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A Graphical Approach to Precalculus

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This edition has evolved to address the needs of today's student. While maintaining its unique table of contents and functions-based approach, the text now includes additional components to build skill, address critical thinking, solve applications, and apply technology to support traditional algebraic solutions. It continues to incorporate an open design, helpful features, careful explanations of topics, and a comprehensive package of supplements and study aids to provide new and relevant opportunities for learning and teaching.

Author Biography

John Hornsby- When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics, education, or journalism. His ultimate decision was to become a teacher, but after twenty-five years of teaching at the high school and university levels and fifteen years of writing mathematics textbooks, all three of his goals have been realized; his love for teaching and for mathematics is evident in his passion for working with students and fellow teachers as well. His specific professional interests are recreational mathematics, mathematics history, and incorporating graphing calculators into the curriculum.

John's personal life is busy as he devotes time to his family (wife Gwen, and sons Chris, Jack, and Josh). He has been a rabid baseball fan all of his life. John's other hobbies include numismatics (the study of coins) and record collecting. He loves the music of the 1960s and has an extensive collection of the recorded works of Frankie Valli and the Four Seasons.

Marge Lial has always been interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, is now affiliated with American River College.

Marge is an avid reader and traveler. Her travel experiences often find their way into her books as applications, exercise sets, and feature sets. She is particularly interested in archeology. Trips to various digs and ruin sites have produced some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan.

Gary Rockswold- Dr. Gary Rockswold has been teaching mathematics for 33 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate, and graduate students, and adult education classes. He is currently employed at Minnesota State University, Mankato, where he is a full professor of mathematics. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his lovely wife and two children.

Table of Contents

Prefacep. xiv
Linear Functions, Equations, and Inequalitiesp. 1
Real Numbers and the Rectangular Coordinate Systemp. 2
Sets of Real Numbers
The Rectangular Coordinate System
Viewing Windows
Distance and Midpoint Formulas
Introduction to Relations and Functionsp. 12
Set-Builder Notation and Interval Notation
Relations, Domain, and Range
Function Notation
Reviewing Basic Concepts (Sections 1.1 and 1.2)p. 22
Linear Functionsp. 23
Basic Concepts about Linear Functions
Slope of a Line
Slope-Intercept Form of the Equation of a Line
Equations of Lines and Linear Modelsp. 36
Point-Slope Form of the Equation of a Line
Standard Form of the Equation of a Line
Parallel and Perpendicular Lines
Linear Models and Regression
Reviewing Basic Concepts (Sections 1.3 and 1.4)p. 50
Linear Equations and Inequalitiesp. 51
Solving Linear Equations
Graphical Approaches to Solving Linear Equations
Identities and Contradictions
Solving Linear Inequalities
Graphical Approaches to Solving Linear Inequalities
Three-Part Inequalities
Applications of Linear Functionsp. 66
Problem-Solving Strategies
Applications of Linear Equations
Break-Even Analysis
Direct Variation
Reviewing Basic Concepts (Sections 1.5 and 1.6)p. 78
Chapter 1 Summaryp. 79
Chapter 1 Review Exercisesp. 82
Chapter 1 Testp. 87
Chapter 1 Project: Predicting Heights and Weights of Athletesp. 88
Analysis of Graphs of Functionsp. 89
Graphs of Basic Functions and Relations; Symmetryp. 90
Increasing and Decreasing Functions
The Identity Function
The Squaring Function and Symmetry with Respect to the y-Axis
The Cubing Function and Symmetry with Respect to the Origin
The Square Root and Cube Root Functions
The Absolute Value Function
The Relation x - y[superscript 2] and Symmetry with Respect to the x-Axis
Even and Odd Functions
Vertical and Horizontal Shifts of Graphsp. 103
Vertical Shifts
Horizontal Shifts
Combinations of Vertical and Horizontal Shifts
Effects of Shifts on Domain and Range
Horizontal Shifts Applied to Equations for Modeling
Stretching, Shrinking, and Reflecting Graphsp. 113
Vertical Stretching
Vertical Shrinking
Horizontal Stretching and Shrinking
Reflecting across an Axis
Combining Transformations of Graphs
Reviewing Basic Concepts (Sections 2.1-2.3)p. 125
Absolute Value Functions: Graphs, Equations, Inequalities, and Applicationsp. 127
The Graph of y = [vertical bar]f(x)[vertical bar]
Properties of Absolute Value
Equations and Inequalities Involving Absolute Value
An Application Involving Absolute Value
Piecewise-Defined Functionsp. 138
Graphing Piecewise-Defined Functions
The Greatest Integer Function
Applications of Piecewise-Defined Functions
Operations and Compositionp. 149
Operations on Functions
The Difference Quotient
Composition of Functions
Applications of Operations and Composition
Reviewing Basic Concepts (Sections 2.4-2.6)p. 162
Chapter 2 Summaryp. 163
Chapter 2 Review Exercisesp. 166
Chapter 2 Testp. 169
Chapter 2 Project: Modeling the Movement of a Cold Frontp. 171
Polynomial Functionsp. 173
Complex Numbersp. 174
The Number i
Operations with Complex Numbersp. 174
Quadratic Functions and Graphsp. 181
Completing the Square
Graphs of Quadratic Functions
Vertex Formula
Extreme Values
Applications and Quadratic Models
Quadratic Equations and Inequalitiesp. 194
Zero-Product Property
Solving x[superscript 2] = k
Quadratic Formula and the Discriminant
Solving Quadratic Equations
Solving Quadratic Inequalities
Formulas Involving Quadratics
Another Quadratic Model
Reviewing Basic Concepts (Sections 3.1-3.3)p. 208
Further Applications of Quadratic Functions and Modelsp. 208
Applications of Quadratic Functions
Quadratic Models
Higher-Degree Polynomial Functions and Graphsp. 218
Cubic Functions
Quartic Functions
End Behavior
x-Intercepts (Real Zeros)
Comprehensive Graphs
Curve Fitting and Polynomial Models
Reviewing Basic Concepts (Sections 3.4 and 3.5)p. 231
Topics in the Theory of Polynomial Functions (I)p. 231
Intermediate Value Theorem
Division of Polynomials and Synthetic Division
Remainder and Factor Theorems
Topics in the Theory of Polynomial Functions (II)p. 240
Complex Zeros and the Fundamental Theorem of Algebra
Number of Zeros
Rational Zeros Theorem
Descartes' Rule of Signs
Boundedness Theorem
Polynomial Equations and Inequalities; Further Applications and Modelsp. 251
Polynomial Equations and Inequalities
Complex nth Roots
Applications and Polynomial Models
Reviewing Basic Concepts (Sections 3.6-3.8)p. 259
Chapter 3 Summaryp. 260
Chapter 3 Review Exercisesp. 263
Chapter 3 Testp. 267
Chapter 3 Project: Creating a Social Security Polynomialp. 268
Rational, Power, and Root Functionsp. 270
Rational Functions and Graphsp. 271
The Reciprocal Function
The Rational Function Defined by f(x) = 1 / x[superscript 2]
More on Graphs of Rational Functionsp. 277
Vertical and Horizontal Asymptotes
Graphing Techniques
Oblique Asymptotes
Graphs with Points of Discontinuity
Rational Equations, Inequalities, Applications, and Modelsp. 291
Solving Rational Equations and Inequalities
Applications and Models of Rational Functions
Inverse Variation
Combined and Joint Variation
Reviewing Basic Concepts (Sections 4.1-4.3)p. 305
Functions Defined by Powers and Rootsp. 306
Power and Root Functions
Modeling Using Power Functions
Graphs of f(x) = [characters not reproducible]
Graphing Circles and Horizontal Parabolas Using Root Functions
Equations, Inequalities, and Applications Involving Root Functionsp. 318
Equations and Inequalities
An Application of Root Functions
Reviewing Basic Concepts (Sections 4.4 and 4.5)p. 328
Chapter 4 Summaryp. 329
Chapter 4 Review Exercisesp. 331
Chapter 4 Testp. 335
Chapter 4 Project: How Rugged Is Your Coastline?p. 336
Inverse, Exponential, and Logarithmic Functionsp. 338
Inverse Functionsp. 339
Inverse Operations
One-to-One Functions
Inverse Functions and Their Graphs
Equations of Inverse Functions
An Application of Inverse Functions
Exponential Functionsp. 350
Real-Number Exponents
Graphs of Exponential Functions
Exponential Equations (Type 1)
Compound Interest
The Number e and Continuous Compounding
An Application of Exponential Functions
Logarithms and Their Propertiesp. 363
Definition of Logarithm
Common Logarithms
Natural Logarithms
Properties of Logarithms
Change-of-Base Rule
Reviewing Basic Concepts (Sections 5.1-5.3)p. 373
Logarithmic Functionsp. 374
Graphs of Logarithmic Functions
Applying Earlier Work to Logarithmic Functions
A Logarithmic Model
Exponential and Logarithmic Equations and Inequalitiesp. 384
Exponential Equations and Inequalities (Type 2)
Logarithmic Equations and Inequalities
Equations and Inequalities Involving Both Exponentials and Logarithms
Formulas Involving Exponentials and Logarithms
Reviewing Basic Concepts (Sections 5.4 and 5.5)p. 393
Further Applications and Modeling with Exponential and Logarithmic Functionsp. 394
Physical Science Applications
Financial Applications
Biological and Medical Applications
Modeling Data with Exponential and Logarithmic Functions
Chapter 5 Summaryp. 408
Chapter 5 Review Exercisesp. 411
Chapter 5 Testp. 414
Chapter 5 Project: Modeling Motor Vehicle Sales in the United States (with a lesson about the careless use of mathematical models)p. 415
Analytic Geometryp. 417
Circles and Parabolasp. 418
Conic Sections
Equations and Graphs of Circles
Equations and Graphs of Parabolas
Translations of Parabolas
An Application of Parabolas
Ellipses and Hyperbolasp. 432
Equations and Graphs of Ellipses
Translations of Ellipses
An Application of Ellipses
Equations and Graphs of Hyperbolas
Translations of Hyperbolas
Reviewing Basic Concepts (Sections 6.1 and 6.2)p. 445
Summary of the Conic Sectionsp. 445
Identifying Conic Sections
Parametric Equationsp. 454
Graphs of Parametric Equations and Their Rectangular Equivalents
Alternative Forms of Parametric Equations
An Application of Parametric Equations
Reviewing Basic Concepts (Sections 6.3 and 6.4)p. 458
Chapter 6 Summaryp. 459
Chapter 6 Review Exercisesp. 461
Chapter 6 Testp. 463
Chapter 6 Project: Modeling the Path of a Bouncing Ballp. 464
Systems of Equations and Inequalities; Matricesp. 466
Systems of Equationsp. 467
Linear Systems
Substitution Method
Elimination Method
Special Systems
Nonlinear Systems
Applications of Systems
Solution of Linear Systems in Three Variablesp. 480
Geometric Considerations
Analyttic Solution of Systems in Three Variables
Applications of Systems
Curve Fitting Using a System
Solution of Linear Systems by Row Transformationsp. 488
Matrix Row Transformations
Row Echelon Method
Reduced Row Echelon Method
Special Cases
An Application of Matrices
Reviewing Basic Concepts (Sections 7.1-7.3)p. 499
Matrix Properties and Operationsp. 500
Terminology of Matrices
Operations on Matrices
Applying Matrix Algebra
Determinants and Cramer's Rulep. 513
Determinants of 2 x 2 Matrices
Determinants of Larger Matrices
Derivation of Cramer's Rule
Using Cramer's Rule to Solve Systems
Solution of Linear Systems by Matrix Inversesp. 524
Identity Matrices
Multiplicative Inverses of Square Matrices
Using Determinants to Find Inverses
Solving Linear Systems Using Inverse Matrices
Curve Fitting Using a System
Reviewing Basic Concepts (Sections 7.4-7.6)p. 536
Systems of Inequalities and Linear Programmingp. 537
Solving Linear Inequalities
Solving Systems of Inequalities
Linear Programming
Partial Fractionsp. 547
Decomposition of Rational Expressions
Distinct Linear Factors
Repeated Linear Factors
Distinct Linear and Quadratic Factors
Repeated Quadratic Factors
Reviewing Basic Concepts (Sections 7.7 and 7.8)p. 554
Chapter 7 Summaryp. 554
Chapter 7 Review Exercisesp. 557
Chapter 7 Testp. 561
Chapter 7 Project: Finding a Polynomial Whose Graph Passes through Any Number of Given Pointsp. 562
Trigonometric Functions and Applicationsp. 565
Angles and Their Measuresp. 566
Basic Terminology
Degree Measure
Standard Position and Coterminal Angles
Radian Measure
Arc Lengths and Areas of Sectors
Angular and Linear Speed
Trigonometric Functions and Fundamental Identitiesp. 582
Trigonometric Functions
Quadrantal Angles
Reciprocal Identities
Signs and Ranges of Function Values
Pythagorean Identities
Quotient Identities
An Application of Trigonometric Functions
Reviewing Basic Concepts (Sections 8.1 and 8.2)p. 595
Evaluating Trigonometric Functionsp. 596
Definitions of the Trigonometric Functions
Trigonometric Function Values of Special Angles
Cofunction Identities
Reference Angles
Special Angles as Reference Angles
Finding Function Values with a Calculator
Finding Angle Measures
Applications of Right Trianglesp. 608
Significant Digits
Solving Triangles
Angles of Elevation or Depression
Further Applications of Trigonometric Functions
Reviewing Basic Concepts (Sections 8.3 and 8.4)p. 620
The Circular Functionsp. 620
Circular Functions
Applications of Circular Functions
Graphs of the Sine and Cosine Functionsp. 629
Periodic Functions
Graph of the Sine Function
Graph of the Cosine Function
Graphing Techniques, Amplitude, and Period
Determining a Trigonometric Model Using Curve Fitting
Reviewing Basic Concepts (Sections 8.5 and 8.6)p. 646
Graphs of the Other Circular Functionsp. 646
Graphs of the Cosecant and Secant Functions
Graphs of the Tangent and Cotangent Functions
Addition of Ordinates
Harmonic Motionp. 657
Simple Harmonic Motion
Damped Oscillatory Motion
Reviewing Basic Concepts (Sections 8.7 and 8.8)p. 660
Chapter 8 Summaryp. 661
Chapter 8 Review Exercisesp. 665
Chapter 8 Testp. 670
Chapter 8 Project: Modeling Sunset Timesp. 671
Trigonometric Identities and Equationsp. 672
Trigonometric Identitiesp. 673
Fundamental Identities
Using the Fundamental Identities
Verifying Identities
Sum and Difference Identitiesp. 683
Cosine Sum and Difference Identities
Sine and Tangent Sum and Difference Identities
Reviewing Basic Concepts (Sections 9.1 and 9.2)p. 692
Further Identitiesp. 692
Double-Number Identities
Product-to-Sum and Sum-to-Product Identities
Half-Number Identities
The Inverse Circular Functionsp. 704
Review of Inverse Functions
Inverse Sine Function
Inverse Cosine Function
Inverse Tangent Function
Remaining Inverse Trigonometric Functions
Inverse Function Values
Reviewing Basic Concepts (Sections 9.3 and 9.4)p. 717
Trigonometric Equations and Inequalities (I)p. 717
Equations Solvable by Linear Methods
Equations Solvable by Factoring
Equations Solvable by the Quadratic Formula
Using Trigonometric Identities to Solve Equations
Trigonometric Equations and Inequalities (II)p. 724
Equations and Inequalities Involving Multiple-Number Identities
Equations and Inequalities Involving Half-Number Identities
An Application of Trigonometric Equations
Reviewing Basic Concepts (Sections 9.5 and 9.6)p. 730
Chapter 9 Summaryp. 731
Chapter 9 Review Exercisesp. 733
Chapter 9 Testp. 736
Chapter 9 Project: Modeling a Damped Pendulump. 737
Applications of Trigonometry; Vectorsp. 739
The Law of Sinesp. 740
Congruency and Oblique Triangles
Derivation of the Law of Sines
Applications of Triangles
Ambiguous Case
The Law of Cosines and Area Formulasp. 754
Derivation of the Law of Cosines
Applications of Triangles
Area Formulas
Vectors and Their Applicationsp. 764
Basic Terminology
Algebraic Interpretation of Vectors
Operations with Vectors
Dot Product and the Angle between Vectors
Applications of Vectors
Reviewing Basic Concepts (Sections 10.1-10.3)p. 777
Trigonometric (Polar) Form of Complex Numbersp. 778
The Complex Plane and Vector Representation
Trigonometric (Polar) Form
Products of Complex Numbers in Trigonometric Form
Quotients of Complex Numbers in Trigonometric Form
Powers and Roots of Complex Numbersp. 787
Powers of Complex Numbers (De Moivre's Theorem)
Roots of Complex Numbers
Reviewing Basic Concepts (Sections 10.4 and 10.5)p. 793
Polar Equations and Graphsp. 793
Polar Coordinate System
Graphs of Polar Equations
Classifying Polar Equations
Converting Equations
More Parametric Equationsp. 804
Parametric Equations with Trigonometric Functions
The Cycloid
Applications of Parametric Equations
Reviewing Basic Concepts (Sections 10.6 and 10.7)p. 811
Chapter 10 Summaryp. 811
Chapter 10 Review Exercisesp. 814
Chapter 10 Testp. 817
Chapter 10 Project: When Is a Circle Really a Polygon?p. 818
Further Topics in Algebrap. 820
Sequences and Seriesp. 821
Series and Summation Notation
Summation Properties
Arithmetic Sequences and Seriesp. 831
Arithmetic Sequences
Arithmetic Series
Geometric Sequences and Seriesp. 839
Geometric Sequences
Geometric Series
Infinite Geometric Series
Reviewing Basic Concepts (Sections 11.1-11.3)p. 849
The Binomial Theoremp. 850
A Binomial Expansion Pattern
Pascal's Triangle
Binomial Coefficients
The Binomial Theorem
rth Term of a Binomial Expansion
Mathematical Inductionp. 857
Proof by Mathematical Induction
Proving Statements
Generalized Principle of Mathematical Induction
Proof of the Binomial Theorem
Reviewing Basic Concepts (Sections 11.4 and 11.5)p. 863
Counting Theoryp. 863
Fundamental Principle of Counting
Distinguishing between Permutations and Combinations
Probabilityp. 872
Basic Concepts
Complements and Venn Diagrams
Union of Two Events
Binomial Probability
Reviewing Basic Concepts (Sections 11.6 and 11.7)p. 881
Chapter 11 Summaryp. 882
Chapter 11 Review Exercisesp. 886
Chapter 11 Testp. 888
Chapter 11 Project: Using Experimental Probabilities to Simulate Family Makeupp. 889
Reference: Basic Algebraic Conceptsp. 892
Review of Exponents and Polynomialsp. 893
Rules for Exponents
Terminology for Polynomials
Adding and Subtracting Polynomials
Multiplying Polynomials
Review of Factoringp. 899
Factoring Out the Greatest Common Factor
Factoring by Grouping
Factoring Trinomials
Factoring Special Products
Factoring by Substitution
Review of Rational Expressionsp. 906
Domain of a Rational Expression
Lowest Terms of a Rational Expression
Multipling and Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Complex Fractions
Review of Negative and Rational Exponentsp. 914
Negative Exponents and the Quotient Rule
Rational Exponents
Review of Radicalsp. 920
Radical Notation
Rules for Radicals
Simplifying Radicals
Operations with Radicals
Rationalizing Denominators
Chapter R Testp. 928
Geometry Formulasp. 929
Deciding Which Model Best Fits a Set of Datap. 931
Vectors in Spacep. 936
Polar Form of Conic Sectionsp. 942
Rotation of Axesp. 946
Answers to Selected Exercisesp. 1
Index of Applicationsp. 1
Indexp. 5
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