0 Review: Equations and

Inequalities 1

0.1 Linear Equations 2

0.1.1 Solving Linear Equations in One Variable 2

0.1.2 Applications Involving Linear Equations 6

0.1.3 Interest Problems 8

0.1.4 Mixture Problems 10

0.1.5 Distance–Rate–Time Problems 12

0.2 Quadratic Equations 17

0.2.1 Factoring 17

0.2.2 Square Root Method 19

0.2.3 Completing the Square 21

0.2.4 Quadratic Formula 23

0.2.5 Applications Involving Quadratic

Equations 26

0.3 Other Types of Equations 30

0.3.1 Rational Equations 30

0.3.2 Radical Equations 33

0.3.3 Equations Quadratic in Form:

u-Substitution 35

0.3.4 Factorable Equations 37

0.3.5 Equations Involving Absolute Value 38

0.4 Inequalities 43

0.4.1 Graphing Inequalities and Interval

Notation 43

0.4.2 Linear Inequalities 46

0.4.3 Polynomial Inequalities 48

0.4.4 Rational Inequalities 51

0.4.5 Absolute Value Inequalities 53

0.5 Graphing Equations 58

0.5.1 Cartesian Plane 59

0.5.2 The Distance and Midpoint Formulas 59

0.5.3 Point-Plotting

61

0.5.4 Using Intercepts as Graphing Aids 62

0.5.5 Using Symmetry as a Graphing Aid 64

0.5.6 Circles 67

0.6 Lines 74

0.6.1 General Form of a Line and Slope 74

0.6.2 Equations of Lines 77

0.6.3 Parallel and Perpendicular Lines 80

0.7 Modeling Variation 86

0.7.1 Direct Variation 86

0.7.2 Inverse Variation 88

0.7.3 Joint Variation and Combined Variation 90

0.8* Linear Regression: Best Fit 0.8-1

0.8.1 Scatterplots 0.8-1

0.8.2 Identifying Patterns 0.8-5

0.8.3 Linear Regression 0.8-12

Review 95 | Review Exercises 97 | Practice Test 99

1 Functions and Their Graphs 100

1.1 Functions 101

1.1.1 Relations and Functions 101

1.1.2 Functions Defined by Equations 104

1.1.3 Function Notation 106

1.1.4 Domain of a Function 110

1.2 Graphs of Functions; Piecewise-Defined Functions;

Increasing and Decreasing Functions; Average Rate

of Change 118

1.2.1 Recognizing and Classifying Functions 118

1.2.2 Increasing and Decreasing Functions 122

1.2.3 Average Rate of Change 125

1.2.4 Piecewise-Defined Functions 128

1.3 Graphing Techniques: Transformations 138

1.3.1 Horizontal and Vertical Shifts 138

1.3.2 Reflection About the Axes 143

1.3.3 Stretching and Compressing 146

1.4 Operations on Functions and Composition of

Functions 152

1.4.1 Adding, Subtracting, Multiplying, and

Dividing Functions 153

1.4.2 Composition of Functions 154

1.5 One-to-One Functions and Inverse Functions 163

1.5.1 Determine Whether a Function Is

One-to-One 163

1.5.2 Inverse Functions 166

1.5.3 Graphical Interpretation of Inverse

Functions 168

1.5.4 Finding the Inverse Function 170

Review 179 | Review Exercises 181 | Practice Test 184

2 Polynomial and Rational

Functions 185

2.1 Quadratic Functions 186

2.1.1 Graphs of Quadratic Functions:

Parabolas 186

2.1.2 Finding the Equation of a Parabola 195

2.2 Polynomial Functions of Higher Degree 203

2.2.1 Identifying Polynomial Functions 203

2.2.2 Graphing Polynomial Functions Using

Transformations of Power Functions 206

2.2.3 Real Zeros of a Polynomial Function 207

2.2.4 Graphing General Polynomial

Functions 210

2.3 Dividing Polynomials: Long Division and Synthetic

Division 219

2.3.1 Long Division of Polynomials 219

2.3.2 Synthetic Division of Polynomials 223

2.4 The Real Zeros of a Polynomial Function 227

2.4.1 The Remainder Theorem and the Factor

Theorem 227

2.4.2 The Rational Zero Theorem and Descartes’

Rule of Signs 230

2.4.3 Factoring Polynomials 234

2.4.4 The Intermediate Value Theorem 236

2.4.5 Graphing Polynomial Functions 238

2 .5 Complex Zeros: The Fundamental Theorem of

Algebra 242

2.5.1 Complex Zeros 243

2.5.2 Factoring Polynomials 247

2.6 Rational Functions 251

2.6.1 Domain of Rational Functions 251

2.6.2 Vertical, Horizontal, and Slant

Asymptotes 253

2.6.3 Graphing Rational Functions 259

Review 271 | Review Exercises 273 | Practice

Test 276 | Cumulative Test 277

3 Exponential and Logarithmic

Functions 278

3.1 Exponential Functions and Their Graphs 279

3.1.1 Evaluating Exponential Functions 279

3.1.2 Graphs of Exponential Functions 281

3.1.3 The Natural Base e 285

3.1.4 Applications of Exponential Functions 287

3.2 Logarithmic Functions and Their Graphs 295

3.2.1 Evaluating Logarithms 295

3.2.2 Common and Natural Logarithms 298

3.2.3 Graphs of Logarithmic Functions 298

3.2.4 Applications of Logarithms 303

3.3 Properties of Logarithms 311

3.3.1 Properties of Logarithmic Functions 311

3.3.2 Change-of-

Base

Formula 316

3.4 Exponential and Logarithmic Equations 320

3.4.1 Exponential Equations 320

3.4.2 Solving Logarithmic Equations 324

3.4.3 Applications 326

3.5 Exponential and Logarithmic Models 331

3.5.1 Exponential Growth Models 332

3.5.2 Exponential Decay Models 333

3.5.3 Gaussian (Normal) Distribution Models 335

3.5.4 Logistic Growth Models 336

3.5.5 Logarithmic Models 337

Review 343 | Review Exercises 345 | Practice

Test 348 | Cumulative Test 349

4 Trigonometric Functions of

Angles 350

4.1 Angle Measure 351

4.1.1 Angles and their Measure 352

4.1.2 Radian Measure 354

4.1.3 Angles in Standard Position 357

4.1.4 Coterminal Angles 358

4.1.5 Arc Length 359

4.1.6 Area of a Circular Sector 360

4.1.7 Linear and Angular Speed 361

4.1.8 Relationship Between Linear and

Angular Speeds 363

4.2 Right Triangle Trigonometry 369

4.2.1 Right Triangle Ratios 370

4.2.2 Trigonometric Functions: Right

Triangle Ratios 371

4.2.3 Reciprocal Identities 372

4.2.4 Evaluating Trigonometric Functions

Exactly for Special Angle Measures 373

4.2.5 Using Calculators to Evaluate (Approximate)

Trigonometric Function Values 377

4.2.6 Solving a Right Triangle Given an Acute Angle

Measure and a Side Length 378

4.3 Trigonometric Functions of Angles 387

4.3.1 Trigonometric Functions: The Cartesian

Plane 387

4.3.2 Algebraic Signs of the Trigonometric

Functions 390

4.3.3 Ranges of the Trigonometric Functions 393

4.3.4 Reference Angles and Reference Right

Triangles 395

4.3.5 Evaluating Trigonometric Functions for

Nonacute Angles 398

4.4 The Law of Sines 405

4.4.1 Solving Oblique Triangles 405

4.5 The Law of Cosines 418

4.5.1 Solving Oblique Triangles Using the

Law of Cosines 419

4.5.2 The Area of a Triangle 423

Review 432 | Review Exercises 436 | Practice

Test 438 | Cumulative Test 439

5 Trigonometric Functions of Real

Numbers 440

5.1 Trigonometric Functions: The Unit Circle

Approach 441

5.1.1 Trigonometric Functions and the Unit Circle

(Circular Functions) 441

5.1.2 Properties of Trigonometric

(Circular) Functions 444

5.2 Graphs of Sine and Cosine Functions 451

5.2.1 The Graphs of Sinusoidal Functions 451

5.2.2 Graphing a Shifted Sinusoidal Function:

y = A sin(Bx + C) + D and y =

A cos(Bx + C) + D 464

5.2.3 Harmonic Motion 467

5.2.4 Graphing Sums of Functions: Addition of

Ordinates 471

5.3 Graphs of Other Trigonometric Functions 481

5.3.1 Graphing the Tangent, Cotangent, Secant,

and Cosecant Functions 481

5.3.2 Translations of Trigonometric

Functions 492

Review 500 | Review Exercises 504 | Practice

Test 505 | Cumulative Test 506

6 Analytic Trigonometry 507

6.1 Trigonometric Identities 508

6.1.1 Fundamental Identities 509

6.1.2 Simplifying Trigonometric Identities 511

6.1.3 Verifying Identities 513

6.2 Sum and Difference Identities 519

6.2.1 Sum and Difference Identities

for the Cosine Function 520

6.2.2 Sum and Difference Identities

for the Sine Function 523

6.2.3 Sum and Difference Identities for the

Tangent Function 526

6.3 Double-Angle and Half-Angle Identities 531

6.3.1 Applying Double-Angle Identities 531

6.3.2 Applying Half-Angle Identities 536

6.4 Product-to-Sum and Sum-to-Product

Identities 547

6.4.1 Product-to-Sum Identities 547

6.4.2 Sum-to-Product Identities 549

6.5 Inverse Trigonometric Functions 555

6.5.1 Inverse Sine Function 556

6.5.2 Inverse Cosine Function 560

6.5.3 Inverse Tangent Function 563

6.5.4 Remaining Inverse Trigonometric

Functions 565

6.5.5 Finding Exact Values for Expressions

Involving Inverse Trigonometric

Functions 568

6.6 Trigonometric Equations 576

6.6.1 Solving Trigonometric Equations by

Inspection 577

6.6.2 Solving Trigonometric Equations Using

Algebraic Techniques 580

6.6.3 Solving Trigonometric Equations That

Require the Use of Inverse Functions 582

6.6 .4 Using Trigonometric Identities to Solve

Trigonometric Equations 584

Review 593 | Review Exercises 597 | Practice

Test 600 | Cumulative Test 601

7 Vectors, the Complex Plane, and

Polar Coordinates 602

7.1 Vectors 603

7.1.1 Vectors: Magnitude and Direction 603

7.1.2 Vector Operations 607

7.1.3 Horizontal and Vertical Components of a

Vector 609

7.1.4 Unit Vectors 610

7.1.5 Resultant Vectors 611

7.2 The Dot Product 619

7.2.1 Multiplying Two Vectors: The Dot

Product 619

7.2.2 Angle Between Two Vectors 620

7.2.3 Work 623

7.3 Polar (Trigonometric) Form of Complex

Numbers 628

7.3.1 Complex Numbers in

Rectangular Form 629

7.3.2 Complex Numbers in Polar Form 630

7.4 Products, Quotients, Powers, and Roots of

Complex Numbers 638

7.4.1 Products of Complex Numbers 638

7.4.2 Quotients of Complex Numbers 640

7.4.3 Powers of Complex Numbers 641

7.4.4 Roots of Complex Numbers 643

7.5 Polar Equations and Graphs 649

7.5.1 Polar Coordinates 650

7.5.2 Converting Between Polar and

Rectangular Coordinates 651

7.5.3 Graphs of Polar Equations 653

Review 667 | Review Exercises 670 | Practice

Test 672 | Cumulative Test 673

8 Systems of Linear Equations and

Inequalities 674

8.1 Systems of Linear Equations in Two Variables 675

8.1.1 Solving Systems of Linear Equations 676

8.1.2 Three Methods and Three Types

of Solutions 685

8.2 Systems of Linear Equations in

Three Variables 691

8.2.1 Solving Systems of Linear Equations

in Three Variables 691

8.2.2 Types of Solutions 694

8.3 Systems of Linear Equations and Matrices 703

8.3.1 Matrices 703

8.3.2 Augmented Matrices 705

8.3.3 Row Operations on a Matrix 707

8.3.4 Row–Echelon

Form of a Matrix 708

8.3.5 Gaussian Elimination with Back-Substitution

709

8.3.6 Gauss–Jordan

Elimination 711

8.3.7 Inconsistent and Dependent Systems 714

8.4 Matrix Algebra 726

8.4.1 Equality of Matrices 726

8.4.2 Matrix Addition and Subtraction 727

8.4.3 Scalar and Matrix Multiplication 729

8.4.4 Matrix Equations 735

8.4.5 Finding the Inverse of a Matrix 737

8.4.6 Solving Systems of Linear Equations Using

Matrix Algebra and Inverses of Square

Matrices 742

8.5 The Determinant of a Square Matrix and Cramer’s

Rule 751

8.5.1 Determinant of a 2 × 2 Matrix 752

8.5.2 Determinant of an n × n Matrix 753

8.5.3 Cramer’s Rule: Systems of Linear Equations

in Two Variables 756

8.5.4 Cramer’s Rule: Systems of Linear Equations

in Three Variables 758

8.6 Partial Fractions 765

8.6.1 Performing Partial-Fraction

Decomposition 765

8.7 Systems of Linear Inequalities

in Two Variables 776

8.7.1 Linear Inequalities in Two Variables 776

8.7.2 Systems of Linear Inequalities in Two

Variables 779

8.7.3 The Linear Programming Model 784

Review 792 | Review Exercises 795 | Practice

Test 798 | Cumulative Test 799

9 Conics and Systems of Nonlinear

Equations and Inequalities 800

9.1 Conic Basics 801

9.1.1 Three Types of Conics 801

9.2 The Parabola 804

9.2.1 Parabola with a Vertex at the Origin 804

9.2.2 Parabola with a Vertex at the Point

(h, k) 809

9.2.3 Applications 812

9.3 The Ellipse 818

9.3.1 Ellipse Centered at the Origin 818

9.3.2 Ellipse Centered at the Point (h, k) 823

9.3.3 Applications 825

9.4 The Hyperbola 832

9.4.1 Hyperbola Centered at the Origin 832

9.4.2 Hyperbola Centered at the Point (h, k) 837

9.4.3 Applications 839

9.5 Systems of Nonlinear Equations 844

9.5.1 Solving a System of Nonlinear

Equations 844

9.6 Systems of Nonlinear Inequalities 854

9.6.1 Nonlinear Inequalities in Two Variables 854

9.6.2 Systems of Nonlinear Inequalities 857

9.7 Rotation of Axes 864

9.7.1 Transforming Second-Degree Equations

Using Rotation of Axes 864

9.7.2 Determine the Angle of Rotation Necessary

to Transform a General Second-Degree

Equation into an Equation of a Conic 867

9.8 Polar Equations of Conics 872

9.8.1 Equations of Conics in Polar

Coordinates 872

9.9 Parametric Equations and Graphs 881

9.9.1 Parametric Equations of a Curve 881

9.9.2 Applications of Parametric Equations 885

Review 890 | Review Exercises 893 | Practice

Test 895 | Cumulative Test 896

10 Sequences and Series 897

10.1 Sequences and Series 898

10.1.1 Sequences 898

10.1.2 Factorial Notation 901

10.1.3 Recursion Formulas 902

10.1.4 Sums and Series 903

10.2 Arithmetic Sequences and Series 909

10.2.1 Arithmetic Sequences 910

10.2.2 The General (nth) Term of an Arithmetic

Sequence 911

10.2.3 The Sum of an Arithmetic Sequence 912

10.3 Geometric Sequences and Series 918

10.3.1 Geometric Sequences 918

10.3.2 The General (nth) Term of a

Geometric Sequence 920

10.3.3 Geometric Series 921

10.4 Mathematical Induction 929

10.4.1 Mathematical Induction 929

10.5 The Binomial Theorem 933

10.5.1 Binomial Coefficients 934

10.5.2 Binomial Expansion 936

10.5.3 Pascal’s Triangle 937

10.5.4 Finding a Particular Term of a Binomial

Expansion 939

Review 943 | Review Exercises 944 | Practice

Test 946 | Cumulative Test 947

11 Limits: A Preview to Calculus 948

11.1 Introduction to Limits: Estimating Limits

Numerically and Graphically 949

11.1.1 Definition of a Limit 949

11.1.2 Estimating Limits Numerically and

Graphically 951

11.1.3 Limits That Fail to Exist 952

11.1.4 One-Sided Limits 955

11.2 Techniques for Finding Limits 961

11.2.1 Limit Laws 962

11.2.2 Finding Limits Using Limit Laws 963

11.2.3 Finding Limits Using Direct

Substitution 967

11.2.4 Finding Limits Using Algebraic

Techniques 969

11.2.5 Finding Limits Using Left-Hand

and Right-Hand Limits 971

11.3 Tangent Lines and Derivatives 975

11.3.1 Tangent Lines 975

11.3.2 The Derivative of a Function 980

11.3.3 Instantaneous Rates of Change 983

11.4 Limits at Infinity; Limits of Sequences 986

11.4.1 Limits at Infinity 986

11.4.2 Limits of Sequences 991

11.5 Finding the Area Under a Curve 995

11.5.1 Limits of Summations 996

11.5.2 The Area Problem 998

Review 1010 | Review Exercises 1012 | Practice

Test 1014 | Cumulative Test 1015

APPENDIX 1016

ANSWERS TO ALL EXERCISES 1111

APPLICATION INDEX XXX

SUBJECT INDEX XXX

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