Graphical Approach to College Algebra & Trigonometry, A

by Hornsby, John; Lial, Margaret L.; Rockswold, Gary K.

• ISBN13: 9780201735109
• ISBN10: 0201735105
• Edition: 3rd
• Format: Hardcover
• Copyright: 2003-01-01
• Publisher: Addison Wesley

Summary

This series is the culmination of many years of teaching experience with the graphing calculator. the books were written from the beginning for use with the graphing calculator. Throughout the text, the authors emphasize the power of technology but provide numerous warnings of its limitations: they stress that only through understanding the mathematical concepts can students fully appreciate the power of graphing calculators and use technology appropriately. Additionally, the authors consistently use the same four-step process when introducing the different classes of functions. This allows students to easily make connections between graphs of functions and their associated equations and inequalities.

Table of Contents

Linear Functions, Equations, and Inequalities	1	(89)
Real Numbers and Coordinate Systems	2	(9)
Sets of Real Numbers
Coordinate Systems
Viewing Windows
Roots
Distance and Midpoint Formulas
Introduction to Relations and Functions	11	(12)
Set-Builder Notation and Interval Notation
Relations, Domain, and Range
Functions
Tables
Function Notation
Reviewing Basic Concepts (Sections 1.1 and 1.2)	23	(1)
Linear Functions	23	(13)
Basic Concepts about Linear Functions
Slope of a Line
Slope-Intercept Form of the Equation of a Line
Equations of Lines and Linear Models	36	(15)
Point-Slope Form of the Equation of a Line
Other Forms of the Equation of a Line
Parallel and Perpendicular Lines
Linear Models and Regression
Reviewing Basic Concepts (Sections 1.3 and 1.4)	51	(1)
Linear Equations and Inequalities	51	(16)
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 Functions	67	(22)
Problem-Solving Strategies
Applications of Linear Equations
Break-Even Analysis
Direct Variation
Formulas
Reviewing Basic Concepts (Sections 1.5 and 1.6)	77	(1)
Chapter 1 Summary	78	(4)
Chapter 1 Review Exercises	82	(4)
Chapter 1 Test	86	(2)
Chapter 1 Project Predicting Heights and Weights of Athletes	88	(1)
Analysis of Graphs of Functions	89	(83)
Graphs of Basic Functions and Relations; Symmetry	90	(14)
Continuity
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 = y2 and Symmetry with Respect to the x-Axis
Even and Odd Functions
Vertical and Horizontal Shifts of Graphs	104	(10)
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 Graphs	114	(12)
Vertical Stretching
Vertical Shrinking
Reflecting Across an Axis
Combining Transformations of Graphs
Reviewing Basic Concepts (Sections 2.1 -- 2.3)	124	(2)
Absolute Value Functions: Graphs, Equations, Inequalities, and Applications	126	(11)
The Graph of y = |f(x)|
Properties of Absolute Value
Equations and Inequalities Involving Absolute Value
An Application Involving Absolute Value
Piecewise-Defined Functions	137	(11)
Graphing Piecewise-Defined Functions
The Greatest Integer Function
Applications of Piecewise-Defined Functions
Operations and Composition	148	(24)
Operations on Functions
The Difference Quotient
Composition of Functions
Applications of Operations and Composition
Reviewing Basic Concepts (Sections 2.4--2.6)	161	(1)
Chapter 2 Summary	162	(3)
Chapter 2 Review Exercises	165	(3)
Chapter 2 Test	168	(1)
Chapter 2 Project Modeling the Movement of a Cold Front	169	(3)
Polynomial Functions	172	(100)
Complex Numbers	173	(7)
The Number i
Operations with Complex Numbers
Quadratic Functions and Graphs	180	(13)
Completing the Square
Graphs of Quadratic Functions
Vertex Formula
Extreme Values
Applications and Modeling
Quadratic Equations and Inequalities	193	(14)
Zero-Product Property
Solving x2 = 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)	207	(1)
Further Applications of Quadratic Functions and Models	207	(10)
Applications of Quadratic Functions
Quadratic Models
Higher-Degree Polynomial Functions and Graphs	217	(15)
Cubic Functions
Quartic Functions
Extrema
End Behavior
x-Intercepts (Real Zeros)
Comprehensive Graphs
Curve Fitting and Polynomial Models
Reviewing Basic Concepts (Sections 3.4 and 3.5)	231	(1)
Topics in the Theory of Polynomial Functions (I)	232	(10)
Intermediate Value Theorem
Division of Polynomials and Synthetic Division
Remainder and Factor Theorems
Topics in the Theory of Polynomial Functions (II)	242	(9)
Complex Zeros and the Fundamental Theorem of Algebra
Number of Zeros
Rational Zeros Theorem
Polynomial Equations and Inequalities; Further Applications and Models	251	(21)
Polynomial Equations and Inequalities
Complex nth Roots
Applications and Polynomial Models
Reviewing Basic Concepts (Sections 3.6--3.8)	261	(1)
Chapter 3 Summary	262	(3)
Chapter 3 Review Exercises	265	(4)
Chapter 3 Test	269	(1)
Chapter 3 Project Creating a Social Security Polynomial	270	(2)
Rational, Power, and Root Functions	272	(70)
Rational Functions and Graphs	273	(6)
The Reciprocal Function
The Rational Function Defined by f(x) = 1/x2
Mode and Window Choices for Calculator Graphs
More on Graphs of Rational Functions	279	(14)
Vertical and Horizontal Asymptotes
Graphing Techniques
Oblique Asymptotes
Graphs with Points of Discontinuity
Rational Equations, Inequalities, Applications, and Models	293	(16)
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)	308	(1)
Functions Defined by Powers and Roots	309	(11)
Power and Root Functions
Modeling Using Power Functions
Graphs of
Graphing Circles and Horizontal Parabolas Using Root Functions
Equations, Inequalities, and Applications Involving Root Functions	320	(22)
Equations and Inequalities
Applications
Reviewing Basic Concepts (Sections 4.4 and 4.5)	330	(1)
Chapter 4 Summary	331	(2)
Chapter 4 Review Exercises	333	(4)
Chapter 4 Test	337	(2)
Chapter 4 Project How Rugged Is Your Coastline?	339	(3)
Inverse, Exponential, and Logarithmic Functions	342	(77)
Inverse Functions	343	(10)
Inverse Operations
One-to-One Functions
Inverse Functions and Their Graphs
An Application of Inverse Functions
Exponential Functions	353	(12)
Real-Number Exponents
Graphs of Exponential Functions
Exponential Equations (Type I)
The Number e
Compound Interest
Logarithms and Their Properties	365	(12)
Definition of Logarithm
Common Logarithms
Natural Logarithms
Properties of Logarithms
Change-of-Base Rule
Reviewing Basic Concepts (Sections 5.1--5.3)	376	(1)
Logarithmic Functions	377	(9)
Graphs of Logarithmic Functions
Applying Earlier Work to Logarithmic Functions
A Logarithmic Model
Exponential and Logarithmic Equations and Inequalities	386	(10)
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)	395	(1)
Further Applications and Modeling with Exponential and Logarithmic Functions	396	(23)
Physical Science Applications
Financial Applications
Biological and Medical Applications
Modeling Data with Exponential and Logarithmic Functions
Chapter 5 Summary	410	(3)
Chapter 5 Review Exercises	413	(3)
Chapter 5 Test	416	(1)
Chapter 5 Project Modeling Motor Vehicle Sales in the United States (with a lesson about careless use of mathematical models)	417	(2)
Analytic Geometry	419	(47)
Circles and Parabolas	420	(13)
Conic Sections
Equations and Graphs of Circles
An Application of Circles
Equations and Graphs of Parabolas
An Application of Parabolas
Ellipses and Hyperbolas	433	(13)
Equations and Graphs of Ellipses
Applications of Ellipses
Equations and Graphs of Hyperbolas
Reviewing Basic Concepts (Sections 6.1 and 6.2)	446	(1)
Summary of the Conic Sections	446	(8)
Characteristics
Identifying Conic Sections
Eccentricity
Parametric Equations	454	(12)
Graphs of Parametric Equations and Their Rectangular Equivalents
Alternative Forms of Parametric Equations
An Application
Reviewing Basic Concepts (Sections 6.3 and 6.4)	459	(1)
Chapter 6 Summary	459	(2)
Chapter 6 Review Exercises	461	(2)
Chapter 6 Test	463	(1)
Chapter 6 Project Modeling the Path of a Bouncing Ball	464	(2)
Matrices and Systems of Equations and Inequalities	466	(100)
Systems of Equations	467	(13)
Linear Systems
Substitution Method
Elimination Method
Special Systems
Nonlinear Systems
Applications of Systems
Solution of Linear Systems by the Echelon Method	480	(8)
Geometric Considerations
Analytic Solution of Systems in Three Variables
Applications of Systems
Curve Fitting Using a System
Solution of Linear Systems by Row Transformations	488	(13)
Matrices and Technology
Matrix Row Transformations
Row Echelon Method
Reduced Row Echelon Method
Special Cases
An Application
Reviewing Basic Concepts (Sections 7.1--7.3)	501	(1)
Matrix Properties and Operations	501	(15)
Terminology of Matrices
Operations on Matrices
Applying Matrix Algebra
Determinants and Cramer's Rule	516	(11)
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 Inverses	527	(11)
Identity Matrices
Multiplicative Inverses of Square Matrices
Solving Linear Systems Using Inverse Matrices
Curve Fitting Using a System
Reviewing Basic Concepts (Sections 7.4--7.6)	537	(1)
Systems of Inequalities and Linear Programming	538	(10)
Solving Linear Inequalities
Solving Systems of Inequalities
Linear Programming
Partial Fractions	548	(18)
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)	555	(1)
Chapter 7 Summary	556	(2)
Chapter 7 Review Exercises	558	(4)
Chapter 7 Test	562	(1)
Chapter 7 Project Finding a Polynomial Whose Graph Passes Through Any Number of Given Points	563	(3)
Trigonometric Functions and Applications	566	(107)
Angles and Their Measures	567	(16)
Basic Terminology
Degree Measure
Standard Position and Coterminal Angles
Radian Measure
Arc Lengths and Sectors
Angular and Linear Speed
Trigonometric Functions and Fundamental Identities	583	(12)
Trigonometric Functions
Quadrantal Angles
Reciprocal Identities
Signs and Ranges of Function Values
Pythagorean Identities
Quotient Identities
Reviewing Basic Concepts (Sections 8.1 and 8.2)	594	(1)
Evaluating Trigonometric Functions	595	(12)
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 Triangles	607	(13)
Significant Digits
Solving Triangles
Angles of Elevation or Depression
Bearing
Further Applications
Reviewing Basic Concepts (Sections 8.3 and 8.4)	619	(1)
The Circular Functions	620	(8)
Circular Functions
Applications of Circular Functions
Graphs of the Sine and Cosine Functions	628	(18)
Periodic Functions
Graph of the Sine Function
Graph of the Cosine Function
Graphing Techniques, Amplitude, and Period
Translations
Determining a Trigonometric Model Using Curve Fitting
Reviewing Basic Concepts (Sections 8.5 and 8.6)	645	(1)
Graphs of the Other Circular Functions	646	(10)
Graphs of the Cosecant and Secant Functions
Graphs of the Tangent and Cotangent Functions
Addition of Ordinates
Harmonic Motion	656	(17)
Simple Harmonic Motion
Damped Oscillatory Motion
Reviewing Basic Concepts (Sections 8.7 and 8.8)	659	(1)
Chapter 8 Summary	660	(4)
Chapter 8 Review Exercises	664	(5)
Chapter 8 Test	669	(2)
Chapter 8 Project Modeling Sunset Times	671	(2)
Trigonometric Identities and Equations	673	(66)
Trigonometric Identities	674	(10)
Fundamental Identities
Using the Fundamental Identities
Verifying Identities
Sum and Difference Identities	684	(10)
Cosine Sum and Difference Identities
Sine and Tangent Sum and Difference Identities
Reviewing Basic Concepts (Sections 9.1 and 9.2)	693	(1)
Further Identities	694	(12)
Double-Number Identities
Product-to-Sum and Sum-to-Product Identities
Half-Number Identities
The Inverse Circular Functions	706	(12)
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)	717	(1)
Trigonometric Equations and Inequalities (I)	718	(6)
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)	724	(15)
Equations and Inequalities Involving Multiple-Number Identities
Equations and Inequalities Involving Half-Number Identities
An Application
Reviewing Basic Concepts (Sections 9.5 and 9.6)	730	(1)
Chapter 9 Summary	731	(2)
Chapter 9 Review Exercises	733	(3)
Chapter 9 Test	736	(1)
Chapter 9 Project Modeling a Damped Pendulum	737	(2)
Applications of Trigonometry; Vectors	739	(77)
The Law of Sines	740	(13)
Congruency and Oblique Triangles
Derivation of the Law of Sines
Applications
Ambiguous Case
The Law of Cosines and Area Formulas	753	(10)
Derivation of the Law of Cosines
Applications
Area Formulas
Vectors and Their Applications	763	(13)
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)	775	(1)
Trigonometric (Polar) Form of Complex Numbers	776	(8)
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 Numbers	784	(7)
Powers of Complex Numbers (De Moivre's Theorem)
Roots of Complex Numbers
Reviewing Basic Concepts (Sections 10.4 and 10.5)	790	(1)
Polar Equations and Graphs	791	(10)
Polar Coordinate System
Graphs of Polar Equations
Classifying Polar Equations
Converting Equations
More Parametric Equations	801	(15)
Parametric Equations with Trigonometric Functions
The Cycloid
Applications of Parametric Equations
Reviewing Basic Concepts (Sections 10.6 and 10.7)	808	(1)
Chapter 10 Summary	808	(3)
Chapter 10 Review Exercises	811	(3)
Chapter 10 Test	814	(1)
Chapter 10 Project When Is a Circle Really a Polygon?	814	(2)
Further Topics in Algebra	816	(74)
Sequences and Series	817	(10)
Sequences
Series and Summation Notation
Summation Properties
Arithmetic Sequences and Series	827	(8)
Arithmetic Sequences
Arithmetic Series
Geometric Sequences and Series	835	(12)
Geometric Sequences
Geometric Series
Infinite Geometric Series
Annuities
Reviewing Basic Concepts (Sections 11.1--11.3)	846	(1)
The Binomial Theorem	847	(7)
A Binomial Expansion Pattern
Pascal's Triangle
n-Factorial
Binomial Coefficients
The Binomial Theorem
rth Term of a Binomial Expansion
Mathematical Induction	854	(7)
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)	861	(1)
Counting Theory	861	(9)
Fundamental Principle of Counting
Permutations
Combinations
Distinguishing between Permutations and Combinations
Probability	870	(20)
Basic Concepts
Complements and Venn Diagrams
Odds
Union of Two Events
Binomial Probability
Reviewing Basic Concepts (Sections 11.6 and 11.7)	881	(1)
Chapter 11 Summary	881	(4)
Chapter 11 Review Exercises	885	(2)
Chapter 11 Test	887	(1)
Chapter 11 Project Using Experimental Probabilities to Simulate Family Makeup	888	(2)
Reference: Basic Algebraic Concepts and Geometry Formulas	890	(36)
Review of Exponents and Polynomials	891	(6)
Rules for Exponents
Terminology for Polynomials
Adding and Subtracting Polynomials
Multiplying Polynomials
Review of Factoring	897	(7)
Factoring Out the Greatest Common Factor
Factoring by Grouping
Factoring Trinomials
Factoring Special Products
Factoring by Substitution
Review of Rational Expressions	904	(7)
Domain of a Rational Expression
Lowest Terms of a Rational Expression
Multiplying and Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Complex Fractions
Review of Negative and Rational Exponents	911	(6)
Negative Exponents and the Quotient Rule
Rational Exponents
Review of Radicals	917	(7)
Radical Notation
Rules for Radicals
Simplifying Radicals
Operations with Radicals
Rationalizing Denominators
Geometry Formulas	924	(2)
Appendix A: Vectors in Space	926	(6)
Appendix B: Polar Form of Conic Sections	932	(4)
Appendix C: Rotation of Axes	936
Answers to Selected Exercises	1	(1)
Index of Applications	1	(6)
Index	7