Precalculus Demystified

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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

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

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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.

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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

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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