Algebra and Trigonometry

1114 Pages·2011·15.51 MB·English

by  Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger

Algebra and
t h
4
Trigonometry
E D I T I O N
Judith A. Beecher
Indiana University Purdue University Indianapolis
Judith A. Penna
Indiana University Purdue University Indianapolis
Marvin L. Bittinger
Indiana University Purdue University Indianapolis
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Algebra and trigonometry / Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger.—4th ed. student learning.
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Contents
Preface xi
R Basic Concepts of Algebra 1
R.1 The Real-Number System 2
Real Numbers / Interval Notation / Properties of the Real Numbers /
Absolute Value
R.2 Integer Exponents, Scientific Notation, and Order of Operations 8
Integers as Exponents / Scientific Notation / Order of Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials 17
Polynomials / Addition and Subtraction / Multiplication
R.4 Factoring 22
Terms with Common Factors / Factoring by Grouping / Trinomials of the Type
x2 + bx + c / Trinomials of the Type ax2 + bx + c, a Z 1 / Special Factorizations
R.5 The Basics of Equation Solving 31
Linear Equations and Quadratic Equations / Formulas
R.6 Rational Expressions 36
The Domain of a Rational Expression / Simplifying, Multiplying, and Dividing
Rational Expressions / Adding and Subtracting Rational Expressions /
Complex Rational Expressions
R.7 Radical Notation and Rational Exponents 45
Simplifying Radical Expressions / An Application / Rationalizing Denominators or
Numerators / Rational Exponents
Study Guide 53
Review Exercises 54
Chapter Test 56
1 Graphs, Functions, and Models 59
1.1 Introduction to Graphing 60
Graphs / Solutions of Equations / Graphs of Equations / The Distance Formula /
Midpoints of Segments / Circles
VISUALIZING THE GRAPH 71
1.2 Functions and Graphs 75
Functions / Notation for Functions / Graphs of Functions / Finding Domains of
Functions / Visualizing Domain and Range / Applications of Functions
C O N T E N T S iii
iv C O N T E N T S
1.3 Linear Functions, Slope, and Applications 91
Linear Functions / The Linear Function f1x2 = mx + b and Slope /
Applications of Slope / Slope–Intercept Equations of Lines /
Graphing f1x2 = mx + b Using m and b / Applications of Linear Functions
VISUALIZING THE GRAPH 102
MID-CHAPTER MIXED REVIEW 107
1.4 Equations of Lines and Modeling 109
Slope–Intercept Equations of Lines / Point–Slope Equations of Lines /
Parallel Lines / Perpendicular Lines / Mathematical Models / Curve Fitting
1.5 Linear Equations, Functions, Zeros, and Applications 121
Linear Equations / Applications Using Linear Models / Zeros of Linear Functions
1.6 Solving Linear Inequalities 138
Linear Inequalities / Compound Inequalities / An Application
Study Guide 144
Review Exercises 150
Chapter Test 154
2 More on Functions 157
2.1 Increasing, Decreasing, and Piecewise Functions; Applications 158
Increasing, Decreasing, and Constant Functions / Relative Maximum
and Minimum Values / Applications of Functions / Functions
Defined Piecewise
2.2 The Algebra of Functions 171
The Algebra of Functions: Sums, Differences, Products, and Quotients /
Difference Quotients
2.3 The Composition of Functions 179
The Composition of Functions / Decomposing a Function as a Composition
MID-CHAPTER MIXED REVIEW 187
2.4 Symmetry and Transformations 189
Symmetry / Even Functions and Odd Functions / Transformations of Functions /
Vertical Translations and Horizontal Translations / Reflections / Vertical and
Horizontal Stretchings and Shrinkings
VISUALIZING THE GRAPH 205
2.5 Variation and Applications 211
Direct Variation / Inverse Variation / Combined Variation
Study Guide 220
Review Exercises 227
Chapter Test 231
C O N T E N T S v
3 Quadratic Functions and Equations; Inequalities 233
3.1 The Complex Numbers 234
The Complex-Number System / Addition and Subtraction /
Multiplication / Conjugates and Division
3.2 Quadratic Equations, Functions, Zeros, and Models 241
Quadratic Equations and Quadratic Functions / Completing the Square /
Using the Quadratic Formula / The Discriminant / Equations Reducible to
Quadratic / Applications
3.3 Analyzing Graphs of Quadratic Functions 257
Graphing Quadratic Functions of the Type f1x2 = a1x - h22 + k /
Graphing Quadratic Functions of the Type f1x2 = ax2 + bx + c, a Z 0 /
Applications
VISUALIZING THE GRAPH 266
MID-CHAPTER MIXED REVIEW 270
3.4 Solving Rational Equations and Radical Equations 272
Rational Equations / Radical Equations
3.5 Solving Equations and Inequalities with Absolute Value 280
Equations with Absolute Value / Inequalities with Absolute Value
Study Guide 285
Review Exercises 289
Chapter Test 293
4 Polynomial Functions and Rational Functions 295
4.1 Polynomial Functions and Models 296
The Leading-Term Test / Finding Zeros of Factored Polynomial Functions /
Polynomial Models
4.2 Graphing Polynomial Functions 309
Graphing Polynomial Functions / The Intermediate Value Theorem
VISUALIZING THE GRAPH 317
4.3 Polynomial Division; The Remainder Theorem
and the Factor Theorem 320
Division and Factors / The Remainder Theorem and Synthetic Division /
Finding Factors of Polynomials
MID-CHAPTER MIXED REVIEW 328
4.4 Theorems about Zeros of Polynomial Functions 330
The Fundamental Theorem of Algebra / Finding Polynomials with
Given Zeros / Zeros of Polynomial Functions with Real Coefficients /
Rational Coefficients / Integer Coefficients and the Rational Zeros Theorem /
Descartes’ Rule of Signs
vi C O N T E N T S
4.5 Rational Functions 340
The Domain of a Rational Function / Asymptotes / Applications
VISUALIZING THE GRAPH 356
4.6 Polynomial Inequalities and Rational Inequalities 360
Polynomial Inequalities / Rational Inequalities
Study Guide 372
Review Exercises 381
Chapter Test 385
5 Exponential Functions and Logarithmic Functions 387
5.1 Inverse Functions 388
Inverses / Inverses and One-to-One Functions / Finding Formulas for Inverses /
Inverse Functions and Composition / Restricting a Domain
5.2 Exponential Functions and Graphs 400
Graphing Exponential Functions / Applications / The Number e /
Graphs of Exponential Functions, Base e
5.3 Logarithmic Functions and Graphs 413
Logarithmic Functions / Finding Certain Logarithms / Converting Between Exponential
Equations and Logarithmic Equations / Finding Logarithms on a Calculator / Natural
Logarithms / Changing Logarithmic Bases / Graphs of Logarithmic Functions /
Applications
VISUALIZING THE GRAPH 425
MID-CHAPTER MIXED REVIEW 429
5.4 Properties of Logarithmic Functions 430
Logarithms of Products / Logarithms of Powers / Logarithms of Quotients /
Applying the Properties / Simplifying Expressions of the Type loga ax and aloga x
5.5 Solving Exponential Equations and Logarithmic Equations 439
Solving Exponential Equations / Solving Logarithmic Equations
5.6 Applications and Models: Growth and Decay; Compound Interest 450
Population Growth / Interest Compounded Continuously / Models of
Limited Growth / Exponential Decay
Study Guide 465
Review Exercises 471
Chapter Test 475
6 The Trigonometric Functions 477
6.1 Trigonometric Functions of Acute Angles 478
The Trigonometric Ratios / The Six Functions Related /
Function Values of 30°, 45°, and 60° / Function Values of Any
Acute Angle / Cofunctions and Complements
C O N T E N T S vii
6.2 Applications of Right Triangles 490
Solving Right Triangles / Applications
6.3 Trigonometric Functions of Any Angle 503
Angles, Rotations, and Degree Measure / Trigonometric Functions
of Angles or Rotations / The Six Functions Related / Terminal
Side on an Axis / Reference Angles: 30°, 45°, and 60° /
Function Values for Any Angle
MID-CHAPTER MIXED REVIEW 519
6.4 Radians, Arc Length, and Angular Speed 520
Distances on the Unit Circle / Radian Measure / Arc Length
and Central Angles / Linear Speed and Angular Speed
6.5 Circular Functions: Graphs and Properties 536
Reflections on the Unit Circle / Finding Function Values /
Graphs of the Sine and Cosine Functions / Graphs of the Tangent,
Cotangent, Cosecant, and Secant Functions
6.6 Graphs of Transformed Sine and Cosine Functions 551
Variations of Basic Graphs / Graphs of Sums: Addition of
Ordinates / Damped Oscillation: Multiplication of Ordinates
VISUALIZING THE GRAPH 565
Study Guide 569
Review Exercises 578
Chapter Test 581
Trigonometric Identities, Inverse Functions,
7
and Equations 583
7.1 Identities: Pythagorean and Sum and Difference 584
Pythagorean Identities / Simplifying Trigonometric
Expressions / Sum and Difference Identities
7.2 Identities: Cofunction, Double-Angle, and Half-Angle 597
Cofunction Identities / Double-Angle Identities / Half-Angle Identities
7.3 Proving Trigonometric Identities 605
The Logic of Proving Identities / Proving Identities /
Product-to-Sum and Sum-to-Product Identities
MID-CHAPTER MIXED REVIEW 613
7.4 Inverses of the Trigonometric Functions 614
Restricting Ranges to Define Inverse Functions / Composition of
Trigonometric Functions and Their Inverses
7.5 Solving Trigonometric Equations 625
VISUALIZING THE GRAPH 635
Study Guide 639
Review Exercises 645
Chapter Test 648
viii C O N T E N T S
8 Applications of Trigonometry 649
8.1 The Law of Sines 650
Solving Oblique Triangles / The Law of Sines / Solving
Triangles (AAS and ASA) / Solving Triangles (SSA) /
The Area of a Triangle
8.2 The Law of Cosines 663
The Law of Cosines / Solving Triangles (SAS) /
Solving Triangles (SSS)
8.3 Complex Numbers: Trigonometric Form 673
Graphical Representation / Trigonometric Notation for
Complex Numbers / Multiplication and Division with
Trigonometric Notation / Powers of Complex Numbers /
Roots of Complex Numbers
MID-CHAPTER MIXED REVIEW 684
8.4 Polar Coordinates and Graphs 686
Polar Coordinates / Polar Equations and Rectangular Equations /
Graphing Polar Equations
VISUALIZING THE GRAPH 694
8.5 Vectors and Applications 697
Vectors / Vector Addition / Applications / Components
8.6 Vector Operations 705
Position Vectors / Operations on Vectors /
Unit Vectors / Direction Angles / Angle Between Vectors /
Forces in Equilibrium
Study Guide 720
Review Exercises 731
Chapter Test 735
9 Systems of Equations and Matrices 737
9.1 Systems of Equations in Two Variables 738
Solving Systems of Equations Graphically / The Substitution Method /
The Elimination Method / Applications
VISUALIZING THE GRAPH 749
9.2 Systems of Equations in Three Variables 754
Solving Systems of Equations in Three Variables / Applications /
Mathematical Models and Applications
9.3 Matrices and Systems of Equations 764
Matrices and Row-Equivalent Operations / Gaussian Elimination with Matrices /
Gauss–Jordan Elimination
C O N T E N T S ix
9.4 Matrix Operations 772
Matrix Addition and Subtraction / Scalar Multiplication /
Products of Matrices / Matrix Equations
MID-CHAPTER MIXED REVIEW 783
9.5 Inverses of Matrices 785
The Identity Matrix / The Inverse of a Matrix / Solving Systems of Equations
9.6 Determinants and Cramer’s Rule 792
Determinants of Square Matrices / Evaluating Determinants Using Cofactors /
Cramer’s Rule
9.7 Systems of Inequalities and Linear Programming 800
Graphs of Linear Inequalities / Systems of Linear Inequalities /
Applications: Linear Programming
9.8 Partial Fractions 813
Partial Fraction Decompositions
Study Guide 819
Review Exercises 825
Chapter Test 829
10 Analytic Geometry Topics 831
10.1 The Parabola 832
Parabolas / Finding Standard Form by Completing the Square / Applications
10.2 The Circle and the Ellipse 840
Circles / Ellipses / Applications
10.3 The Hyperbola 851
Standard Equations of Hyperbolas / Applications
10.4 Nonlinear Systems of Equations and Inequalities 861
Nonlinear Systems of Equations / Modeling and Problem Solving /
Nonlinear Systems of Inequalities
VISUALIZING THE GRAPH 869
MID-CHAPTER MIXED REVIEW 874
10.5 Rotation of Axes 875
Rotation of Axes / The Discriminant
10.6 Polar Equations of Conics 885
Polar Equations of Conics / Converting from Polar Equations to Rectangular
Equations / Finding Polar Equations of Conics
10.7 Parametric Equations 891
Graphing Parametric Equations / Determining a Rectangular Equation
for Given Parametric Equations / Determining Parametric Equations
for a Given Rectangular Equation / Applications
Study Guide 898
Review Exercises 905
Chapter Test 908