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Contents

Preface xiii To the Student xxi Calculators and Calculations

xxii

1

Fundamentals


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


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


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


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


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


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


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


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


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...
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