College Algebra: Graphs & Models

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 30
Preface......Page 9
Index of Applications......Page 35
CHAPTER R: A Review of Basic Concepts and Skills......Page 42
B. Translating Written or Verbal Information into a Mathematical Model......Page 43
C. Evaluating Algebraic Expressions......Page 44
D. Properties of Real Numbers......Page 46
E. Simplifying Algebraic Expressions......Page 48
A. The Properties of Exponents......Page 52
B. Exponents and Scientific Notation......Page 56
C. Identifying and Classifying Polynomial Expressions......Page 57
D. Adding and Subtracting Polynomials......Page 58
E. The Product of Two Polynomials......Page 59
F. Special Polynomial Products......Page 60
A. Solving Linear Equations Using Properties of Equality......Page 66
C. Solving Linear Inequalities......Page 68
D. Solving Compound Inequalities......Page 70
E. Solving Basic Applications of Linear Equations and Inequalities......Page 72
F. Solving Applications of Basic Geometry......Page 73
A. The Greatest Common Factor......Page 80
B. Common Binomial Factors and Factoring by Grouping......Page 81
C. Factoring Quadratic Polynomials......Page 82
D. Factoring Special Forms and Quadratic Forms......Page 84
E. Polynomial Equations and the Zero Product Property......Page 87
A. Writing a Rational Expression in Simplest Form......Page 94
B. Multiplication and Division of Rational Expressions......Page 96
C. Addition and Subtraction of Rational Expressions......Page 97
D. Simplifying Compound Fractions......Page 99
E. Solving Rational Equations......Page 100
A. Simplifying Radical Expressions of the Form n√\na[sup(n)]]......Page 106
B. Radical Expressions and Rational Exponents......Page 108
C. Using Properties of Radicals to Simplify Radical Expressions......Page 109
E. Multiplication and Division of Radical Expressions; Radical Expressions in Simplest Form......Page 111
F. Equations and Formulas Involving Radicals......Page 113
Overview of Chapter R: Prerequisite Definitions, Properties, Formulas, and Relationships......Page 122
CHAPTER 1 Relations, Functions, and Graphs......Page 126
A. Relations, Mapping Notation, and Ordered Pairs......Page 127
B. The Graph of a Relation......Page 128
C. Graphing Relations on a Calculator......Page 131
D. The Equation and Graph of a Circle......Page 133
A. The Graph of a Linear Equation......Page 144
B. The Slope of a Line and Rates of Change......Page 145
C. Horizontal Lines and Vertical Lines......Page 148
D. Parallel and Perpendicular Lines......Page 150
E. Applications of Linear Equations......Page 152
A. Functions and Relations......Page 158
B. The Domain and Range of a Function......Page 161
C. Function Notation......Page 164
D. Reading and Interpreting Information Given Graphically......Page 166
Reinforcing Basic Concepts: Finding the Domain and Range of a Relation from Its Graph......Page 173
A. Linear Equations, Slope-Intercept Form and Function Form......Page 175
B. Slope-Intercept Form and the Graph of a Line......Page 177
C. Linear Equations in Point-Slope Form......Page 180
D. Applications of Linear Equations......Page 181
A. Solving Equations Graphically Using the Intersect Method......Page 189
B. Solving Equations Graphically Using the x-Intercept/Zeroes Method......Page 191
C. Solving Linear Inequalities Graphically......Page 193
D. Solving for a Specified Variable in Literal Equations......Page 194
E. Using a Problem-Solving Guide......Page 196
A. Scatterplots and Positive/Negative Associations......Page 204
B. Scatterplots and Linear/Nonlinear Associations......Page 205
C. Identifying Strong and Weak Correlations......Page 206
D. Linear Functions That Model Relationships Observed in a Set of Data......Page 208
E. Linear Regression and the Line of Best Fit......Page 210
Making Connections......Page 218
Summary and Concept Review......Page 219
Practice Test......Page 224
Strengthening Core Skills: The Various Forms of a Linear Equation......Page 225
Calculator Exploration and Discovery: Evaluating Expressions and Looking for Patterns......Page 226
CHAPTER 2 More on Functions......Page 228
A. Graphs and Symmetry......Page 229
B. Intervals Where a Function Is Positive or Negative......Page 231
C. Intervals Where a Function Is Increasing or Decreasing......Page 233
E. Locating Maximum and Minimum Values Using Technology......Page 235
A. The Toolbox Functions......Page 243
B. Vertical and Horizontal Shifts......Page 245
C. Vertical and Horizontal Reflections......Page 247
D. Vertically Stretching/Compressing a Basic Graph......Page 249
E. Transformations of a General Function......Page 250
A. Solving Absolute Value Equations......Page 259
B. Solving “Less Than” Absolute Value Inequalities......Page 262
C. Solving “Greater Than” Absolute Value Inequalities......Page 263
E. Applications Involving Absolute Value......Page 265
Mid-Chapter Check......Page 269
Reinforcing Basic Concepts: Using Distance to Understand Absolute Value Equations and Inequalities......Page 270
A. Rational Functions and Asymptotes......Page 271
B. Using Asymptotes to Graph Basic Rational Functions......Page 274
C. Graphs of Basic Power Functions......Page 275
D. Applications of Rational and Power Functions......Page 278
2.5 Piecewise-Defined Functions......Page 286
A. The Domain of a Piecewise-Defined Function......Page 287
B. Graphing Piecewise-Defined Functions......Page 288
C. Applications of Piecewise-Defined Functions......Page 293
A. Toolbox Functions and Direct Variation......Page 300
B. Inverse Variation......Page 303
C. Joint or Combined Variations......Page 305
Making Connections......Page 311
Summary and Concept Review......Page 312
Practice Test......Page 316
Calculator Exploration and Discovery: Studying Joint Variations......Page 318
Strengthening Core Skills: Variation and Power Functions: y = kx[sup(p)]......Page 319
Cumulative Review: Chapters R–2......Page 320
CHAPTER 3 Quadratic Functions and Operations on Functions......Page 322
A. Identifying and Simplifying Imaginary and Complex Numbers......Page 323
B. Adding and Subtracting Complex Numbers......Page 325
C. Multiplying Complex Numbers; Powers of i......Page 326
D. Division of Complex Numbers......Page 329
A. Zeroes of Quadratic Functions and x-Intercepts of Quadratic Graphs......Page 333
B. Quadratic Equations and the Square Root Property of Equality......Page 336
C. Solving Quadratic Equations by Completing the Square......Page 338
D. The Quadratic Formula and the Discriminant......Page 340
E. Quadratic Inequalities......Page 344
F. Applications of Quadratic Functions and Inequalities......Page 347
A. Graphing Quadratic Functions by Completing the Square......Page 354
B. Graphing Quadratic Functions Using the Vertex Formula......Page 356
D. Quadratic Functions and Extreme Values......Page 357
Reinforcing Basic Concepts: An Alternative Method for Checking Solutions to Quadratic Equations......Page 368
A. Quadratic Equation Models......Page 369
B. Nonlinear Functions and Rates of Change......Page 372
C. The Average Rate of Change Formula......Page 373
A. Sums and Differences of Functions......Page 381
B. Products and Quotients of Functions......Page 382
C. Graphical and Numerical Views of Operations on Functions......Page 384
D. Applications of the Algebra of Functions......Page 386
A. The Composition of Functions......Page 393
B. A Numerical and Graphical View of the Composition of Functions......Page 398
C. Average Rates of Change and the Difference Quotient......Page 400
D. Applications of Composition and the Difference Quotient......Page 403
Summary and Concept Review......Page 411
Practice Test......Page 416
Calculator Exploration and Discovery: Residuals, Correlation Coefficients, and Goodness of Fit......Page 417
Strengthening Core Skills: Base Functions and Quadratic Graphs......Page 419
Cumulative Review: Chapters R–3......Page 420
CHAPTER 4 Polynomial and Rational Functions......Page 422
A. Long Division and Synthetic Division......Page 423
B. The Remainder Theorem......Page 427
C. The Factor Theorem......Page 428
D. Applications......Page 430
A. The Fundamental Theorem of Algebra......Page 435
B. Real Polynomials and the Intermediate Value Theorem......Page 438
C. The Rational Zeroes Theorem......Page 440
D. Descartes’ Rule of Signs and Upper/Lower Bounds......Page 443
E. Applications of Polynomial Functions......Page 446
A. Identifying the Graph of a Polynomial Function......Page 452
B. The End-Behavior of a Polynomial Graph......Page 453
C. Attributes of Polynomial Graphs with Zeroes of Multiplicity......Page 456
D. The Graph of a Polynomial Function......Page 460
E. Applications of Polynomials and Polynomial Modeling......Page 462
Mid-Chapter Check......Page 469
Reinforcing Basic Concepts: Approximating Real Zeroes......Page 470
A. Rational Functions and Vertical Asymptotes......Page 471
B. Vertical Asymptotes and Multiplicities......Page 473
C. Finding Horizontal Asymptotes......Page 474
D. The Graph of a Rational Function......Page 476
E. Applications of Rational Functions......Page 480
4.5 Additional Insights into Rational Functions......Page 486
A. Rational Functions and Removable Discontinuities......Page 487
B. Rational Functions with Oblique or Nonlinear Asymptotes......Page 489
C. Applications of Rational Functions......Page 492
A. Polynomial Inequalities......Page 500
B. Rational Inequalities......Page 503
C. Applications of Inequalities......Page 505
Summary and Concept Review......Page 511
Practice Test......Page 515
Calculator Exploration and Discovery: Complex Zeroes, Repeated Zeroes, and Inequalities......Page 516
Strengthening Core Skills: Solving Inequalities Using the Push Principle......Page 517
Cumulative Review: Chapters R–4......Page 518
CHAPTER 5 Exponential and Logarithmic Functions......Page 520
A. Identifying One-to-One Functions......Page 521
B. Inverse Functions and Ordered Pairs......Page 522
C. Finding Inverse Functions Using an Algebraic Method......Page 523
D. The Graph of a Function and Its Inverse......Page 526
E. Applications of Inverse Functions......Page 528
A. Evaluating Exponential Functions......Page 533
B. Graphing Exponential Functions......Page 534
C. The Base-e Exponential Function: f(x) = e[sup(x)]......Page 536
D. Solving Exponential Equations Using the Uniqueness Property......Page 537
A. Exponential Equations and Logarithmic Form......Page 544
B. Finding Common Logarithms and Natural Logarithms......Page 546
C. Graphing Logarithmic Functions......Page 547
D. Finding the Domain of a Logarithmic Function......Page 548
E. Applications of Logarithms......Page 549
A. Solving Equations Using the Fundamental Properties of Logarithms......Page 557
B. The Product, Quotient, and Power Properties of Logarithms......Page 560
D. Solving Applications of Logarithms......Page 563
Mid-Chapter Check......Page 567
A. Solving Logarithmic and Exponential Equations......Page 568
B. Applications of Logistic, Exponential, and Logarithmic Functions......Page 574
A. Simple and Compound Interest......Page 580
B. Interest Compounded Continuously......Page 582
C. Applications Involving Annuities and Amortization......Page 583
D. Applications Involving Exponential Growth and Decay......Page 586
A. Choosing an Appropriate Form of Regression......Page 593
B. Exponential and Logarithmic Regression Models......Page 594
C. Logistic Equations and Regression Models......Page 596
D. Applications of Regression......Page 597
Making Connections......Page 606
Summary and Concept Review......Page 607
Practice Test......Page 612
Calculator Exploration and Discovery: Investigating Logistic Equations......Page 613
Strengthening Core Skills: The HerdBurn Scale—What’s Hot and What’s Not......Page 614
Cumulative Review: Chapters R–5......Page 615
CHAPTER 6 Systems of Equations and Inequalities......Page 616
A. Solutions to a System of Equations......Page 617
B. Solving Systems Graphically......Page 618
C. Solving Systems by Substitution......Page 619
D. Solving Systems Using Elimination......Page 620
E. Inconsistent and Dependent Systems......Page 622
F. Systems and Modeling......Page 623
A. Visualizing Solutions in Three Dimensions......Page 632
B. Solutions to a System of Three Equations in Three Variables......Page 633
C. Solving Systems of Three Equations in Three Variables Using Elimination......Page 634
D. Inconsistent and Dependent Systems......Page 637
E. Applications......Page 639
Reinforcing Basic Concepts: Window Size and Graphing Technology......Page 644
A. Possible Solutions for a Nonlinear System......Page 645
B. Solving Nonlinear Systems by Substitution......Page 646
C. Solving Nonlinear Systems by Elimination......Page 648
D. Solving Systems of Nonlinear Inequalities......Page 650
E. Applications of Nonlinear Systems......Page 651
A. Linear Inequalities in Two Variables......Page 656
B. Solving Systems of Linear Inequalities......Page 659
C. Applications of Systems of Linear Inequalities......Page 660
D. Linear Programming......Page 661
Making Connections......Page 670
Summary and Concept Review......Page 671
Practice Test......Page 673
Calculator Exploration and Discovery: Optimal Solutions and Linear Programming......Page 674
Strengthening Core Skills: Understanding Why Elimination and Substitution “Work”......Page 675
Cumulative Review: Chapters R–6......Page 676
CHAPTER 7 Matrices and Matrix Applications......Page 678
B. The Augmented Matrix of a System of Equations......Page 679
C. Solving a System Using Matrices......Page 681
D. Solving Systems of Equations Using Technology......Page 684
E. Inconsistent and Dependent Systems......Page 685
F. Solving Applications Using Matrices......Page 686
A. Equality of Matrices......Page 691
B. Addition and Subtraction of Matrices......Page 693
C. Matrices and Multiplication......Page 694
Mid-Chapter Check......Page 703
A. Multiplication and Identity Matrices......Page 704
B. The Inverse of a Matrix......Page 706
C. Solving Systems Using Matrix Equations......Page 708
D. Determinants and Singular Matrices......Page 709
A. Solving Systems Using Determinants and Cramer’s Rule......Page 720
B. Rational Expressions and Partial Fractions......Page 723
C. Determinants, Geometry, and the Coordinate Plane......Page 729
A. Solving Static Systems with Varying Constraints......Page 733
B. Using Matrices to Encrypt Messages......Page 735
Making Connections......Page 740
Summary and Concept Review......Page 741
Practice Test......Page 743
Strengthening Core Skills: Augmented Matrices and Matrix Inverses......Page 745
Cumulative Review: Chapters R–7......Page 746
CHAPTER 8 Analytic Geometry and the Conic Sections......Page 748
A. Verifying Relationships from Plane Geometry......Page 749
B. The Distance between a Point and a Line......Page 750
C. Characteristics of the Conic Sections......Page 751
A. The Equation and Graph of a Circle......Page 756
B. The Equation of an Ellipse......Page 757
C. The Foci of an Ellipse......Page 761
D. Applications Involving Foci......Page 765
Mid-Chapter Check......Page 770
Reinforcing Basic Concepts: More on Completing the Square......Page 771
A. The Equation of a Hyperbola......Page 772
B. Distinguishing between the Equations of Circles, Ellipses, and Hyperbolas......Page 777
C. The Foci of a Hyperbola......Page 778
D. Applications Involving Foci......Page 780
A. Parabolas with a Horizontal Axis......Page 785
B. The Focus-Directrix Form of the Equation of a Parabola......Page 787
C. Nonlinear Systems and the Conic Sections......Page 790
D. Application of the Analytic Parabola......Page 791
Making Connections......Page 795
Summary and Concept Review......Page 796
Practice Test......Page 798
Calculator Exploration and Discovery: Elongation and Eccentricity......Page 799
Strengthening Core Skills: Ellipses and Hyperbolas with Rational/Irrational Values of a and b......Page 800
Cumulative Review: Chapters R–8......Page 801
CHAPTER 9 Additional Topics in Algebra......Page 802
A. Finding the Terms of a Sequence Given the General Term......Page 803
B. Recursive Sequences and Factorial Notation......Page 805
C. Series and Partial Sums......Page 806
D. Summation Notation......Page 807
E. Applications of Sequences......Page 809
A. Identifying an Arithmetic Sequence and Finding the Common Difference......Page 814
B. Finding the n th Term of an Arithmetic Sequence......Page 815
C. Finding the n th Partial Sum of an Arithmetic Sequence......Page 818
D. Applications......Page 820
A. Geometric Sequences......Page 823
B. Find the nth Term of a Geometric Sequence......Page 824
C. Find the nth Partial Sum of a Geometric Sequence......Page 828
D. The Sum of an Infinite Geometric Series......Page 829
E. Applications Involving Geometric Sequences and Series......Page 830
B. Mathematical Induction Applied to Sums......Page 837
C. The General Principle of Mathematical Induction......Page 840
Reinforcing Basic Concepts: Applications of Summation......Page 844
A. Counting by Listing and Tree Diagrams......Page 845
B. Fundamental Principle of Counting......Page 847
C. Distinguishable Permutations......Page 848
E. Combinations......Page 850
A. Defining an Event......Page 857
B. Elementary Probability......Page 858
C. Properties of Probability......Page 859
D. Probability and Quick-Counting......Page 861
E. Probability and Nonexclusive Events......Page 862
A. Binomial Powers and Pascal’s Triangle......Page 870
B. Binomial Coefficients and Factorials......Page 872
C. The Binomial Theorem......Page 874
E. Applications......Page 875
Making Connections......Page 878
Summary and Concept Review......Page 879
Practice Test......Page 883
Calculator Exploration and Discovery: Infinite Series, Finite Results......Page 885
Strengthening Core Skills: Probability, Quick-Counting, and Card Games......Page 886
Cumulative Review: Chapters R–9......Page 887
Appendix I: The Language, Notation, and Numbers of Mathematics......Page 890
Appendix II: Geometry Review with Unit Conversions......Page 903
Appendix III: More on Synthetic Division......Page 917
Appendix IV: More on Matrices......Page 919
Appendix V: Deriving the Equation of a Conic......Page 921
Appendix VI: Proof Positive—A Selection of Proofs from College Algebra......Page 923
Student Answer Appendix (SE only)......Page 926
Index......Page 978