Precalculus: graphical, numerical, algebraic 9780133518450, 1292079452, 9781292079455, 0133518450

Table of contents :
Cover......Page 1
Title......Page 2
Copyright......Page 3
Foreword......Page 4
Contents......Page 6
About the Authors......Page 15
Preface......Page 17
Acknowledgments......Page 23
CHAPTER P Prerequisites......Page 26
Representing Real Numbers......Page 27
Order and Interval Notation......Page 28
Basic Properties of Algebra......Page 30
Integer Exponents......Page 32
Scientific Notation......Page 33
Absolute Value of a Real Number......Page 37
Distance Formulas......Page 38
Midpoint Formulas......Page 39
Equations of Circles......Page 40
Applications......Page 41
Linear Equations in One Variable......Page 46
Linear Inequalities in One Variable......Page 48
Slope of a Line......Page 53
Slope-Intercept Form Equation of a Line......Page 54
Graphing Linear Equations in Two Variables......Page 55
Parallel and Perpendicular Lines......Page 56
Applying Linear Equations in Two Variables......Page 58
Solving Equations Graphically......Page 65
Solving Quadratic Equations......Page 66
Approximating Solutions of Equations Numerically with Tables......Page 68
Solving Equations by Finding Intersections......Page 70
Operations with Complex Numbers......Page 74
Complex Solutions of Quadratic Equations......Page 76
Solving Absolute Value Inequalities......Page 79
Solving Quadratic Inequalities......Page 80
Approximating Solutions to Inequalities......Page 81
Projectile Motion......Page 82
Key Ideas......Page 84
Review Exercises......Page 85
CHAPTER 1 Functions and Graphs......Page 88
Numerical Models......Page 89
Algebraic Models......Page 90
Graphical Models......Page 91
The Zero Factor Property......Page 93
Problem Solving......Page 95
Grapher Failure and Hidden Behavior......Page 96
A Word About Proof......Page 98
Function Definition and Notation......Page 105
Domain and Range......Page 106
Continuity......Page 109
Increasing and Decreasing Functions......Page 111
Boundedness......Page 113
Local and Absolute Extrema......Page 114
Symmetry......Page 115
Asymptotes......Page 117
End Behavior......Page 118
Twelve Basic Functions......Page 124
Analyzing Functions Graphically......Page 128
Combining Functions Algebraically......Page 135
Composition of Functions......Page 136
Relations and Implicitly Defined Functions......Page 139
Relations Defined Parametrically......Page 144
Inverse Relations and Inverse Functions......Page 146
Vertical and Horizontal Translations......Page 154
Reflections Across Axes......Page 156
Vertical and Horizontal Stretches and Shrinks......Page 158
Combining Transformations......Page 160
Functions from Formulas......Page 165
Functions from Graphs......Page 166
Functions from Verbal Descriptions......Page 167
Functions from Data......Page 168
Review Exercises......Page 177
Chapter Project......Page 181
CHAPTER 2 Polynomial, Power, and Rational Functions......Page 182
Polynomial Functions......Page 183
Linear Functions and Their Graphs......Page 184
Average Rate of Change......Page 185
Association, Correlation, and Linear Modeling......Page 186
Quadratic Functions and Their Graphs......Page 189
Applications of Quadratic Functions......Page 191
Power Functions and Variation......Page 199
Monomial Functions and Their Graphs......Page 200
Graphs of Power Functions......Page 202
Modeling with Power Functions......Page 204
Graphs of Polynomial Functions......Page 210
End Behavior of Polynomial Functions......Page 212
Zeros of Polynomial Functions......Page 214
Intermediate Value Theorem......Page 215
Modeling......Page 216
Long Division and the Division Algorithm......Page 222
Remainder and Factor Theorems......Page 223
Synthetic Division......Page 224
Rational Zeros Theorem......Page 226
Upper and Lower Bounds......Page 227
Two Major Theorems......Page 235
Complex Conjugate Zeros......Page 236
Factoring with Real Number Coefficients......Page 238
Rational Functions......Page 243
Transformations of the Reciprocal Function......Page 244
Limits and Asymptotes......Page 245
Analyzing Graphs of Rational Functions......Page 247
Exploring Relative Humidity......Page 249
Solving Rational Equations......Page 253
Extraneous Solutions......Page 254
Applications......Page 255
Polynomial Inequalities......Page 261
Rational Inequalities......Page 264
Other Inequalities......Page 265
Applications......Page 266
Key Ideas......Page 270
Review Exercises......Page 271
Chapter Project......Page 275
CHAPTER 3 Exponential, Logistic, and Logarithmic Functions......Page 276
Exponential Functions and Their Graphs......Page 277
The Natural Base e......Page 281
Logistic Functions and Their Graphs......Page 283
Population Models......Page 285
Constant Percentage Rate and Exponential Functions......Page 290
Exponential Growth and Decay Models......Page 291
Using Regression to Model Population......Page 292
Other Logistic Models......Page 295
Inverses of Exponential Functions......Page 299
Common Logarithms—Base 10......Page 300
Graphs of Logarithmic Functions......Page 302
Measuring Sound Using Decibels......Page 305
Properties of Logarithms......Page 308
Change of Base......Page 310
Graphs of Logarithmic Functions with Base b......Page 311
Re-expressing Data......Page 312
Solving Exponential Equations......Page 317
Solving Logarithmic Equations......Page 318
Orders of Magnitude and Logarithmic Models......Page 319
Newton’s Law of Cooling......Page 321
Logarithmic Re-expression......Page 323
Simple and Compound Interest......Page 329
Interest Compounded k Times per Year......Page 330
Interest Compounded Continuously......Page 331
Annual Percentage Yield......Page 332
Annuities—Future Value......Page 333
Loans and Mortgages—Present Value......Page 334
Key Ideas......Page 338
Review Exercises......Page 339
Chapter Project......Page 343
CHAPTER 4 Trigonometric Functions......Page 344
Degrees and Radians......Page 345
Circular Arc Length......Page 347
Angular and Linear Motion......Page 348
Right Triangle Trigonometry......Page 354
Two Famous Triangles......Page 355
Evaluating Trigonometric Functions with a Calculator......Page 356
Common Calculator Errors when Evaluating Trig Functions......Page 357
Applications of Right Triangle Trigonometry......Page 358
Trigonometric Functions of Any Angle......Page 363
Trigonometric Functions of Real Numbers......Page 369
Periodic Functions......Page 370
The 16-Point Unit Circle......Page 371
The Basic Waves Revisited......Page 375
Sinusoids and Transformations......Page 377
Modeling Periodic Behavior with Sinusoids......Page 380
The Tangent Function......Page 386
The Cotangent Function......Page 387
The Secant Function......Page 388
The Cosecant Function......Page 389
Combining Trigonometric and Algebraic Functions......Page 394
Sums and Differences of Sinusoids......Page 396
Damped Oscillation......Page 398
Inverse Sine Function......Page 403
Inverse Cosine and Tangent Functions......Page 405
Composing Trigonometric and Inverse Trigonometric Functions......Page 407
Applications of Inverse Trigonometric Functions......Page 409
More Right Triangle Problems......Page 413
Simple Harmonic Motion......Page 415
Review Exercises......Page 424
Chapter Project......Page 427
CHAPTER 5 Analytic Trigonometry......Page 428
Basic Trigonometric Identities......Page 429
Pythagorean Identities......Page 430
Odd-Even Identities......Page 431
Simplifying Trigonometric Expressions......Page 432
Solving Trigonometric Equations......Page 433
Proving Identities......Page 438
Disproving Non-Identities......Page 441
Identities in Calculus......Page 442
Cosine of a Difference......Page 446
Cosine of a Sum......Page 447
Sine of a Difference or Sum......Page 448
Verifying a Sinusoid Algebraically......Page 449
Power-Reducing Identities......Page 453
Half-Angle Identities......Page 454
Solving Trigonometric Equations......Page 455
Deriving the Law of Sines......Page 459
The Ambiguous Case (SSA)......Page 460
Applications......Page 462
Deriving the Law of Cosines......Page 467
Solving Triangles (SAS, SSS)......Page 468
Triangle Area and Heron’s Formula......Page 469
Applications......Page 470
Review Exercises......Page 475
Chapter Project......Page 479
CHAPTER 6 Applications of Trigonometry......Page 480
Two-Dimensional Vectors......Page 481
Vector Operations......Page 483
Unit Vectors......Page 484
Direction Angles......Page 485
Applications of Vectors......Page 486
The Dot Product......Page 492
Angle Between Vectors......Page 493
Projecting One Vector onto Another......Page 495
Work......Page 496
Parametric Curves......Page 500
Eliminating the Parameter......Page 501
Lines and Line Segments......Page 502
Simulating Motion with a Grapher......Page 503
Polar Coordinate System......Page 512
Coordinate Conversion......Page 513
Equation Conversion......Page 514
Finding Distance Using Polar Coordinates......Page 516
Symmetry......Page 519
Analyzing Polar Graphs......Page 520
Rose Curves......Page 521
Limaçon Curves......Page 522
Other Polar Curves......Page 524
The Complex Plane......Page 528
Polar Form of Complex Numbers......Page 529
Multiplication and Division of Complex Numbers......Page 530
Powers of Complex Numbers......Page 531
Roots of Complex Numbers......Page 533
Key Ideas......Page 538
Review Exercises......Page 539
Chapter Project......Page 542
CHAPTER 7 Systems and Matrices......Page 543
The Method of Substitution......Page 544
The Method of Elimination......Page 546
Applications......Page 548
Matrix Addition and Subtraction......Page 554
Matrix Multiplication......Page 556
Identity and Inverse Matrices......Page 558
Determinant of a Square Matrix......Page 559
Applications......Page 561
Gaussian Elimination......Page 568
Elementary Row Operations and Row Echelon Form......Page 570
Reduced Row Echelon Form......Page 572
Solving Systems Using Inverse Matrices......Page 574
Partial Fraction Decomposition......Page 575
Other Applications......Page 576
Graph of an Inequality......Page 582
Systems of Inequalities......Page 583
Linear Programming......Page 584
Review Exercises......Page 590
Chapter Project......Page 594
CHAPTER 8 Analytic Geometry in Two and Three Dimensions......Page 595
Conic Sections......Page 596
Geometry of a Parabola......Page 597
Translations of Parabolas......Page 600
Reflective Property of a Parabola......Page 602
Geometry of an Ellipse......Page 607
Translations of Ellipses......Page 610
Orbits and Eccentricity......Page 612
Reflective Property of an Ellipse......Page 614
Geometry of a Hyperbola......Page 618
Translations of Hyperbolas......Page 621
Eccentricity and Orbits......Page 622
Reflective Property of a Hyperbola......Page 623
Long-Range Navigation......Page 624
Quadratic Equations Revisited......Page 628
Axis Rotation Formulas......Page 629
Discriminant Test......Page 633
Eccentricity Revisited......Page 637
Writing Polar Equations for Conics......Page 638
Analyzing Polar Equations of Conics......Page 640
Orbits Revisited......Page 641
Three-Dimensional Cartesian Coordinates......Page 646
Distance and Midpoint Formulas......Page 647
Planes and Other Surfaces......Page 649
Vectors in Space......Page 650
Lines in Space......Page 651
Key Ideas......Page 654
Review Exercises......Page 655
Chapter Project......Page 657
CHAPTER 9 Discrete Mathematics......Page 658
The Importance of Counting......Page 659
The Multiplication Principle of Counting......Page 660
Permutations......Page 661
Combinations......Page 663
Subsets of an n-Set......Page 664
Powers of Binomials......Page 669
Pascal’s Triangle......Page 670
Binomial Theorem......Page 671
Factorial Identities......Page 672
Infinite Sequences......Page 675
Limits of Infinite Sequences......Page 676
Arithmetic and Geometric Sequences......Page 677
Sequences and Technology......Page 679
Sums of Arithmetic and Geometric Sequences......Page 683
Infinite Series......Page 686
Convergence of Geometric Series......Page 688
Tower of Hanoi Problem......Page 692
Principle of Mathematical Induction......Page 693
Induction and Deduction......Page 695
Review Exercises......Page 698
Chapter Project......Page 700
CHAPTER 10 Statistics and Probability......Page 701
Sample Spaces and Probability Functions......Page 702
Determining Probabilities......Page 705
Venn Diagrams......Page 707
Tree Diagrams......Page 708
Conditional Probability......Page 709
Categorical Data......Page 716
Quantitative Data: Stemplots......Page 718
Histograms......Page 721
Time Plots......Page 722
Describing and Comparing Distributions......Page 729
Five-Number Summary......Page 730
Boxplots......Page 731
The Mean (and When to Use It)......Page 733
Variance and Standard Deviation......Page 735
Normal Distributions......Page 736
Probability Models and Expected Values......Page 742
Binomial Probability Models......Page 745
Normal Model......Page 748
Normal Approximation for Binomial Distributions......Page 750
Correlation Revisited......Page 757
Importance of Randomness......Page 759
Samples, Surveys, and Observational Studies......Page 760
Experimental Design......Page 761
Using Randomness......Page 763
Simulations......Page 765
Review Exercises......Page 772
Chapter Project......Page 777
CHAPTER 11 An Introduction to Calculus: Limits, Derivatives, and Integrals......Page 778
Average Velocity......Page 779
Limits Revisited......Page 780
The Connection to Tangent Lines......Page 781
The Derivative......Page 783
Distance from a Constant Velocity......Page 790
Limits at Infinity......Page 791
The Connection to Areas......Page 792
The Definite Integral......Page 793
Defining a Limit Informally......Page 798
Properties of Limits......Page 799
One-Sided and Two-Sided Limits......Page 801
Limits Involving Infinity......Page 803
Derivatives on a Calculator......Page 809
Definite Integrals on a Calculator......Page 810
Computing a Derivative from Data......Page 811
Computing a Definite Integral from Data......Page 813
Review Exercises......Page 818
Chapter Project......Page 820
Radicals......Page 822
Simplifying Radical Expressions......Page 823
Rational Exponents......Page 824
Adding, Subtracting, and Multiplying Polynomials......Page 827
Factoring Polynomials Using Special Products......Page 828
Factoring Trinomials......Page 830
Factoring by Grouping......Page 831
Reducing Rational Expressions......Page 834
Operations with Rational Expressions......Page 835
Compound Rational Expressions......Page 836
Statements......Page 839
Compound Statements......Page 841
Forms of Statements......Page 845
Valid Reasoning......Page 847
C.1 Formulas from Algebra......Page 852
C.3 Formulas from Trigonometry......Page 853
C.4 Formulas from Analytic Geometry......Page 855
C.5 Gallery of Basic Functions......Page 856
Bibliography......Page 857
Glossary......Page 858
Selected Answers......Page 876
Applications Index......Page 978
B......Page 982
D......Page 983
E......Page 984
F......Page 985
I......Page 986
L......Page 987
O......Page 988
Q......Page 989
S......Page 990
T......Page 991
W......Page 992
Z......Page 993