math You MAY NOT use a calculator. 1. Without using a calculator, can you complete these problems in 45 seconds? 6x4 9x6 7x8 9x9 0x6 6x9 8 x 10 9x4 6x7 7x2 9x0 6x2 4x7 8x9 9x7 EGC COMPASS Placement Test Review Packet 5 6x5 8x9 8x4 3x6 8x8 12 4 56 8 72 9 40 5 36 6 NOTE: If you miss more than 5 problems, then you should consider taking the previous math course – ABE 0750 or ALC 60, 61, 62, 63. 2. Without using a calculator, can you get at least 8 correct answers on the following problems? a) 20 x 30 d) 4984 8 g) 305 x 27 b) 25 + 4 + 125 c) 872 - 431 e) 68 x 34 f) 17575 h) 5843 - 2338 i) 4590 25 15 j) 45 + 2,341 + 8 + 124 3. Without using a calculator, can you get at least 4 correct answers on the following problems? a) Find the change from a $20 bill after purchasing 2 records at $6 each, and 1 pair of earrings that cost $3. b) A computer screen consists of small rectangular dots called pixels. How many pixels are there on a screen that has 600 rows with 800 pixels in each row? c) Before going back to college, David buys 4 shirts at $59 each and 6 pairs of pants at $78 each. What is the total cost of the purchase? d) Portland community college is constructing new dorms. Each dorm room has a small kitchen. If someone buys 85 microwave ovens at $90 each, what is the total cost of the purchase? e) Hershey Chocolate USA makes small, fun-size chocolate bars. How many 20-bar packages can be filled with 8,110 bars? How many bars will be left over? EGC COMPASS Placement Test Review Packet 5 Answers QUESTION 2: a) 600 b) 154 c) 441 d) 623 e) 2,312 f) 703 g) 8,235 h) 3,505 i) 306 j) 2,518 QUESTION 3: a) $5 b) 480,000 PIXELS c) $704 d) $7,650 e) 405 Packages with 10 bars left over How many of these problems can you miss and still succeed in MTH 20? Ideally, NONE. These problems are just a sample of the larger number of skills that you should be familiar with BEFORE taking this course. If some of these ideas are not familiar to you, you should consider enrolling in one of the previous math courses (ABE 0750 or ALC 60, 61, 62, or 63). Below is a sample of some skills you should have BEFORE entering MTH 60 – Introductory Algebra I You MAY NOT use a calculator. 1.Without using a calculator, can you get at least 16 correct answers on the following problems? a) Round 6.8449 to the nearest hundredth. EGC COMPASS Placement Test Review Packet b) Round 7.995 to the nearest tenth. 5 c) Round 37,328 to the nearest hundred. d) Change 0.625 to a fraction e) Write 70% as a fraction and reduce to the lowest terms. g) Multiply: f) Change 9 2 x2 16 3 2 to a decimal. 5 h) Divide: 1 i) Find the average of 1 1 7 , 12.5, 8, 10 4 4 2 10 3 j) Perform the indicated operations. 7 3 2 10 5 l) Perform the indicated operations. 18 2(3) 2 2 5 k) Subtract: 8.3 .973 m) List these numbers from smallest to largest: n) Solve the proportion: 5 7 , , 0.555, 0.583 9 12 2.5 4 1.1 x o) How many inches equal 2 yd? p) Change 72 mg to grams. q) If 1 km is approximately 0.6 miles, how many miles in 18 km? r) Find the area of a circle whose diameter is 6 cm. s) Find the perimeter of this figure: 16 m 10 m 10m 8m EGC COMPASS Placement Test Review Packet t) Find the volume of this figure: 18 in 25 m 5 in 5 2. Without using a calculator, can you get at least 4 correct answers on the following problems? a) A family’s monthly income is $1,200. It is spent as follows: 20% on food, 35% on rent, 17% on utilities, 8% on automobile, and the rest on miscellaneous expense. What dollar amount is spent on miscellaneous expenses? b) A TV is priced to sell at $585. What is the sale price if the sale sign says ‚ c) A machinist needs a bar that is is the bar? 1 off‛? 3 3 3 in. thick. If she cuts off in. thick, how thick 8 32 d) A teacher assigns problems 96 to 128 that are multiples of 8. Which problems should the students do? e) Find the unit price if the total cost of a 5-lb. steak is $21. Answers Question 1: a) 6.84 b) 8.0 g) 3 2 m) 0.555, h) c) 37,300 1 6 I) 9 1 2 5 7 , 0.583, 9 12 p) 0.072 g q) 10.8 mi d) 5 8 e) j) 3 7 10 f) 0.4 k) 7.327 n) 1.76 l) 26 o) 72 inches r) 28. 26 cm2 s) 44 m t) 2,250 in 3 Question 2: a) $240 b) $390 d) 96, 104, 112, 120, 128 15 in. thick 32 e) $4.20 per lb c) How many of these problems can you miss and still succeed in MTH 60? Ideally, NONE. EGC COMPASS Placement Test Review Packet These problems are just a sample of the larger number of skills that you should be familiar with BEFORE taking this course. If some of these ideas are not familiar to you, you should enroll in the previous course (MTH 20 or ALC 60, 61, 62, or 63) Below is a sample of some skills you should have BEFORE entering MTH 65 – Introductory Algebra II You MAY NOT use a calculator. a) Perform the indicated operations: 18 2( 3) ( 2) 3 5 c) Simplify: (12 x 2 4 x 1) 3( 2 x 2 5 x 3) e) Solve for x: x 1 x 2 10 5 1 2 f) Solve for W: P 2L 2 W i) Given two points on a line, find the slope and indicate whether the line rises, falls, (is 3horizontal, ,4) ( or 5, is2)vertical. and 2, k) Given the slope, ( 1,4), and a point passing through write an equation in the point-slope form and slope-intercept form. EGC COMPASS Placement Test Review Packet 2 b) Evaluate 7 x x , when x d) Solve for x: 5( x 2 ) 2 3 6( x 7 ) f) Solve for x and graph on a number line. 2 6 x 2 (5 x ) h) Find the slope and the y-intercept of the 2x y 6 line when j) Write an equation for the following graph. l) Graph the inequality on a rectangular coordinate system. 4 y < x 1 3 This is a graph of Frank’s body temperature from 8 a.m. to 3 p.m. Let x represent the number of hours after 8 a.m. and y equal Frank’s temperature (in F) at time x. m) What is the y-intercept? What does this mean about Frank’s temperature at 8 a.m.? n) During which period of time is Frank’s temperature decreasing? o) Estimate Frank’s minimum temperature during the time period shown. How many hours after 8 a.m. does this occur? At what time does this occur? p) How many grams of an alloy that is 80% gold should be melted with 40 grams of an alloy that is 50% gold to produce an alloy that is 70% gold? q) Vikki has $200 to spend on clothing. She buys a skirt for $68. She would like to buy some sweaters that sell for $15.50 each. How many sweaters can she buy and stay within her budget? r) The pressure of water on an object below the surface is proportional to its distance below the surface. If a submarine experiences a pressure of 25 pounds per square inch 60 feet below the surface, how much pressure will it experience 330 feet below the surface? Answers a) g) j) b) 24 P 2L 2 W x l) y 18 3 c) 6x2 11x 8 d) x 5 h) Slope =2, y-intercept = ( 0, 6 ) e) x 2 f) x 2 i) Slope = 3, rises k) Point-slope form: y – 4 = - 2(x + 1) and slope-intercept form: y = - 2x + 2 4 x 1 3 m) The y-intercept is (0,101). At 8:00 a.m. Frank’s body temperature is at 101 F. EGC COMPASS Placement Test Review Packet n) Frank’s temperature is decreasing during the time from 8:00 a.m. to 11:00 a.m. o) Frank’s minimum temperature is 98.6 F. This occurs about 3 hours afterwards and the time would be 11:00 a.m. p) Eighty grams of an alloy that is 80% gold should be melted with 40 grams of an alloy that is 50% gold to produce an alloy that is 70% gold. q) Vikki can buy at most eight sweaters. r) A submarine will experience a pressure of 137.5 pounds per square inch 330 feet below the surface. How many of these problems can you miss and still succeed in MTH 65? Ideally, NONE. These problems are just a sample of the larger number of skills which you should be familiar with BEFORE taking this course If some of these ideas are not familiar to you, you should consider enrolling in the previous course (MTH 60 or ALC 60, 61, 62, or 63) Below is a sample of some skills you should have BEFORE entering MTH 70 – Review of Introductory Algebra Part I Work with positive and negative real numbers, fractions, and the order of operations. a) 100 4 5 b) ( 3)( 4) 3 2 4 6 c) 2 3 1 3 3 8 EGC COMPASS Placement Test Review Packet 6 Part II 1. Simplify expressions: 3(2 x 2 3xy y ) ( y a) b) 12 2( x 2) c) 27 x 2 y 5 9x 6 y 2 x2 2 xy ) 3 2. Factor: a) x 2 5 x 14 b) 6a 2 b 3 3a 2 b 3. Solve for x: 3x x 4 5 5( x 4) 4 a) b) 3x 5 y 6 0 c) x2 5x 14 0 4. Evaluate expressions: If x 3 , evaluate x 2 2x 1 5. Graph by HAND and on your GRAPHING CALCULATOR* a) 4x 3 y 12 2 y 6 x 90 x 600 b) 6. Find the equation of the line passing through 2 given points: (2, 1) ( 1, 7) 7. Solve a first-degree inequality in one variable: Given: 8 5x 3x 9, solve for x 8. Given f ( x) 3x 2 a) Evaluate f ( 2) b) Solve for x if f ( x) 2 Answers Part I a) 125 b) 3 2 EGC COMPASS Placement Test Review Packet c) 16 17 6 Part II 1. a) 7 x 11 xy 2 y b) 16 2 x 2 2. a) ( x 7)(x 2) 3. a) x 4. 5y 6 3 c) x 7, x 2 4 a) 5. 6. y b) 3a 2 b(2b 2 1) b) x 5 27 y 9 c) x 24 2x 5 8. a) f ( 2) 8 b) 7. x 1 or 8 b) x 4 3 1 8 x *Students with no graphing calculator experience should enroll concurrently in MATH 93. MATH 70 IS AN OPTIONAL COURSE CONSULT A MATH ADVISOR How many of these problems can you miss and still succeed in MTH 70? a) If you missed any of the problems in Part I you should consider enrolling in MTH 60. b) If you missed several of the problems in Part II, MTH 70 is the course for you. These topics will be reviewed in MTH 70. c) If you missed none of the problems, enroll in MTH 95. EGC COMPASS Placement Test Review Packet 6 Below is a sample of some skills you should have BEFORE entering MTH 95 – Intermediate Algebra You MAY NOT use a calculator, except where indicated. 1. Work with positive and negative real numbers, and the order of operations. 5 ( 4)( 3) 3 2 Simplify 2. Simplify expressions: 3(2 x 2 3xy y ) ( y a) b) 5 x2 2 xy ) 2 12 a b 8a 3b7 3. Expand and collect like terms: a) (3x 5)(6 x 7) b) (2 x 3) 2 4. Factor: x 2 5 x 14 a) b) 6a 2 b 3 3a 2 b 5. Solve for x: a) 3x ( x 4) 5 5( x 4) 4 b) 3x 5 y 6 0 c) x2 5x 14 0 6. Evaluate expressions: If x 3 , evaluate x 2 2 x 1 7. Graph by HAND and on your GRAPHING CALCULATOR* 4x 3 y 12 a) 2 y x 5 x 14 b) 8. Find the equation of the line passing through 2 given points: (2, 1) ( 1, 7) EGC COMPASS Placement Test Review Packet 6 9. Solve a system of equations by all of the following methods: substitution, elimination by addition (linear combinations), and graphically. 2x y 3 Given: 3x 4 y 2 10: Solve a first degree inequality in one variable: Given: 8 5x 3x 9 , solve for x Answers -2 1. 3. 2. a) 7 x 2 11 xy 2 y a) 18x 2 4. a) 6. 14 7. a) b) x 5 b) 3a 2 b(2b 2 1) 5y 6 3 c) x y 15 1 -4 -3 -2 -1 10 x 1 5 2 -1 -4 -3 -2 -1 -5 -2 -1 5 -4 -2 0 -5 8. y 10. x 4x -6 3y x 1 2 3 4 5 6 7 8 9 10 -1 0 -3 Figure 1: 2 20 2 -5 7, x b) y -6 3a 8 2b 9 b) 4 x 2 12 x 9 9 x 35 ( x 7)(x 2) 5. a) x b) -2 5 -3 0 12 Figure 2: y 9. x 2x 5 1 / 8 or 1/ 8 x2 5x 14 2, y 1 x * Students with no graphing calculator experience should enroll concurrently in MTH 93. EGC COMPASS Placement Test Review Packet 6 How many of these problems can you miss and still succeed in MTH 95? Ideally, NONE. These problems are just a sample of the larger number of skills that you should be familiar with BEFORE taking this course. If some of these ideas are not familiar to you, you should consider enrolling in one of the prerequisite courses (MTH 65 or MTH 70 or ALC 60, 61, 62, or 63). Below is a sample of some skills you should have BEFORE entering MTH 111 – College Algebra (MTH 111B or MTH 111C) 1. What is the equation of a line with slope m ( 6, 4) ? 1 which passes through the point 2 2. Write each of these inequalities using interval notation: a) 2 x 7 b) x 1 c) 5 x 3 3. Find the x-intercepts, the y-intercepts and the vertex of y the equation. x2 8 x 7 then graph 4. Simplify these exponential expressions: 2x 3 y 2 z 5 a) 8x 5 y 3 z 7 y 2 b) 2 3 1 4 5 3 5 x y x6z 1 2 5. Given the points (0,2) and (2,18), find the equation for an exponential function of the form f (t ) a b t which passes through both points. 6. Find the inverse of the function f ( x) EGC COMPASS Placement Test Review Packet 2x 5 . 6 7. Given the function y f (x) in Figure 1, find the domain and range of the function. What is the value of f (1) ? Estimate the horizontal and vertical intercepts. 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 Figure 1: y 8. Given f ( x) x 3 x and g ( x) y x 1 2 3 4 5 f x 2 3x 2 , evaluate the composition ( f g ) 5 x 9. Find the value of f (g (2)) from the table below. For the function h , which function type best describes its graph: linear; quadratic, or exponential? x f (x) g (x) h(x) 1 3 -1 0.5 2 0 3 2 3 -5 7 4.5 4 -14 11 8 EGC COMPASS Placement Test Review Packet 6 Answers 1 x 1 2 1. y 2) (a) (2,7] 3. x -intercepts: (1,0), (7,0) (b) (1, ) (c) 3,5) y -intercept: (0,7) Vertex: (4, 9) y 10 8 6 4 2 x -2 -1 -2 1 2 3 4 5 6 7 8 9 10 -4 -6 -8 -1 0 Figure 2: y 4. (a) 7. 8. 16 z x 16 4 (b) 1 8 y z 3 10 1 12 5. f (t ) ( f g) 2 5 6 8x 6. f 1 ( x) 2 3t x Domain: ( ,3] Range: ( Horizontal intercept is (1,0) x2 7 x 5 2 , 4] f (1) 2 Vertical intercept is 9. f ( g (2)) 5, (0,1.8) h is quadratic How many of these problems can you miss and still succeed in MTH 111B or MTH 111C? Ideally, NONE. These problems are just a sample of the larger number of skills that you should be familiar with BEFORE taking this course. If some of these ideas are not familiar to you, you should enroll in the prerequisite course (MATH 95). EGC COMPASS Placement Test Review Packet 7 Below is a sample of some skills you should have BEFORE entering MTH 112 – Elementary Functions (Trigonometry) You MAY use a calculator. 1. Find the inverse function for f ( x) 23 x . 2. If an initial sample of 50 mg of a radioactive substance decays to 40 mg in 235 years, find the half-life of the substance. 3. Solve ln x + ln (x 2) 3. 4. On January 1, 1995, a park ranger estimates that there are 65 wolves in a wilderness area and that the wolf population is growing at an annual rate of 2.3%. When will there be 100 wolves in the area? 5. Draw a graph of a 5th degree polynomial with a negative leading coefficient, three single zeroes and a double zero. 6. Graph g ( x) 4 x2 and label all asymptotes and intercepts. x2 9 7. Given h( x) 2 x 3 5 x 2 14 x 8 , a) find intervals where h is increasing and intervals where h is decreasing. Solve for x if h(x) 10. 17 cm 8 cm 8. Solve for x given the similar triangles shown in Figure 1. 12 cm x Figure 1: Triangles for #8 y 9. Given the function y f (x) in Figure 2, graph the following transformations: y f (x) 2 a) y f (x 3) b) y 2 f ( x) c) 5 4 3 2 1 -1 -1 x 1 2 3 4 5 6 7 8 9 -2 -3 -4 -5 Figure 2: y EGC COMPASS Placement Test Review Packet f x 10. From a common location, Car A heads north at 55 mph at the same time as Car B heads east at 45 mph. Assuming the roads are straight, how far apart are the two cars after 20 minutes? Answers 1. f 1 ( x) 3 log 2 x 2. The half life is almost 730 years ( 729.977) 3. x 5.592 4. There will be 100 wolves in December of 2013. 5. y 5 y 3 4 2 3 1 2 1 -3 -2 -1 -1 x 1 2 -5 -4 -3 -2 -1 -1 7. a) b) 8. x 3 4 5 y 1 -4 -5 -3 Figure 3: A Solution to #5 2 -3 -2 -5 x 1 -2 3 -4 2, 0 4 y-int.: 0, 9 x-int.: 4 5 x 3 Figure 4: y x 3 4 x2 x2 9 h is increasing on ( , 0.907) (2.573, ) h is decreasing on ( 0.907, 2.573) h(x) 10 when x 1.565, 0.152, or 4.216 25.5cm EGC COMPASS Placement Test Review Packet 9. y y 5 5 5 4 4 4 3 3 2 2 1 y 1 -1 -1 3 2 x 1 2 3 4 5 6 7 8 9 -1 -1 -2 -3 -4 1 x 1 2 3 4 5 6 7 8 -1 -1 9 -2 -2 -3 -3 -4 -4 Figure 5: y f x Figure 6: y 2 2 3 4 5 6 7 8 9 -5 -5 -5 x 1 f x 3 Figure 7: y 2f x 10. The cars are approximately 23.688 miles apart in 20 minutes. How many of these problems can you miss and still succeed in MTH 112? Ideally, NONE. These problems are just a sample of the larger number of skills that you should be familiar with BEFORE taking this course. If some of these ideas are not familiar to you, you should consider enrolling in one of the prerequisite courses (MTH 111B or MTH 111C) Below is a sample of some skills you should have BEFORE entering MTH 211 – Foundations of Elementary Math I You MAY use a calculator. 1. The temperature at 10:00 pm in West Yellowstone was 5 degrees below zero. By 3:00am the temperature had dropped 8 degrees. What was the temperature at 3:00am? a) 3 b) 3 c) 12 d) 13 e) 13 2. What is the equation of a line with slope (6, 4) ? a) 6 x 4 y 1 b) y 2 1 x 1 c) 2 x 4 y 2 EGC COMPASS Placement Test Review Packet 1 which passes through the point 2 2 c) y 6x 1 2 7 3. A roast is to be cooked 20 minutes per pound. If the roast weighs 6 pounds and the cook wants it to finish cooking by 5:30pm, what is the latest time he can begin cooking the roast? a) 11:30am b) 2:30pm c) 3:30pm 4. If x 2 y a) 6 5. b) 8 e) 12 c) 12% d) 20% e) 32% The acceleration A that results when force F is applied to a body of mass M can be calculated from the formula F = MA. What is the value of A if M = 1200 and F = 90,000? If a) 8. d) 10 ? c) 9 b) 8% a) 75 7. e) 4:10pm Les saved $8 on the purchase of a tire whose regular price was $40. What percent of the regular price did he save? a) 5% 6. 6, then 2 x 4 y d) 4:00pm 4 x b) 750 c) 7500 d) 1,080,000 e) 108,000,000 8, then x 1 ? 1 1 2 2 3 b) c) 1 2 d) 1 2 e) 1 Consider the problem: ‚Fr ank’s average speed riding a bicycle is 4 miles per hour less than twice Liz’s. If Frank’s average speed is 12 miles per hour, what is Liz’s average speed?‛ If s represents Liz’s average speed riding a bicycle, which of the following equations can be used to solve the problem. a) 4 2s 12 d) s b) 2s 4 12 2(12) 4 e) s EGC COMPASS Placement Test Review Packet c) 2s 4 12 2(12) 4 7 9. If a a) 2, then the value of 4a 2 65 10. If y a) 1 64 b) 17 c) 15 2a 3 is d) 23 e) 71 1 , then what is the value of y? 4 1 1 1 3 b) c) d) e) 16 12 4 4 x 3 and x 11. Given the function y f (x) in Figure 1, find the domain and range of the function. What is the value of f (1) ? Estimate the horizontal and vertical intercepts. 5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 Figure 1: y y x 1 2 3 4 5 f x Answers 1. d 6. a 2 b 7. c 3. c 8. c 4. e 9. d 5. d 10. a ,3 , Range: ,4 , f (1) 2 . 11. Domain: Horizontal intercept is 1, vertical intercept is 1.8. How many of these problems can you miss and still succeed in MTH 211? Ideally, NONE. These problems are just a sample of the larger number of skills that you should be familiar with BEFORE taking this course. If some of these ideas are not familiar to you, you should enroll in the prerequisite course (MTH 95). EGC COMPASS Placement Test Review Packet 7 Math Review – Part II Refreshing your Math Please note that the following information is meant for review only. If the material or part of it is unfamiliar to you, it is recommended that you take the corresponding math class. Under each topic you will find the math class in which that particular material is taught. Multiplication can be symbolized in different ways. For example: 5 • 2, 5 x 2, or (5)(2). The use of ‚x‛ is not useful for algebra and will therefore not be used here. The other two variations will be used interchangeably. If variables are used multiplication is assumed if no sign appears. For example: 3a or ab. Integers (Math 20) Definitions Integers are counting numbers, their negative counterparts, and zero: The distance of a number from zero is called the absolute value. The absolute value is always positive: |5| = 5 and |-5| = 5 EGC COMPASS Placement Test Review Packet 7 Multiplying, Dividing, Adding, and Subtracting Integers Examples: 12 4 = 3 -2 • (-3) = 6 8 (-2) = -4 -5 • 6 = 30 Practice Problems: 1. 10 • (-7) 2. -8 • (-5) 3. -3 • (-15) 4. (-1)(15) 4 + 9 = 13 -6 + (-11) = -17 9 + (-7) = 2 -14 + 6 = -8 5. 6. 7. 8. (0)(-8) 80 (-10) -63 7 -81 (-9) Answers to Practice Problems: 1. -70 5. 0 2. 40 6. -8 3. 45 7. -9 4. -15 8. 9 9. 0 (-5) 10. -7 0 11. -3 + (-8) 12. 10 + (-4) 9. 0 10. undefined 11. -11 12. 6 EGC COMPASS Placement Test Review Packet 9 – 12 = 9 + (-12) = -3 -14 – 7 = -14 + (-7) = -21 15 – (-3) = 15 + 3 = 18 -4 – (-5) = -4 + 5 = 1 13. 5 + (-9) 14. -7 + 2 15. -6 + 8 16. 8 – 13 13. -4 14. -5 15. 2 16. -5 17. -7 – 10 18. 12 – (-4) 19. -5 – (-1) 20. -9 – (-9) 17. -17 18. 16 19. -4 20. 0 7 Fractions (Math 20) Definitions Fraction = Numerator Denominator When the numerator is smaller than the denominator we call the fraction proper. If the numerator is greater than the denominator we call the fraction improper. Improper fractions can be written as mixed numbers, which is as an addition of a whole number and a proper fraction. For example: 2 is a proper fraction; 3 4 1 is an improper fraction and can be written as a mixed number: 1 3 3 5 Whole Numbers such as 5 can be written as 1 The reciprocal of a fraction has the numerator and denominator switched. For example: 3 2 is the reciprocal of 2 3 Mixed Numbers Mixed numbers can be converted to improper fractions like this: 3 4 5 3 5 4 5 19 5 Improper fractions can be converted to mixed numbers by dividing with remainder: 4 19 5 3 R 4 which translates into 3 5 Simplifying Fractions When simplifying fractions we divide the numerator and the denominator by a common factor. Like this: 28 28 4 7 48 48 4 12 EGC COMPASS Placement Test Review Packet 7 28 48 This can also be done in several steps: 28 2 48 2 14 24 14 2 24 2 7 12 At the end of a calculation fractions should always be simplified. Multiplying, Dividing, Adding, and Subtracting Fractions Examples: 5 9 45 3 20 60 3 4 5 15 3 7 or 1 5 13 93 204 2 3 5 7 35 45 7 9 2 3 1 7 2 7 3 7 or 1 3 1 4 Practice Problems: 3 5 1. 4 11 2. 5 7 3 15 3 4 5 15 3 7 1 3 7 3 14 3 17 21 21 21 and 1 5 7 3 153 1 7 3 3 7 9 5 1 6 8 5 4 1 3 6 4 8 3 20 24 3 24 3. 7 9 3 5 5. 9 5 6 7. 2 5 3 4 9. 8 2 15 5 4. 5 21 14 25 6. 1 5 3 4 8. 7 10 5 9 10. 4 4 5 EGC COMPASS Placement Test Review Packet 17 24 7 11. 2 3 1 4 12. 1 3 5 10 1 6 14. 8 15 72 32 8. 1 50 50 4 1 9. 1 3 3 1 5 11 11. 12 1 12. 2 13. Answers to Practice Problems: 15 2 1. 4. 44 15 6 15 1 2. 5. 7 35 2 2 21 4 3. 6. 45 15 7 9 7. 10. 5 1 8 12 15. 5 12 3 10 11 18 17 14. 24 43 15. 60 13. Order of Operations (Math 20) When evaluating numerical expressions we follow the Order of Operations: 1. Evaluate the inside of the parentheses or other grouping symbols first. Grouping symbols include also brackets, absolute value, square roots, and complex numerators and denominators. 2. Evaluate exponents. 3. Multiply or divide, whichever comes first as you read left to right. 4. Add or subtract, whichever comes first as you read left to right. Example 1: 25 (2 4) 2 4 2 1 Evaluate inside of parantheses first. 25 62 4 2 1 25 36 4 2 1 25 9 2 1 25 18 1 7 1 8 Evaluate exponents next. Divide first since the division is further left than the multiplication Multiply. Subtract first since the subtraction is further left than the addition. Add. EGC COMPASS Placement Test Review Packet Example 2: 5 1 ) Evaluate inside of parantheses first. 3 2 5 1 25 ( 6) 2 ( 8) • ( ) Evaluate exponents next. 3 2 5 1 25 36 ( 8) • ( ) Divide first since the division is further left than the multiplication. 3 2 3 9 5 1 25 •( ) Multiplication next by first simplifying the fractions. 2 31 2 25 (2 8) 2 ( 8) • ( 25 25 25 1 50 2 35 2 36 2 18 3 •( 2 15 2 15 2 15 2 1 2 5 1 ) 1 2 1 2 1 2 1 2 Multiply. Subtract first since the subtraction is further left than the addition. The common denominator is 2. Add. Simplify. Practice Problems: 1. 10 (9 2 2) 2 5 3 2. 2 3 3. 7 2 1 3 4. 25 36 32 22 24 2 3 (5 ( 6)) 2 3 5 2 4 3 23 5 8 4 5 4(4 1) Answers to Practice Problems: 1. 8 23 1 2. 11 2 2 5. (36 42 2 2) 2 ( 5 30 2 3 40) 2 6. 25 1 12 2 2 4. 75 3. EGC COMPASS Placement Test Review Packet 10 15 3 27 36 48 30 45 36 24 5. 300 27 6. 4 6 3 4 8 Solving Linear Equations (Math 60) Definitions A variable is a place holder for a number. It is represented by a letter. Example: x, y, a, b A term is a number, variable, or a combination of both if multiplied together. Different terms are separated by addition and subtraction. Example: In the expression 5 + 7x – 7(x+2) the terms are 5 and 7x, and -7(x+2). Like terms are terms that have the same variables with the same exponents. In an equation like terms can be combined. Example: 7x and 2x are like terms, 8x and 4x2 are not like terms. Distributive property: a(x+y) = ax + ay Example: 2(3x-4) = 6x – 8 The Golden Rule of Algebra What you do to one side of an equation or inequality you have to do to the other side of the equation or inequality as well. The objective for solving equations or inequalities is to isolate the variable on one side of the equation or inequality. To achieve that we can do a combination of the following operations (‚s omething‛ c an be a number, variable, or a combination of both): Add something to both sides. Subtract something from both sides. Multiply something to both sides. Divide both sides by something. Square both sides. Take the square root of both sides. In case of an inequality, if multiplied or divided by a negative number the sign will turn around (for instance from < to >). EGC COMPASS Placement Test Review Packet Example 1: 2x 8 7 x 2 2x 8 7 x 7 x 2 7 x 5x 8 2 5x 8 8 2 8 5 x 10 5x 5 x 10 5 2 subtracting 7 x from both sides to bring variables to the same side combining like terms adding 8 to both sides to isolate variable combining like terms dividing both sides by 5 to isolate variable simplify fractions Example 2: 7 a (a 1) 8(a 4) 3(7 a 12) 3 7 a a 1 8a 32 21a 36 3 distributing 14a 31 21a 39 simplifying like terms 14a 31 21a 21a 39 21a subtracting 21a from both sides to bring variables to the same side 7 a 31 39 7 a 31 31 39 31 7 a 70 7 a 70 7 7 a 10 combining like terms adding 31 to both sides to isolate variable combining like terms dividing both sides by 7, turning around the inequality sign simplify fractions Practice Problems: Solve each equation or inequality. 1. 5x – 3 + 2x = 15 + 3x + 2 2. 9b – 8 + 8b > 17 + 2b + 5 3. 41y – 53 + 38y = 46 + 73y + 81 4. 4 + 7a – 11 = 24a – 6 – 13a 5. 2 + 8z – 5 < 8z – 9 – 4z Answers to Practice Problems: 1. x = 5 2. b > 2 3. y = 30 1 4. x 4 3 5. x 2 EGC COMPASS Placement Test Review Packet 6. 54 + 79k – 91 = 34k – 37 + 45k 7. 15 – 3(b+7) = 2(b+2) 8. 23(x+1) + 7(2x–1) = 43x – 6(x–2) 9. 4n – 7(n–5) +10 < 8 – 15(n+2) – 6x 10. 54x + 6(7x–4) ≥ 7(8x+7) – 9(4x–8x) 6. all real numbers 7. x = -2 8. no solutions 67 9. x 18 73 10. x 4 8 Graphing Lines (Math 60) Definitions A line is the graphic representation of a linear equation in two variables. Example: The linear equation y = 2x + 1 can be graphically represented as: y = 2x+1 y x The slope is a measure of how steep the line is. The x-intercept is the intersection of the line and the x-axis. The y-intercept is the intersection of the line and the y-axis. If the line is in the form y = mx + b we call it the slope-intercept form. With m representing the slope and b representing the y-intercept (0,b). If the line is in the form ax + bx = c we call it the standard form. We get the slope-intercept form from the standard form by solving for y. We get the standard form from the slope-intercept form by subtracting mx from both sides (add if m is negative) and multiply by the common denominator (if there are fractions). Graphing a Line To graph a line in slope-intercept form we make a table of values by choosing several values for x and solving the equation for y respectively. EGC COMPASS Placement Test Review Packet 8 1 x 1 we choose 0, 2, and -2 for our x values (2 was 2 chosen so the fraction simplifies easily). We then substitute these values for x and solve for y. In the case of x=2 this is how: Example: For the equation y y 1 x 1 2 1 (2) 1 2 2 1 2 1 1 y 2 y y y Finding the other points the same way we get the following table: x 0 2 -2 y -1 -2 0 Graphing each of those points and connecting the dots, we get the following graph: y = -1/2*x-1 y x To graph a line in standard form we make a table of values by choosing 0 for x and solving the equation for y, and then choosing 0 for y and solving the equation for x. Example: For the equation x 2 y 2 we choose 0 for x and y and then solve for the other variable respectively. For x=0 this is how: x 2y 0 2y 2y y y 2 2 2 2 2 1 Finding the other points the same way we get the following table: EGC COMPASS Placement Test Review Packet x 0 -2 y -1 0 8 Graphing each of those points and connecting the dots, we get the following graph: y x Finding the Slope and Intercepts If the equation appears in slope-intercept form y = mx + b then m represents the slope and b is the y-intercept (0,b). Another way of finding the slope is by using two points from the line and the slope y2 y1 formula: m x2 x1 The y-intercept can also be found by choosing 0 (zero) for x and solving for y. The x-intercept can be found by choosing 0 (zero) for y and solving for x. 1 1 and the y-intercept is x 1 we know the slope is 2 2 (0,-1). We find the x-intercept by choosing 0 for y and solving for x: Example: In the example of y 0 2(0) 0 0 0 x x 1 x 1 2 1 2( x 1) 2 1 2( x) 2(1) 2 x 2 So the x-intercept it (-2,0). x 2 x 2 EGC COMPASS Placement Test Review Packet 8 Practice Problems: Graph each line and find its slope and intercepts. 1. y = x – 1 4. 2x + 3y = 5 2. y = -3x – 4 5. x – y = -2 1 6. 3x – y = 4 x 2 3. y 3 Answers to Practice Problems: 1. m=1 x-intercept: (1,0) y-intercept: (0,1) 1 3 x-intercept: (-6,0) y-intercept: (0,2) 3. m= y y x x 2. m=-3 4. m= 4 ,0) 3 y-intercept: (0,-4) x-intercept: ( 2 3 5 ,0) 2 5 y-intercept: (0, ) 3 x-intercept: ( y y x x EGC COMPASS Placement Test Review Packet 8 5. m= 1 x-intercept: (-2,0) y-intercept: (0,2) 6. m= 3 2 ,0) 3 y-intercept: (0,-2) x-intercept: ( y y x x Laws of Exponents (Math 65) When simplifying expressions with exponents we follow these laws: aman am an (a m ) n 8 n am n a mn a0 1 am 1 (a (ab) m a mb m a Examples: 1. x5x3 = x5+3 = x8 h6 2. 14 h6 14 h h am m 0) 1 h8 EGC COMPASS Placement Test Review Packet 8 3. 83 4 83 4 812 4. 54 x 7 y13 z 5 y12 56 x 2 z 5 2 54 6 x 7 2 y13 12 z 5 5 2 x5 y 25 1 52( 2) 2 2 5 2 x5 y 25 z 0 5 2 x5 y 25 5 2 2 x 2(5) y 2(25) 5 4 x10 y 50 x10 y 50 54 Practice Problems: Simplify: 1. 52•54 x 11 2. x3 3. (a4)5 4. c–7 5. (yz3)6 73 x 4 7 2 x3 6. 74 x5 Answers to Practice Problems: 1. 56 2. x8 3. a20 1 4. 7 c 5. y6z18 7. 8. a 2b3 3 a 5b10 47 s 2t 2 43 st 4 32 x 3 y 4 z 4 313 x 6 z 9. 310 xy12 35 y 3 z 4 10. 6. 7x2 a 7. b 411 s 3 8. t2 EGC COMPASS Placement Test Review Packet 4 3 a 3b 4c5 2 10 a 6 c 2 8 a 1c 12 4 5 b 3c 4 9. x8 z y11 10. 4c14 a 2b 8 Functions Consider a function f(x). Then x is the input and f(x) the output. All eligible inputs make up the domain. All outputs make up the range. Unless otherwise noted the domain is usually all real numbers. The two most common exceptions are: 1. If the function contains a fraction the domain will be restricted because the denominator cannot be zero. The function has a vertical asymptote at that point. 2. If the function contains an even root the domain will be restricted because the radicand has to be greater or equal to zero. To find the range it is often helpful to graph the function by solving f(x) for as many x as needed to see what the function looks like. Example1: For the function f(x) = 2x+1 the domain is all real numbers and so is the range. Example 2: For the function f ( x) cannot be zero. 5x+2=0 when x 3 the domain is restricted by the fact that 5x+2 5x 2 2 . Therefore the domain is all real numbers with the 5 2 . We can write that mathematically in different ways: 5 2 2 2 , , 1. Domain: 2. Domain: x | x 5 5 5 2 . We there for have a vertical asymptote x 5 To find the range we graph the function. We will start by choosing 0, 1, -1, 2, -2 for x and solve f(x). For x=2 this is how: exception of 3 5x 2 3 f (2) 5(2) 2 3 12 f ( x) EGC COMPASS Placement Test Review Packet Finding the other points the same way we get the following table: x 0 1 -1 2 -2 y 3 1.5 2 3 0.4 7 -1 3 0.25 12 3 0.4 8 If we put everything we have so far in a picture we get: y x It can be helpful to choose a few more points and we will find this graph: y x Knowing that we have a vertical and a horizontal asymptote we can see from here that the range is ,0 0, . EGC COMPASS Placement Test Review Packet Function Transformation We relate many functions back to a few basic function types by using transformations. That can be very helpful in graphing the function and finding its range. Function f(x) + k f(x) – k f(x+k) f(x-k) –f(x) f(–x) k•f(x) f(k•x) Transformation Shift vertically up k units Shift vertically down k units Shift horizontally left k units Shift horizontally right k units Reflect vertically about the x-axis Reflect horizontally about the y-axis Stretch/Shrink vertically by a factor of k Stretch/Shrink horizontally by a factor of 1/k Example: The function f(x) = (x-2)2 –3 has the function g(x) = x2 as the base. Looking at the above transformation table we can see that we can find the graph of f(x) by shifting g(x) 3 units down and 2 units to the right. If we know that g(x) has a range of [0,∞) we know that the range of f(x) is [-3,∞). And we can graph f(x) easily: y x Practice Problems: For each of the functions: a) Find the domain. 1. f ( x) 2. f ( x) 3. f ( x) 2x2 4 1 x 2 4 x 4 x3 5 x 2 b) Find the range. x 1 EGC COMPASS Placement Test Review Packet c) Graph the function. 4. f ( x) 5. f ( x) 6. f ( x) 2x 6 x 4 2 x 2 x x 2 9 Answers to Practice Problems: 1. a) all real numbers b) [-4,∞) c) 4. a) [-3,∞) b) *0,∞) c) y y x 2. a) b) , 2 ,0 x 5. a) (-∞,4+ b) [-2,∞) c) 2, 0, c) y y x x 6. a) 3. a) all real numbers b) (-∞,4) c) , 1 1, (2, ) b) all real numbers c) y y x EGC COMPASS Placement Test Review Packet x 9 Laws of Logarithms (Math 111) The logarithm is defined as the inverse of the exponent. If we want to solve b x then the logarithm is defined as the solution: x log b m m for x When simplifying expressions with logarithms we follow the laws of logarithms, just like we did with the laws of exponents. It is important to notice that the laws of logarithms are different, inverse to be exact. That is because the logarithm is the inverse function of the exponent. log b m log b n log b (m n) log b m logb n logb m n r log b m log b (m r ) log b 1 0 logb b 1 log b m log a m log a b The logarithms to the base 10 and e have special expressions: log m log10 m ln x log e x Practice Problems: Simplify, using the laws of logarithms. 1. log10 log 3 2. log 28 log 4 3. 4ln 3 Answers to Practice Problems: 1. log 30 2. log 7 3. ln81 EGC COMPASS Placement Test Review Packet 4. log9 2log3 5. log 20 4 log 20 5 6. ln 20 ln3 ln 6 4. 0 5. 1 6. ln10 9 Additional Math Review Resources Online Websites http://www.coolmath.com/ http://www.testprepreview.com/modules/algebra1c.htm http://www.testprepreview.com/modules/algebra2.htm http://www.testprepreview.com/modules/fractionsandsquareroots.htm http://www.testprepreview.com/modules/mathematics3.htm http://www.testprepreview.com/modules/exponents.htm http://www.testprepreview.com/modules/percentandratios.htm http://www.khanacademy.org EGC COMPASS Placement Test Review Packet 9Ř

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