Algebra and Trigonometry

Cover......Page 1
Title Page......Page 8
Copyright Page......Page 9
Acknowledgments......Page 25
Contents......Page 10
Three Distinct Series......Page 21
The Contemporary Series......Page 22
Preface to the Instructor......Page 23
Resources for Success......Page 27
B......Page 29
D......Page 30
E......Page 31
G......Page 32
N......Page 33
S......Page 34
W......Page 35
R: Review......Page 36
Work with Sets......Page 37
Classify Numbers......Page 39
Evaluate Numerical Expressions......Page 43
Work with Properties of Real Numbers......Page 44
R.2 Algebra Essentials......Page 52
Graph Inequalities......Page 53
Find Distance on the Real Number Line......Page 54
Evaluate Algebraic Expressions......Page 55
Use the Laws of Exponents......Page 56
Evaluate Square Roots......Page 58
Use Scientific Notation......Page 59
Use the Pythagorean Theorem and Its Converse......Page 65
Know Geometry Formulas......Page 66
Understand Congruent Triangles and Similar Triangles......Page 67
Recognize Monomials......Page 74
Recognize Polynomials......Page 75
Add and Subtract Polynomials......Page 76
Multiply Polynomials......Page 77
Know Formulas for Special Products......Page 78
Divide Polynomials Using Long Division......Page 79
Work with Polynomials in Two Variables......Page 82
R.5 Factoring Polynomials......Page 84
Factor the Difference of Two Squares and the Sum and Difference of Two Cubes......Page 85
Factor Perfect Squares......Page 86
Factor a Second-Degree Polynomial: x[sup(2)] + Bx + C......Page 87
Factor by Grouping......Page 88
Factor a Second-Degree Polynomial: Ax[sup(2)] + Bx + C, A ≠ 1......Page 89
Complete the Square......Page 91
Divide Polynomials Using Synthetic Division......Page 93
Reduce a Rational Expression to Lowest Terms......Page 97
Multiply and Divide Rational Expressions......Page 98
Add and Subtract Rational Expressions......Page 99
Use the Least Common Multiple Method......Page 101
Simplify Complex Rational Expressions......Page 103
Work with nth Roots......Page 108
Simplify Radicals......Page 109
Rationalize Denominators......Page 110
Simplify Expressions with Rational Exponents......Page 111
1 Equations and Inequalities......Page 116
1.1 Linear Equations......Page 117
Solve a Linear Equation......Page 119
Solve Equations That Lead to Linear Equations......Page 121
Solve Problems That Can Be Modeled by Linear Equations......Page 122
1.2 Quadratic Equations......Page 127
Solve a Quadratic Equation by Factoring......Page 128
Solve a Quadratic Equation by Completing the Square......Page 130
Solve a Quadratic Equation Using the Quadratic Formula......Page 131
Solve Problems That Can Be Modeled by Quadratic Equations......Page 134
1.3 Complex Numbers; Quadratic Equations in the Complex Number System......Page 139
Add, Subtract, Multiply, and Divide Complex Numbers......Page 140
Solve Quadratic Equations in the Complex Number System......Page 144
Solve Radical Equations......Page 148
Solve Equations Quadratic in Form......Page 149
Solve Equations by Factoring......Page 151
1.5 Solving Inequalities......Page 154
Use Interval Notation......Page 155
Use Properties of Inequalities......Page 156
Solve Inequalities......Page 158
Solve Combined Inequalities......Page 159
Solve Inequalities Involving Absolute Value......Page 165
1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications......Page 169
Translate Verbal Descriptions into Mathematical Expressions......Page 170
Solve Interest Problems......Page 171
Solve Mixture Problems......Page 172
Solve Uniform Motion Problems......Page 173
Solve Constant Rate Job Problems......Page 175
Chapter Review......Page 178
Chapter Projects......Page 182
2 Graphs......Page 184
2.1 The Distance and Midpoint Formulas......Page 185
Use the Distance Formula......Page 186
Use the Midpoint Formula......Page 188
Graph Equations by Plotting Points......Page 192
Find Intercepts from a Graph......Page 194
Test an Equation for Symmetry with Respect to the x-Axis, the y-Axis, and the Origin......Page 195
Know How to Graph Key Equations......Page 198
Calculate and Interpret the Slope of a Line......Page 202
Find the Equation of a Vertical Line......Page 205
Use the Point–Slope Form of a Line; Identify Horizontal Lines......Page 206
Write the Equation of a Line in Slope–Intercept Form......Page 207
Identify the Slope and y-Intercept of a Line from Its Equation......Page 208
Graph Lines Written in General Form Using Intercepts......Page 209
Find Equations of Parallel Lines......Page 210
Find Equations of Perpendicular Lines......Page 211
Write the Standard Form of the Equation of a Circle......Page 217
Graph a Circle......Page 218
Work with the General Form of the Equation of a Circle......Page 219
2.5 Variation......Page 223
Construct a Model Using Inverse Variation......Page 224
Construct a Model Using Joint Variation or Combined Variation......Page 225
Chapter Review......Page 229
Cumulative Review......Page 231
Chapter Project......Page 232
3 Functions and Their Graphs......Page 233
Determine Whether a Relation Represents a Function......Page 234
Find the Value of a Function......Page 237
Find the Difference Quotient of a Function......Page 240
Find the Domain of a Function Defined by an Equation......Page 241
Form the Sum, Difference, Product, and Quotient of Two Functions......Page 243
Identify the Graph of a Function......Page 249
Obtain Information from or about the Graph of a Function......Page 250
Determine Even and Odd Functions from a Graph......Page 258
Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant......Page 260
Use a Graph to Locate Local Maxima and Local Minima......Page 261
Use a Graph to Locate the Absolute Maximum and the Absolute Minimum......Page 262
Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing......Page 264
Find the Average Rate of Change of a Function......Page 265
Graph the Functions Listed in the Library of Functions......Page 272
Graph Piecewise-defined Functions......Page 277
3.5 Graphing Techniques: Transformations......Page 282
Graph Functions Using Vertical and Horizontal Shifts......Page 283
Graph Functions Using Compressions and Stretches......Page 286
Graph Functions Using Reflections about the x-Axis and the y-Axis......Page 288
Build and Analyze Functions......Page 295
Chapter Review......Page 301
Chapter Test......Page 305
Chapter Projects......Page 306
4 Linear and Quadratic Functions......Page 308
Use Average Rate of Change to Identify Linear Functions......Page 309
Determine Whether a Linear Function Is Increasing, Decreasing, or Constant......Page 312
Build Linear Models from Verbal Descriptions......Page 313
Draw and Interpret Scatter Diagrams......Page 319
Distinguish between Linear and Nonlinear Relations......Page 320
Use a Graphing Utility to Find the Line of Best Fit......Page 321
4.3 Quadratic Functions and Their Properties......Page 325
Graph a Quadratic Function Using Transformations......Page 327
Graph a Quadratic Function Using Its Vertex, Axis, and Intercepts......Page 329
Find a Quadratic Function Given Its Vertex and One Other Point......Page 332
Find the Maximum or Minimum Value of a Quadratic Function......Page 333
Build Quadratic Models from Verbal Descriptions......Page 337
Build Quadratic Models from Data......Page 341
Solve Inequalities Involving a Quadratic Function......Page 347
Chapter Review......Page 350
Chapter Test......Page 353
Cumulative Review......Page 354
Chapter Projects......Page 355
5 Polynomial and Rational Functions......Page 356
Identify Polynomial Functions and Their Degree......Page 357
Graph Polynomial Functions Using Transformations......Page 361
Identify the Real Zeros of a Polynomial Function and Their Multiplicity......Page 362
Analyze the Graph of a Polynomial Function......Page 369
Build Cubic Models from Data......Page 371
Find the Domain of a Rational Function......Page 378
Find the Vertical Asymptotes of a Rational Function......Page 381
Find the Horizontal or Oblique Asymptote of a Rational Function......Page 383
Analyze the Graph of a Rational Function......Page 388
Solve Applied Problems Involving Rational Functions......Page 399
Solve Polynomial Inequalities......Page 403
Solve Rational Inequalities......Page 405
Use the Remainder and Factor Theorems......Page 410
Use Descartes' Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function......Page 413
Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function......Page 414
Find the Real Zeros of a Polynomial Function......Page 415
Solve Polynomial Equations......Page 417
Use the Theorem for Bounds on Zeros......Page 418
Use the Intermediate Value Theorem......Page 419
5.6 Complex Zeros; Fundamental Theorem of Algebra......Page 425
Use the Conjugate Pairs Theorem......Page 426
Find a Polynomial Function with Specified Zeros......Page 427
Find the Complex Zeros of a Polynomial Function......Page 428
Chapter Review......Page 431
Cumulative Review......Page 434
Chapter Projects......Page 435
6 Exponential and Logarithmic Functions......Page 437
Form a Composite Function......Page 438
Find the Domain of a Composite Function......Page 439
Determine Whether a Function Is One-to-One......Page 446
Determine the Inverse of a Function Defined by a Map or a Set of Ordered Pairs......Page 448
Obtain the Graph of the Inverse Function from the Graph of the Function......Page 451
Find the Inverse of a Function Defined by an Equation......Page 452
Evaluate Exponential Functions......Page 458
Graph Exponential Functions......Page 462
Define the Number e......Page 465
Solve Exponential Equations......Page 467
Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements......Page 475
Determine the Domain of a Logarithmic Function......Page 476
Graph Logarithmic Functions......Page 477
Solve Logarithmic Equations......Page 481
Work with the Properties of Logarithms......Page 487
Write a Logarithmic Expression as a Sum or Difference of Logarithms......Page 489
Write a Logarithmic Expression as a Single Logarithm......Page 490
Evaluate Logarithms Whose Base Is Neither 10 Nor e......Page 492
Solve Logarithmic Equations......Page 496
Solve Exponential Equations......Page 498
Solve Logarithmic and Exponential Equations Using a Graphing Utility......Page 499
Determine the Future Value of a Lump Sum of Money......Page 503
Calculate Effective Rates of Return......Page 506
Determine the Present Value of a Lump Sum of Money......Page 507
Determine the Rate of Interest or the Time Required to Double a Lump Sum of Money......Page 508
Find Equations of Populations That Obey the Law of Uninhibited Growth......Page 513
Find Equations of Populations That Obey the Law of Decay......Page 515
Use Newton's Law of Cooling......Page 516
Use Logistic Models......Page 518
Build an Exponential Model from Data......Page 524
Build a Logistic Model from Data......Page 526
Chapter Review......Page 532
Cumulative Review......Page 537
Chapter Projects......Page 538
7 Trigonometric Functions......Page 540
7.1 Angles and Their Measure......Page 541
Convert between Decimal and Degree, Minute, Second Measures for Angles......Page 543
Find the Length of an Arc of a Circle......Page 544
Convert from Degrees to Radians and from Radians to Degrees......Page 545
Find the Area of a Sector of a Circle......Page 547
Find the Linear Speed of an Object Traveling in Circular Motion......Page 548
Find the Values of Trigonometric Functions of Acute Angles......Page 554
Use Fundamental Identities......Page 556
Find the Values of the Remaining Trigonometric Functions, Given the Value of One of Them......Page 558
Use the Complementary Angle Theorem......Page 560
Find the Exact Values of the Trigonometric Functions of (Omitted)......Page 566
Find the Exact Values of the Trigonometric Functions of (Omitted) and (Omitted)......Page 567
Model and Solve Applied Problems Involving Right Triangles......Page 569
Find the Exact Values of the Trigonometric Functions for Any Angle......Page 578
Use Coterminal Angles to Find the Exact Value of a Trigonometric Function......Page 580
Find the Reference Angle of an Angle......Page 582
Use a Reference Angle to Find the Exact Value of a Trigonometric Function......Page 583
Find the Exact Values of the Trigonometric Functions of an Angle, Given Information about the Functions......Page 585
Find the Exact Values of the Trigonometric Functions Using the Unit Circle......Page 588
Know the Domain and Range of the Trigonometric Functions......Page 592
Use Periodic Properties to Find the Exact Values of the Trigonometric Functions......Page 594
Use Even–Odd Properties to Find the Exact Values of the Trigonometric Functions......Page 595
7.6 Graphs of the Sine and Cosine Functions......Page 599
Graph Functions of the Form y = A sin (ωx) Using Transformations......Page 600
Graph Functions of the Form y = A cos (ωx) Using Transformations......Page 602
Determine the Amplitude and Period of Sinusoidal Functions......Page 603
Graph Sinusoidal Functions Using Key Points......Page 605
Find an Equation for a Sinusoidal Graph......Page 608
7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions......Page 614
Graph Functions of the Form y = A tan (ωx) + B and y = A cot (ωx) + B......Page 616
Graph Functions of the Form y = A csc (ωx) + B and y = A sec (ωx) + B......Page 618
Graph Sinusoidal Functions of the Form y = A sin (ωx – ϕ) + B......Page 622
Build Sinusoidal Models from Data......Page 625
Chapter Review......Page 632
Cumulative Review......Page 638
Chapter Projects......Page 639
8 Analytic Trigonometry......Page 641
8.1 The Inverse Sine, Cosine, and Tangent Functions......Page 642
Find the Exact Value of an Inverse Sine Function......Page 643
Find an Approximate Value of an Inverse Sine Function......Page 644
Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions......Page 645
Find the Inverse Function of a Trigonometric Function......Page 650
Solve Equations Involving Inverse Trigonometric Functions......Page 651
Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions......Page 655
Define the Inverse Secant, Cosecant, and Cotangent Functions......Page 656
Use a Calculator to Evaluate sec[sup(-1)]x, csc[sup(-1)]x, and cot[sup(-1)]x......Page 657
Write a Trigonometric Expression as an Algebraic Expression......Page 658
Solve Equations Involving a Single Trigonometric Function......Page 660
Solve Trigonometric Equations Using a Calculator......Page 663
Solve Trigonometric Equations Using Fundamental Identities......Page 664
Solve Trigonometric Equations Using a Graphing Utility......Page 665
8.4 Trigonometric Identities......Page 670
Use Algebra to Simplify Trigonometric Expressions......Page 671
Establish Identities......Page 672
8.5 Sum and Difference Formulas......Page 678
Use Sum and Difference Formulas to Find Exact Values......Page 679
Use Sum and Difference Formulas to Establish Identities......Page 682
Use Sum and Difference Formulas Involving Inverse Trigonometric Functions......Page 684
Solve Trigonometric Equations Linear in Sine and Cosine......Page 685
8.6 Double-angle and Half-angle Formulas......Page 690
Use Double-angle Formulas to Establish Identities......Page 691
Use Half-angle Formulas to Find Exact Values......Page 694
Express Products as Sums......Page 700
Express Sums as Products......Page 701
Chapter Review......Page 704
Chapter Test......Page 707
Cumulative Review......Page 708
Chapter Projects......Page 709
9 Applications of Trigonometric Functions......Page 710
Solve Right Triangles......Page 711
Solve Applied Problems......Page 712
9.2 The Law of Sines......Page 717
Solve SAA or ASA Triangles......Page 718
Solve SSA Triangles......Page 719
Solve Applied Problems......Page 721
9.3 The Law of Cosines......Page 727
Solve SAS Triangles......Page 728
Solve Applied Problems......Page 729
Find the Area of SAS Triangles......Page 734
Find the Area of SSS Triangles......Page 735
Build a Model for an Object in Simple Harmonic Motion......Page 740
Analyze Simple Harmonic Motion......Page 742
Analyze an Object in Damped Motion......Page 743
Graph the Sum of Two Functions......Page 744
Chapter Review......Page 749
Chapter Test......Page 751
Cumulative Review......Page 752
Chapter Projects......Page 753
10 Polar Coordinates; Vectors......Page 754
Plot Points Using Polar Coordinates......Page 755
Convert from Polar Coordinates to Rectangular Coordinates......Page 757
Convert from Rectangular Coordinates to Polar Coordinates......Page 759
Transform Equations between Polar and Rectangular Forms......Page 761
10.2 Polar Equations and Graphs......Page 764
Identify and Graph Polar Equations by Converting to Rectangular Equations......Page 765
Test Polar Equations for Symmetry......Page 768
Graph Polar Equations by Plotting Points......Page 769
Plot Points in the Complex Plane......Page 779
Convert a Complex Number between Rectangular Form and Polar Form......Page 780
Find Products and Quotients of Complex Numbers in Polar Form......Page 781
Use De Moivre's Theorem......Page 782
Find Complex Roots......Page 783
10.4 Vectors......Page 787
Find a Position Vector......Page 789
Add and Subtract Vectors Algebraically......Page 791
Find a Unit Vector......Page 792
Find a Vector from Its Direction and Magnitude......Page 793
Model with Vectors......Page 794
Find the Dot Product of Two Vectors......Page 801
Find the Angle between Two Vectors......Page 802
Decompose a Vector into Two Orthogonal Vectors......Page 803
Compute Work......Page 805
Chapter Review......Page 808
Cumulative Review......Page 811
Chapter Projects......Page 812
11 Analytic Geometry......Page 813
Know the Names of the Conics......Page 814
Analyze Parabolas with Vertex at the Origin......Page 815
Analyze Parabolas with Vertex at (h, k)......Page 818
Solve Applied Problems Involving Parabolas......Page 820
Analyze Ellipses with Center at the Origin......Page 824
Analyze Ellipses with Center at (h, k)......Page 828
Solve Applied Problems Involving Ellipses......Page 830
Analyze Hyperbolas with Center at the Origin......Page 834
Find the Asymptotes of a Hyperbola......Page 839
Analyze Hyperbolas with Center at (h, k)......Page 841
Solve Applied Problems Involving Hyperbolas......Page 842
Identify a Conic......Page 847
Use a Rotation of Axes to Transform Equations......Page 848
Analyze an Equation Using a Rotation of Axes......Page 850
Identify Conics without a Rotation of Axes......Page 852
Analyze and Graph Polar Equations of Conics......Page 855
Convert the Polar Equation of a Conic to a Rectangular Equation......Page 859
Graph Parametric Equations......Page 861
Find a Rectangular Equation for a Curve Defined Parametrically......Page 862
Use Time as a Parameter in Parametric Equations......Page 864
Find Parametric Equations for Curves Defined by Rectangular Equations......Page 867
Chapter Review......Page 873
Cumulative Review......Page 876
Chapter Projects......Page 877
12 Systems of Equations and Inequalities......Page 878
12.1 Systems of Linear Equations: Substitution and Elimination......Page 879
Solve Systems of Equations by Substitution......Page 881
Solve Systems of Equations by Elimination......Page 882
Express the Solution of a System of Dependent Equations Containing Two Variables......Page 884
Solve Systems of Three Equations Containing Three Variables......Page 885
Identify Inconsistent Systems of Equations Containing Three Variables......Page 887
Express the Solution of a System of Dependent Equations Containing Three Variables......Page 888
Write the Augmented Matrix of a System of Linear Equations......Page 894
Write the System of Equations from the Augmented Matrix......Page 895
Perform Row Operations on a Matrix......Page 896
Solve a System of Linear Equations Using Matrices......Page 897
Evaluate 2 by 2 Determinants......Page 909
Use Cramer's Rule to Solve a System of Two Equations Containing Two Variables......Page 910
Evaluate 3 by 3 Determinants......Page 912
Use Cramer's Rule to Solve a System of Three Equations Containing Three Variables......Page 914
Know Properties of Determinants......Page 915
12.4 Matrix Algebra......Page 919
Find the Sum and Difference of Two Matrices......Page 920
Find Scalar Multiples of a Matrix......Page 922
Find the Product of Two Matrices......Page 923
Find the Inverse of a Matrix......Page 927
Solve a System of Linear Equations Using an inverse Matrix......Page 931
12.5 Partial Fraction Decomposition......Page 936
Decompose (Omitted), Where Q Has Only Nonrepeated Linear Factors......Page 937
Decompose (Omitted), Where Q Has Repeated Linear Factors......Page 939
Decompose (Omitted), Where Q Has a Nonrepeated Irreducible Quadratic Factor......Page 941
Decompose (Omitted), Where Q Has a Repeated Irreducible Quadratic Factor......Page 942
Solve a System of Nonlinear Equations Using Substitution......Page 944
Solve a System of Nonlinear Equations Using Elimination......Page 945
Graph an Inequality......Page 953
Graph a System of Inequalities......Page 955
12.8 Linear Programming......Page 960
Solve a Linear Programming Problem......Page 961
Chapter Review......Page 967
Chapter Test......Page 971
Chapter Projects......Page 972
13 Sequences; Induction; the Binomial Theorem......Page 974
Write the First Several Terms of a Sequence......Page 975
Write the Terms of a Sequence Defined by a Recursive Formula......Page 978
Use Summation Notation......Page 979
Find the Sum of a Sequence......Page 980
Determine Whether a Sequence Is Arithmetic......Page 985
Find a Formula for an Arithmetic Sequence......Page 986
Find the Sum of an Arithmetic Sequence......Page 987
Determine Whether a Sequence Is Geometric......Page 991
Find a Formula for a Geometric Sequence......Page 992
Find the Sum of a Geometric Sequence......Page 993
Determine Whether a Geometric Series Converges or Diverges......Page 994
Solve Annuity Problems......Page 997
Prove Statements Using Mathematical Induction......Page 1002
Evaluate (Omitted)......Page 1006
Use the Binomial Theorem......Page 1008
Chapter Review......Page 1012
Cumulative Review......Page 1015
Chapter Projects......Page 1016
14 Counting and Probability......Page 1017
Count the Number of Elements in a Set......Page 1018
Solve Counting Problems Using the Multiplication Principle......Page 1020
Solve Counting Problems Using Permutations Involving n Distinct Objects......Page 1023
Solve Counting Problems Using Combinations......Page 1026
Solve Counting Problems Using Permutations Involving n Nondistinct Objects......Page 1028
Construct Probability Models......Page 1032
Compute Probabilities of Equally Likely Outcomes......Page 1034
Find Probabilities of the Union of Two Events......Page 1036
Use the Complement Rule to Find Probabilities......Page 1037
Chapter Review......Page 1042
Chapter Test......Page 1044
Chapter Projects......Page 1045
A.1 The Viewing Rectangle......Page 1048
A.2 Using a Graphing Utility to Graph Equations......Page 1050
A.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry......Page 1052
A.4 Using a Graphing Utility to Solve Equations......Page 1053
A.5 Square Screens......Page 1055
A.7 Using a Graphing Utility to Solve Systems of Linear Equations......Page 1056
A.9 Using a Graphing Utility to Graph Parametric Equations......Page 1058
Answers......Page 1060
Credits......Page 1164
A......Page 1166
C......Page 1167
D......Page 1168
E......Page 1169
G......Page 1170
I......Page 1171
L......Page 1172
N......Page 1173
P......Page 1174
R......Page 1175
S......Page 1176
V......Page 1178
Z......Page 1179