Precalculus Demystified RHONDA HUETTENMUELLER McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-146956-7 The material in this eBook also appears in the print version of this title: 0-07-143927-7. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. 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DOI: 10.1036/0071469567 CONTENTS Preface vii CHAPTER 1 The Slope and Equation of a Line 1 CHAPTER 2 Introduction to Functions 24 CHAPTER 3 Functions and Their Graphs 42 CHAPTER 4 Combinations of Functions and Inverse Functions 64 CHAPTER 5 Translations and Special Functions 88 CHAPTER 6 Quadratic Functions 104 CHAPTER 7 Polynomial Functions 134 CHAPTER 8 Rational Functions 185 v For more information about this title, click here CONTENTS vi CHAPTER 9 Exponents and Logarithms 201 CHAPTER 10 Systems of Equations and Inequalities 262 CHAPTER 11 Matrices 303 CHAPTER 12 Conic Sections 330 CHAPTER 13 Trigonometry 364 CHAPTER 14 Sequences and Series 415 Appendix 439 Final Exam 450 Index 464 PREFACE The goal of this book is to give you the skills and knowledge necessary to succeed in calculus. Much of the difficulty calculus students face is with algebra. They have to solve equations, find equations of lines, study graphs, solve word problems, and rewrite expressions—all of these require a solid background in algebra. You will get experience with all this and more in this book. Not only will you learn about the basic functions in this book, you also will strengthen your algebra skills because all of the examples and most of the solutions are given with a lot of detail. Enough steps are given in the problems to make the reasoning easy to follow. Thebasicfunctionscoveredinthisbookarelinear, polynomial, andrationalfunc- tions, as well as exponential, logarithmic, and trigonometric functions. Because understanding the slope of a line is crucialto making sense of calculus, the interpre- tation of a line’s slope is given extra attention. Other calculus topics introduced in this book are Newton’s Quotient, the average rate of change, increasing/decreasing intervals of a function, and optimizing functions. Your experience with these ideas will help you when you learn calculus. Conceptsare presentedinclear, simplelanguage, followed bydetailedexamples. To make sure you understand the material, each section ends with a set of practice problems. Each chapter ends with a multiple-choice test, and there is a final exam at the end of the book. You will get the most from this book if you work steadily from the beginning to the end. Because much of the material is sequential, you should review the ideas in the previous section. Study for each end-of-chapter test as if it really were a test, and take it without looking at examples and without using notes. This will let you know what you have learned and where, if anywhere, you need to spend more time. Good luck. Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use. vii 1 CHAPTER The Slope and Equation of a Line The slope of a line and the meaning of the slope are important in calculus. In fact, the slope formula is the basis for differential calculus. The slope of a line measures its tilt. The sign of the slope tells us if the line tilts up (if the slope is positive) or tilts down (if the slope is negative). The larger the number, the steeper the slope. We can put any two points on the line, (x 1 ,y 1 ) and (x 2 ,y 2 ), in the slope formula to find the slope of the line. m = y 2 −y 1 x 2 −x 1 1 xi Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use. CHAPTER 1 The Slope and Equation 2 Fig. 1.1. Fig. 1.2. For example, (0, 3), (−2, 2), (6, 6), and (−1, 5 2 ) are all points on the same line. We can pick any pair of points to compute the slope. m = 2 −3 −2 −0 = −1 −2 = 1 2 m = 5 2 −2 −1 −(−2) = 1 2 1 = 1 2 m = 3 −6 0 −6 = −3 −6 = 1 2 A slope of 1 2 means that if we increase the x-value by 2, then we need to increase the y-value by 1 to get another point on the line. For example, knowing that (0, 3) is on the line means that we know (0 +2, 3 + 1) = (2, 4) is also on the line. CHAPTER 1 The Slope and Equation 3 Fig. 1.3. Fig. 1.4. As we can see from Figure 1.4, (−4, −2) and (1, −2) are two points on a horizontal line. We will put these points in the slope formula. m = −2 −(−2) 1 −(−4) = 0 5 = 0 The slope of every horizontal line is 0. The y-values on a horizontal line do not change but the x-values do. What happens to the slope formula for a vertical line? CHAPTER 1 The Slope and Equation 4 Fig. 1.5. The points (3, 2) and (3, −1) are on the vertical line in Figure 1.5. Let’s see what happens when we put them in the slope formula. m = −1 −2 3 −3 = −3 0 This is not a number so the slope of a vertical line does not exist (we also say that it is undefined). The x-values on a vertical line do not change but the y-values do. Any line is the graph of a linear equation. The equation of a horizontal line is y = a (where a is the y-value of every point on the line). Some examples of horizontal lines are y = 4, y = 1, and y =−5. Fig. 1.6. CHAPTER 1 The Slope and Equation 5 The equation of a vertical line is x = a (where a is the x-value of every point on the line). Some examples are x =−3, x = 2, and x = 4. Fig. 1.7. Otherequationsusuallycomeinoneoftwoforms: Ax+By = C andy = mx+b. We will usually use the form y = mx + b in this book. An equation in this form gives us two important pieces of information. The first is m, the slope. The second is b, the y-intercept (where the line crosses the y-axis). For this reason, this form is called the slope-intercept form. In the line y = 2 3 x +4, the slope of the line is 2 3 and the y-intercept is (0, 4), or simply, 4. We can find an equation of a line by knowing its slope and any point on the line. There are two common methods for finding this equation. One is to put m, x, and y (x and y are the coordinates of the point we know) in y = mx +b and use algebra to find b. The other is to put these same numbers in the point-slope form of the line, y −y 1 = m(x −x 1 ). We will use both methods in the next example. EXAMPLES • Find an equation of the line with slope − 3 4 containing the point (8, −2). We will let m =− 3 4 , x = 8, and y =−2iny = mx +b to find b. −2 =− 3 4 (8) +b 4 = b The line is y =− 3 4 x +4. [...]... x − 11 7 7 • (0, 1) and (12, 1) The y-values are the same, making this a horizontal line The equation is y = 1 If a graph is clear enough, we can find two points on the line or even its slope If fact, if the slope and y-intercept are easy enough to see on the graph, we know right away what the equation is EXAMPLES • Fig 1.8 The line in Figure 1.8 crosses the y-axis at 1, so b = 1 From this point, we... a) or vertical (x = a) 1 Find the slope of the line containing the points (4, 12) and (−6, 1) 2 Find the slope of the line with x-intercept 5 and y-intercept −3 3 Find an equation of the line containing the point (−10, 4) with slope −2 5 4 Find an equation of the line with y-intercept −5 and slope 2 5 Find an equation of the line in Figure 1.13 Fig 1.13 6 Find an equation of the line containing the points... = = −6 − 4 −10 10 2 The x-intercept is 5 and the y-intercept is −3 mean that the points (5, 0) and (0, −3) are on the line 1 m = m= −3 − 0 −3 3 = = 0−5 −5 5 3 Put x = −10, y = 4, and m = − 2 in y = mx + b to find b 5 2 4 = − (−10) + b 5 0=b The equation is y = − 2 x + 0, or simply y = − 2 x 5 5 4 m = 2, b = −5, so the line is y = 2x − 5 5 From the graph, we can see that the y-intercept is 3 We can use... inside the parentheses is x and the quantity on the right of the equal sign is y One advantage to this notation is that we have both the x- and y-values without having to say anything about x and y Functions that have no variables in them are called constant functions All y-values for these functions are the same EXAMPLES • • √ Find f (−2), f (0), and f (6) for f (x) = x + 3 We need to substitute −2, 0,... want, so we will let x = −4, y = 5, and m = 2 in y = mx + b We get 5 = 2(−4) + b, so b = 13 The equation of the line we want is y = 2x + 13 • Find an equation of the line with x-intercept 4 that is perpendicular to x − 3y = 12 The x-intercept is 4 means that the point (4, 0) is on the line The slope of the line we want will be the negative reciprocal of the slope of the line x − 3y = 12 We will find the... The equation is y = 1 x + 4000 5 The slope, and sometimes the y-intercept, have important meanings in applied problems In the first example, the household water bill was computed using y = 0.0025x + 15 The slope means that each gallon costs $0.0025 (or 0.25 cents) As the number of gallons increases by 1, the cost increases by $0.0025 The y-intercept is the cost when 0 gallons are used This additional... equation of the line with slope 4, containing the point (0, 3) We know the slope is 4 and we know the y-intercept is 3 (because (0, 3) is on the line), so we can write the equation without having to do any work: y = 4x + 3 • Find an equation of the horizontal line that contains the point (5, −6) Because the y-values are the same on a horizontal line, we know that this equation is y = −6 We can still find... 10) + 110 if 10 < h ≤ 24 CHAPTER 2 Introduction to Functions Below is an example of a piecewise function taken from an Internal Revenue Service (IRS) publication The y-value is the amount of personal income tax for a single person The x-value is the amount of taxable income  4316   4329 f (x) = 4341    4354 if 30,000 ≤ x if 30,050 ≤ x if 30,100 ≤ x if 30,150 ≤ x < 30,050 < 30,100 < 30,150 . prior written permission of the publisher. 0-0 7-1 4695 6-7  The material in this eBook also appears in the print version of this title: 0-0 7-1 4392 7-7 . All trademarks are trademarks of their. Precalculus Demystified RHONDA HUETTENMUELLER McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City. Special Sales, at george_hoare@mcgraw-hill.com or (212) 90 4-4 069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all