Preface xiii To the Student xxi Calculators and Calculations

xxii

1

Fundamentals

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1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

Chapter Overview 1 Real Numbers 2 Exponents and Radicals 12 Algebraic Expressions 24 ● DISCOVERY PROJECT Visualizing a Formula 34 Rational Expressions 35 Equations 44 Modeling with Equations 58 ● DISCOVERY PROJECT Equations through the Ages 75 Inequalities 76 Coordinate Geometry 87 Graphing Calculators; Solving Equations and Inequalities Graphically 101 Lines 111 Modeling Variation 123 Chapter 1 Review 130 Chapter 1 Test 135 ■ FOCUS ON PROBLEM SOLVING General Principles 138

2

Functions

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146

2.1 2.2

Chapter Overview 147 What is a Function? 148 Graphs of Functions 158 ● DISCOVERY PROJECT Relations and Functions

171

vii

viii

Contents

2.3 2.4 2.5 2.6 2.7 2.8

Increasing and Decreasing Functions; Average Rate of Change 173 Transformations of Functions 182 Quadratic Functions; Maxima and Minima 193 Modeling with Functions 203 Combining Functions 214 ● DISCOVERY PROJECT Iteration and Chaos 223 One-to-One Functions and Their Inverses 225 Chapter 2 Review 233 Chapter 2 Test 237 ■ FOCUS ON MODELING Fitting Lines to Data 239

3

Polynomial and Rational Functions

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248

3.1 3.2 3.3 3.4 3.5 3.6

Chapter Overview 249 Polynomial Functions and Their Graphs 250 Dividing Polynomials 265 Real Zeros of Polynomials 272 ● DISCOVERY PROJECT Zeroing in on a Zero 283 Complex Numbers 285 Complex Zeros and the Fundamental Theorem of Algebra 291 Rational Functions 299 Chapter 3 Review 316 Chapter 3 Test 319 ■ FOCUS ON MODELING Fitting Polynomial Curves to Data 320

4

Exponential and Logarithmic 326 Functions

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4.1 4.2 4.3 4.4 4.5

Chapter Overview 327 Exponential Functions 328 ● DISCOVERY PROJECT Exponential Explosion 341 Logarithmic Functions 342 Laws of Logarithms 352 Exponential and Logarithmic Equations 358 Modeling with Exponential and Logarithmic Functions 369 Chapter 4 Review 382 Chapter 4 Test 385 ■ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 386

Contents

ix

5

Trigonometric Functions of Real 398 Numbers

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5.1 5.2 5.3 5.4 5.5

Chapter Overview 399 The Unit Circle 400 Trigonometric Functions of Real Numbers 408 Trigonometric Graphs 418 ● DISCOVERY PROJECT Predator/Prey Models 432 More Trigonometric Graphs 434 Modeling Harmonic Motion 442 Chapter 5 Review 454 Chapter 5 Test 458 ■ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 459

6

Trigonometric Functions of Angles

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466

6.1 6.2 6.3 6.4 6.5

Chapter Overview 467 Angle Measure 468 Trigonometry of Right Triangles 478 Trigonometric Functions of Angles 488 ● DISCOVERY PROJECT Similarity 499 The Law of Sines 501 The Law of Cosines 508 Chapter 6 Review 516 Chapter 6 Test 520 ■ FOCUS ON MODELING Surveying 522

7

Analytic Trigonometry

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526

7.1 7.2 7.3 7.4 7.5

Chapter Overview 527 Trigonometric Identities 528 Addition and Subtraction Formulas 535 Double-Angle, Half-Angle, and Sum-Product Formulas 541 Inverse Trigonometric Functions 550 ● DISCOVERY PROJECT Where to Sit at the Movies Trigonometric Equations 561 Chapter 7 Review 571 Chapter 7 Test 574 ■ FOCUS ON MODELING Traveling and Standing Waves 575

560

x

Contents

8

Polar Coordinates and Vectors

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580

8.1 8.2 8.3

8.4 8.5

Chapter Overview 581 Polar Coordinates 582 Graphs of Polar Equations 587 Polar Form of Complex Numbers; DeMoivre’s Theorem 596 ● DISCOVERY PROJECT Fractals 605 Vectors 607 The Dot Product 617 ● DISCOVERY PROJECT Sailing Against the Wind 626 Chapter 8 Review 627 Chapter 8 Test 629 ■ FOCUS ON MODELING Mapping the World 630

9

Systems of Equations and 634 Inequalities

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9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Chapter Overview 635 Systems of Equations 636 Systems of Linear Equations in Two Variables 644 Systems of Linear Equations in Several Variables 651 ● DISCOVERY PROJECT Best Fit versus Exact Fit 660 Systems of...