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  • Precalculus Demystified

    http://dx.doi.org/10.1036/0071469567

  • Demystified Series Advanced Statistics Demystified Algebra Demystified Anatomy Demystified Astronomy Demystified Biology Demystified Business Statistics Demystified Calculus Demystified Chemistry Demystified College Algebra Demystified Differential Equations Demystified Digital Electronics Demystified Earth Science Demystified Electronics Demystified Everyday Math Demystified Geometry Demystified Math Word Problems Demystified Microbiology Demystified Physics Demystified Physiology Demystified Pre-Algebra Demystified Precalculus Demystified Project Management Demystified Robotics Demystified Statistics Demystified Trigonometry Demystified

  • 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

    http://dx.doi.org/10.1036/0071469567

  • 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. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw- Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/0071469567

    http://dx.doi.org/10.1036/0071469567

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    http://dx.doi.org/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

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  • CONTENTSvi

    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.

    The basic functions covered in this book are linear, polynomial, and rational func- tions, as well as exponential, logarithmic, and trigonometric functions. Because understanding the slope of a line is crucial to 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.

    Concepts are presented in clear, simple language, followed by detailed examples. 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

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  • 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, (x1, y1) and (x2, y2), in the slope formula to find the slope of the line.

    m = y2 − y1 x2 − x1

    1 xi

    Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.

  • CHAPTER 1 The Slope and Equation2

    Fig. 1.1.

    Fig. 1.2.

    For example, (0, 3), (−2, 2), (6, 6), and (−1, 52) 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 12 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 Equation4

    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 i