1

of

MATHEMATICS

RELATIONS, FUNCTIONS, AND GRAPHS

TABLE OF CONTENTS

TELECOMMUNICATIONIntroduction, 3Projects, 61, 123, 201, 271

Matrices, 97

Unit 1 Projects

UNIT

1Chapter 1 LINEAR RELATIONS AND

FUNCTIONS 4

1-1 Relations and Functions ..............................................51-2 Composition of Functions .......................................131-3 Graphing Linear Equations ...................................20

1-3B Graphing Calculator ExplorationAnalyzing Families of Linear Graphs...................26

1-4 Writing Linear Equations ........................................27

Mid-Chapter Quiz ..............................................................31

1-5 Writing Equations of Parallel andPerpendicular Lines ............................................32

1-6 Modeling Real-World Data with LinearFunctions .....................................................................38

1-7 Piecewise Functions ..................................................45

1-8 Graphing Linear Inequalities ................................52

Study Guide and Assessment.......................................57

SAT & ACT Preparation................................................62

Chapter 2 SYSTEMS OF LINEAR EQUATIONSAND INEQUALITIES 66

2-1 Solving Systems of Equations in Two Variables............................................................ 67

2-2 Solving Systems of Equations in Three Variables......................................................... 73

2-3 Modeling Real-World Data with Matrices ........... 78

2-4A Graphing Calculator Exploration TransformationMatrices.......................................................................87

2-4 Modeling Motion with Matrices .......................... 88

Mid-Chapter Quiz.........................................................96

2-5 Determinants and Multiplicative Inverses of Matrices................................................................. 98

2-5B Graphing Calculator Exploration Augmented Matrices and Reduced Row-Echelon Form........106

2-6 Solving Systems of Linear Inequalities................. 107

2-7 Linear Programming............................................... 112

Study Guide and Assessment ......................................119SAT & ACT Preparation...............................................124

vi

GRAPHINGCALCULATOREXPLORATIONS

UNIT 1

Veterinary Medicine, 12Agricultural Manager, 105Biomedical Engineering, 136Environmental Engineering,

221

Self-Check Quizzes, 9, 17,23, 29, 36, 42, 49, 55, 70,76, 83, 93, 102, 110, 116,134, 142, 149, 156, 165,176, 186, 194, 210, 219,226, 240, 247, 255, 262

Data Updates, 84, 263 Career Choices, 12, 105,

136, 221Internet Projects, 61, 123,

201, 271History of Mathematics, 97Graphing Calculator

Programs, 10, 133, 226Review Activities, 57, 119,

197, 267SAT/ACT Practice, 65, 125,

203, 273Chapter Tests, A56–A59

13, 26, 69, 86, 87, 106, 133,169, 232, 265

CAREER CHOICESChapter 3 THE NATURE OF GRAPHS 126

3-1 Symmetry and Coordinate Graphs.........................127

3-2 Families of Graphs.......................................................137

3-3 Graphs of Nonlinear Inequalities............................146

Mid-Chapter Quiz............................................................151

3-4 Inverse Functions and Relations...........................152

3-5 Continuity and End Behavior...................................159

3-5B Graphing Calculator Exploration Gap Discontinuities...........................................................169

3-6 Critical Points and Extrema....................................171

3-7 Graphs of Rational Functions............................... 180

3-8 Direct, Inverse, and Joint Variation ......................189

Study Guide and Assessment .......................................197

SAT & ACT Preparation..............................................202

Chapter 4 POLYNOMIAL AND RATIONALFUNCTIONS 204

4-1 Polynomial Functions............................................205

4-2 Quadratic Equations...............................................213

4-3 The Remainder and Factor Theorems..................222

4-4 The Rational Root Theorem ................................229

Mid-Chapter Quiz.......................................................235

4-5 Locating Zeros of a Polynomial Function ...........236

4-6 Rational Equations and Partial Fractions ............243

4-7 Radical Equations and Inequalities........................251

4-8 Modeling Real-World Data with Polynomial Functions.................................258

4-8B Graphing Calculator Exploration Fitting aPolynomial Function to a Set of Points .............265

Study Guide and Assessment.......................................267

SAT & ACT Preparation..............................................272

vii

TRIGONOMETRY

Unit 2 Projects

THE CYBERCLASSROOMIntroduction, 275Projects, 339, 417, 481, 547

UNIT

2

of

MATHEMATICS

Angles, 319Functions, 367Trigonometry, 462Angle Measure, 534

Chapter 5 THE TRIGONOMETRICFUNCTIONS 276

5-1 Angles and Degree Measure..................................277

5-2 Trigonometric Ratios in Right Triangles.............284

5-3 Trigonometric Functions on the Unit Circle........291

5-4 Applying Trigonometric Functions......................299

Mid-Chapter Quiz.......................................................304

5-5 Solving Right Triangles..........................................305

5-6 The Law of Sines.........................................................313

5-7 The Ambiguous Case for the Law of Sines..........320

5-8 The Law of Cosines ................................................327

5-8B Graphing Calculator ExplorationSolving Triangles..................................................333

Study Guide and Assessment.......................................335

SAT & ACT Preparation ..............................................340

Chapter 6 GRAPHS OF TRIGONOMETRICFUNCTIONS 342

6-1 Angles and Radian Measure..................................343

6-2 Linear and Angular Velocity....................................352

6-3 Graphing Sine and Cosine Functions...................359

6-4 Amplitude and Period of Sine and Cosine Functions ...................................................368

Mid-Chapter Quiz .......................................................377

6-5 Translations of Sine and Cosine Functions....................................................378

6-6 Modeling Real-World Data with SinusoidalFunctions....................................................................387

6-7 Graphing Other Trigonometric Functions...........395

6-7B Graphing Calculator ExplorationSound Beats ..........................................................404

6-8 Trigonometric Inverses and Their Graphs ..........405

Study Guide and Assessment........................................413

SAT & ACT Preparation...............................................418

viii

UNIT 2

CAREER CHOICES

Architecture, 312Audio Recording

Engineer, 358Cartographer, 430Aerospace Engineer, 499

Self-Check Quizzes, 281,288, 296, 302, 309, 316,329, 331, 348, 355, 363,373, 383, 396, 400, 410,427, 434, 442, 453, 459,467, 474, 490, 497, 503,509, 517, 524, 531, 540

Data Updates, 283, 455Research, 357, 388Career Choices, 312, 358,

430, 499Internet Projects, 339, 417,

481, 547History of Mathematics, 319,

367, 462, 534Graphing Calculator

Programs, 331, 333, 470, 512

Review Activities, 335, 413,477, 543

SAT/ACT Practice, 341, 419,483, 549

Chapter Tests, A60–A63

GRAPHINGCALCULATOREXPLORATIONS

284, 323, 333, 369, 378, 404,433, 446, 458, 512, 526

Chapter 7 TRIGONOMETRIC IDENTITIES AND EQUATIONS 420

7-1 Basic Trigonometric Identities..............................421

7-2 Verifying Trigonometric Identities........................431

7-3 Sum and Difference Identities...............................437

Mid-Chapter Quiz.......................................................445

7-3B Graphing Calculator ExplorationReduction Identities...........................................446

7-4 Double-Angle and Half-Angle Identities...............448

7-5 Solving Trigonometric Equations.........................456

7-6 Normal Form of a Linear Equation.......................463

7-7 Distance From a Point to a Line..............................470

Study Guide and Assessment.....................................477

SAT & ACT Preparation..............................................482

Chapter 8 VECTORS AND PARAMETRICEQUATIONS 484

8-1 Geometric Vectors......................................................485

8-2 Algebraic Vectors .................................................493

8-3 Vectors in Three-Dimensional Space ..................500

8-4 Perpendicular Vectors...........................................505

Mid-Chapter Quiz .......................................................5 1 1

8-4B Graphing Calculator ExplorationFinding Cross Products.......................................512

8-5 Applications with Vectors.....................................513

8-6 Vectors and Parametric Equations........................520

8-6B Graphing Calculator Exploration Modeling with Parametric Equations....................................526

8-7 Modeling Motion Using Parametric Equations..................................................................527

8-8 Transformational Matrices in Three-Dimensional Space ...................................535

Study Guide and Assessment.......................................543

SAT & ACT Preparation..............................................548

ix

Chapter 9 POLAR COORDINATES AND COMPLEX NUMBERS 552

9-1 Polar Coordinates.......................................................553

9-2 Graphs of Polar Equations.....................................561

9-3 Polar and Rectangular Coordinates.....................568

9-4 Polar Form of a Linear Equation.............................574

Mid-Chapter Quiz.......................................................579

9-5 Simplifying Complex Numbers...............................580

9-6 The Complex Plane and Polar Form of Complex Numbers ..................................................586

9-6B Graphing Calculator ExplorationGeometry in the Complex Plane..........................592

9-7 Products and Quotients of Complex Numbers in Polar Form........................593

9-8 Powers and Roots of Complex Numbers..............599

Study Guide and Assessment.....................................607

SAT & ACT Preparation...............................................612

Chapter 10 CONICS 614

10-1 Introduction to Analytic Geometry........................615

10-2 Circles......................................................................623

10-3 Ellipses..........................................................................631

10-4 Hyperbolas..............................................................642

10-5 Parabolas......................................................................653

Mid-Chapter Quiz ...........................................................661

10-6 Rectangular and Parametric Forms of Conic Sections ....................................................662

10-7 Transformations of Conics....................................670

10-8 Systems of Second-Degree Equations andInequalities ...............................................................678

10-8B Graphing Calculator ExplorationShading Areas on a Graph .................................685

Study Guide and Assessment.......................................687

SAT & ACT Preparation..............................................692

ADVANCED FUNCTIONS AND GRAPHING

Unit 3 Projects

SPACE—THE FINALFRONTIER

Introduction, 551Projects, 611, 691, 753

UNIT

3

x

Chapter 11 EXPONENTIAL AND LOGARITHMICFUNCTIONS 694

11-1 Real Exponents.......................................................695

11-2 Exponential Functions..............................................704

11-3 The Number e ..........................................................712

Mid-Chapter Quiz........................................................717

11-4 Logarithmic Functions ..............................................718

11-5 Common Logarithms................................................726

11-6 Natural Logarithms ...................................................733

11-6B Graphing Calculator ExplorationNatural Logarithms and Area .............................738

11-7 Modeling Real-World Data with Exponential and Logarithmic Functions...................................740

Study Guide and Assessment.....................................749

SAT & ACT Preparation ..............................................754

GRAPHINGCALCULATOREXPLORATIONS

CAREER CHOICES

Astronomer, 598Meteorologist, 622Food Technologist, 703

592, 602, 641, 665, 685, 695,705, 738

UNIT 3

Self-Check Quizzes, 558,565, 571, 578, 583, 590,596, 605, 620, 627, 638,650, 659, 667, 675, 682,700, 708, 714, 720, 723,730, 735, 744

Data Updates, 741 Career Choices, 598, 622,

703Internet Projects, 611, 691,

753Graphing Calculator

Programs, 582, 604, 620,628, 710

Review Activities, 607, 687,749

SAT/ACT Practice, 613, 693,755

Chapter Tests, A64–A66

Lesson 10-7, page 675

xi

Self-Check Quizzes, 763,771, 781, 791, 798, 804,812, 819, 826, 843, 849,856, 864, 871, 878, 893,903, 914, 923, 930

Data Updates, 895, 906, 908Career Choices, 800, 851,

896Internet Projects, 833, 885,

937Graphing Calculator

Programs, 784Review Activities, 829, 881,

933SAT/ACT Practice, 835, 887,

939 Chapter Tests, A67–A69

Chapter 12 SEQUENCES AND SERIES 758

12-1 Arithmetic Sequences and Series.........................759

12-2 Geometric Sequences and Series .........................766

12-3 Infinite Sequences and Series .................................774

12-3B Graphing Calculator ExplorationContinued Fractions...............................................784

12-4 Convergent and Divergent Series.........................786

Mid-Chapter Quiz.......................................................793

12-5 Sigma Notation and the nth Term ........................794

12-6 The Binomial Theorem...........................................801

12-7 Special Sequences and Series...............................806

12-8 Sequences and Iteration ........................................815

12-9 Mathematical Induction ........................................822

Study Guide and Assessment.......................................829

SAT & ACT Preparation..............................................834

Chapter 13 COMBINATORICS AND PROBABILITY 836

13-1 Permutations and Combinations...........................837

13-2 Permutations with Repetitions and Circular Permutations.........................................846

13-3 Probability and Odds...........................................852

Mid-Chapter Quiz..........................................................858

13-4 Probabilities of Compound Events .....................859

13-5 Conditional Probability............................................868

13-6 The Binomial Theorem and Probability ..............875

Study Guide and Assessment ........................................881

SAT & ACT Preparation..............................................886

DISCRETE MATHEMATICS

Unit 4 Projects

THE UNITED STATESCENSUS BUREAU

Introduction, 757Projects, 833, 885, 937

UNIT

4

CAREER CHOICES

Operation Research Analyst,800

Actuary, 851Accountant, 896

xii

Chapter 14 STATISTICS AND DATA ANALYSIS 888

14-1 The Frequency Distribution....................................889

14-2 Measures of Central Tendency.............................897

14-3 Measures of Variability..........................................908

Mid-Chapter Quiz........................................................917

14-4 The Normal Distribution........................................918

14-4B Graphing Calculator ExplorationThe Standard Normal Curve ................................926

14-5 Sample Sets of Data................................................927

Study Guide and Assessment.......................................933

SAT & ACT Preparation..............................................938

GRAPHINGCALCULATOREXPLORATIONS

784, 809, 877, 926

UNIT 4

Lesson 12-5, page 798

xiii

Self-Check Quizzes, 946,958, 966, 973

Data Updates, 956Research, 943, 970 Career Choices, 976Internet Project, 981History of Mathematics, 969Graphing Calculator

Programs, 961Review Activities, 977SAT/ACT Practice, 983Chapter Test, A70

Chapter 15 INTRODUCTION TO CALCULUS 940

15-1 Limits ........................................................................941

15-2A Graphing Calculator ExplorationThe Slope of a Curve ..............................................949

15-2 Derivatives and Antiderivatives............................951

Mid-Chapter Quiz.......................................................960

15-3 Area Under a Curve...............................................961

15-4 The Fundamental Theorem of Calculus..............970

Study Guide and Assessment.....................................977

SAT & ACT Preparation..............................................982

Student Handbook

Graphing Calculator Appendix.....................................................A2

Extra Practice....................................................................................A26

Chapter Tests ...............................................................................A56

Glossary .........................................................................................A71

Selected Answers..........................................................................A81

Photo Credits ..............................................................................A152

Index.............................................................................................A154

Formulas and Symbols ......................................inside back cover

CALCULUS

Unit 5 Project

DISEASESIntroduction, 940Project, 981

UNIT

5

GRAPHINGCALCULATOREXPLORATIONS

CAREER CHOICES

of

MATHEMATICS

Mathematician, 976

Calculus, 969

945, 949

xiv

#

USING A GRAPHING CALCULATOR

In Glencoe Advanced Mathematical Concepts, the graphing calculator is animportant tool in exploring precalculus topics. A graphing calculator is actuallya specialized hand-held computer. Graphing Calculator Explorations throughoutthis text offer opportunties for you to use a graphing calculator to explore themathematical topics you are studying. You will also see graphing calculatorscreens actually generated by a graphing calculator. If you do not recognize thekeys, commands, or screens shown in this test, you may have a different modelof calculator. Refer to your owner’s manual to verify the proper commands andkeystrokes to use.

Graphing Calculator Appendix

At the back of this text, you will find a Student Handbook. The GraphingCalculator Appendix begins on page A2. This appendix introduces you to someof the common functions of the graphing calculator most widely used in highschool classrooms. It is divided into nine sections:

1 Introduction to the Graphing Calculator

2 Graphing Functions

3 Analyzing Functions

4 Graphing Inequalities

5 Matrices

6 Graphing Trigonometric Functions

7 Graphing Special Functions

8 Parametric and Polar Equations

9 Statistics and Statistical Graphs

In each section, you will find keystrokeinstructions for specific examples thatdemonstrate some of the basic operationsand most common menus available on the calculator. Whenever you have specificquestions about commands or procedures,you should refer to your owner’s manual forinstruction.

SAT and ACT Preparation

It is important to note that graphingcalculators are permitted when taking theACT or SAT and are required for the APCalculus test. Through your experienceswith the graphing calculator in studyingGlencoe Advanced MathematicalConcepts, you should becomecomfortable with its operation andappreciate the power of this tinycomputer.

These sections donot correspondwith the order oftopics in thistextbook.

To findkeystrokeinstructions foreach GraphingCalculatorExploration, visitwww.amc.glencoe.com

Using a Graphing Calculator 1