
Linear Functions, Equations, and Inequalities 


1  (89) 

Real Numbers and Coordinate Systems 


2  (9) 









Distance and Midpoint Formulas 



Introduction to Relations and Functions 


11  (12) 

SetBuilder Notation and Interval Notation 



Relations, Domain, and Range 









Reviewing Basic Concepts (Sections 1.1 and 1.2) 


23  (1) 


23  (13) 

Basic Concepts about Linear Functions 





SlopeIntercept Form of the Equation of a Line 



Equations of Lines and Linear Models 


36  (15) 

PointSlope 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) 



Graphical Approaches to Solving Linear Equations 



Identities and Contradictions 



Solving Linear Inequalities 



Graphical Approaches to Solving Linear Inequalities 





Applications of Linear Functions 


67  (22) 

ProblemSolving Strategies 



Applications of Linear Equations 









Reviewing Basic Concepts (Sections 1.5 and 1.6) 


77  (1) 


78  (4) 

Chapter 1 Review Exercises 


82  (4) 


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) 



Increasing and Decreasing Functions 





The Squaring Function and Symmetry with Respect to the yAxis 



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 xAxis 





Vertical and Horizontal Shifts of Graphs 


104  (10) 





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) 





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) 



Properties of Absolute Value 



Equations and Inequalities Involving Absolute Value 



An Application Involving Absolute Value 



PiecewiseDefined Functions 


137  (11) 

Graphing PiecewiseDefined Functions 



The Greatest Integer Function 



Applications of PiecewiseDefined Functions 



Operations and Composition 


148  (24) 







Applications of Operations and Composition 



Reviewing Basic Concepts (Sections 2.42.6) 


161  (1) 


162  (3) 

Chapter 2 Review Exercises 


165  (3) 


168  (1) 

Chapter 2 Project Modeling the Movement of a Cold Front 


169  (3) 


172  (100) 


173  (7) 



Operations with Complex Numbers 



Quadratic Functions and Graphs 


180  (13) 



Graphs of Quadratic Functions 







Applications and Modeling 



Quadratic Equations and Inequalities 


193  (14) 





Quadratic Formula and the Discriminant 



Solving Quadratic Equations 



Solving Quadratic Inequalities 



Formulas Involving Quadratics 





Reviewing Basic Concepts (Sections 3.13.3) 


207  (1) 

Further Applications of Quadratic Functions and Models 


207  (10) 

Applications of Quadratic Functions 





HigherDegree Polynomial Functions and Graphs 


217  (15) 









xIntercepts (Real Zeros) 





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 







Polynomial Equations and Inequalities; Further Applications and Models 


251  (21) 

Polynomial Equations and Inequalities 





Applications and Polynomial Models 



Reviewing Basic Concepts (Sections 3.63.8) 


261  (1) 


262  (3) 

Chapter 3 Review Exercises 


265  (4) 


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 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 







Graphs with Points of Discontinuity 



Rational Equations, Inequalities, Applications, and Models 


293  (16) 

Solving Rational Equations and Inequalities 



Applications and Models of Rational Functions 





Combined and Joint Variation 



Reviewing Basic Concepts (Sections 4.14.3) 


308  (1) 

Functions Defined by Powers and Roots 


309  (11) 



Modeling Using Power Functions 





Graphing Circles and Horizontal Parabolas Using Root Functions 



Equations, Inequalities, and Applications Involving Root Functions 


320  (22) 

Equations and Inequalities 





Reviewing Basic Concepts (Sections 4.4 and 4.5) 


330  (1) 


331  (2) 

Chapter 4 Review Exercises 


333  (4) 


337  (2) 

Chapter 4 Project How Rugged Is Your Coastline? 


339  (3) 

Inverse, Exponential, and Logarithmic Functions 


342  (77) 


343  (10) 





Inverse Functions and Their Graphs 



An Application of Inverse Functions 




353  (12) 



Graphs of Exponential Functions 



Exponential Equations (Type I) 







Logarithms and Their Properties 


365  (12) 











Reviewing Basic Concepts (Sections 5.15.3) 


376  (1) 


377  (9) 

Graphs of Logarithmic Functions 



Applying Earlier Work to Logarithmic Functions 





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 





Biological and Medical Applications 



Modeling Data with Exponential and Logarithmic Functions 




410  (3) 

Chapter 5 Review Exercises 


413  (3) 


416  (1) 

Chapter 5 Project Modeling Motor Vehicle Sales in the United States (with a lesson about careless use of mathematical models) 


417  (2) 


419  (47) 


420  (13) 



Equations and Graphs of Circles 



An Application of Circles 



Equations and Graphs of Parabolas 



An Application of Parabolas 




433  (13) 

Equations and Graphs 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) 



Identifying Conic Sections 






454  (12) 

Graphs of Parametric Equations and Their Rectangular Equivalents 



Alternative Forms of Parametric Equations 





Reviewing Basic Concepts (Sections 6.3 and 6.4) 


459  (1) 


459  (2) 

Chapter 6 Review Exercises 


461  (2) 


463  (1) 

Chapter 6 Project Modeling the Path of a Bouncing Ball 


464  (2) 

Matrices and Systems of Equations and Inequalities 


466  (100) 


467  (13) 













Solution of Linear Systems by the Echelon Method 


480  (8) 



Analytic Solution of Systems in Three Variables 





Curve Fitting Using a System 



Solution of Linear Systems by Row Transformations 


488  (13) 



Matrix Row Transformations 





Reduced Row Echelon Method 







Reviewing Basic Concepts (Sections 7.17.3) 


501  (1) 

Matrix Properties and Operations 


501  (15) 







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) 



Multiplicative Inverses of Square Matrices 



Solving Linear Systems Using Inverse Matrices 



Curve Fitting Using a System 



Reviewing Basic Concepts (Sections 7.47.6) 


537  (1) 

Systems of Inequalities and Linear Programming 


538  (10) 

Solving Linear Inequalities 



Solving Systems of Inequalities 






548  (18) 

Decomposition of Rational Expressions 







Distinct Linear and Quadratic Factors 



Repeated Quadratic Factors 



Reviewing Basic Concepts (Sections 7.7 and 7.8) 


555  (1) 


556  (2) 

Chapter 7 Review Exercises 


558  (4) 


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) 





Standard Position and Coterminal Angles 









Trigonometric Functions and Fundamental Identities 


583  (12) 







Signs and Ranges of Function Values 







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 







Special Angles as Reference Angles 



Finding Function Values with a Calculator 





Applications of Right Triangles 


607  (13) 





Angles of Elevation or Depression 







Reviewing Basic Concepts (Sections 8.3 and 8.4) 


619  (1) 


620  (8) 



Applications of Circular Functions 



Graphs of the Sine and Cosine Functions 


628  (18) 



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) 


645  (1) 

Graphs of the Other Circular Functions 


646  (10) 

Graphs of the Cosecant and Secant Functions 



Graphs of the Tangent and Cotangent Functions 






656  (17) 



Damped Oscillatory Motion 



Reviewing Basic Concepts (Sections 8.7 and 8.8) 


659  (1) 


660  (4) 

Chapter 8 Review Exercises 


664  (5) 


669  (2) 

Chapter 8 Project Modeling Sunset Times 


671  (2) 

Trigonometric Identities and Equations 


673  (66) 


674  (10) 



Using the Fundamental 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) 


694  (12) 



ProducttoSum and SumtoProduct Identities 





The Inverse Circular Functions 


706  (12) 







Remaining Inverse Trigonometric Functions 





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 MultipleNumber Identities 



Equations and Inequalities Involving HalfNumber Identities 





Reviewing Basic Concepts (Sections 9.5 and 9.6) 


730  (1) 


731  (2) 

Chapter 9 Review Exercises 


733  (3) 


736  (1) 

Chapter 9 Project Modeling a Damped Pendulum 


737  (2) 

Applications of Trigonometry; Vectors 


739  (77) 


740  (13) 

Congruency and Oblique Triangles 



Derivation of the Law of Sines 







The Law of Cosines and Area Formulas 


753  (10) 

Derivation of the Law of Cosines 







Vectors and Their Applications 


763  (13) 



Algebraic Interpretation of Vectors 





Dot Product and the Angle between Vectors 





Reviewing Basic Concepts (Sections 10.110.3) 


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) 





Reviewing Basic Concepts (Sections 10.4 and 10.5) 


790  (1) 

Polar Equations and Graphs 


791  (10) 



Graphs of Polar Equations 



Classifying Polar Equations 





More Parametric Equations 


801  (15) 

Parametric Equations with Trigonometric Functions 





Applications of Parametric Equations 



Reviewing Basic Concepts (Sections 10.6 and 10.7) 


808  (1) 


808  (3) 

Chapter 10 Review Exercises 


811  (3) 


814  (1) 

Chapter 10 Project When Is a Circle Really a Polygon? 


814  (2) 

Further Topics in Algebra 


816  (74) 


817  (10) 



Series and Summation Notation 





Arithmetic Sequences and Series 


827  (8) 





Geometric Sequences and Series 


835  (12) 





Infinite Geometric Series 





Reviewing Basic Concepts (Sections 11.111.3) 


846  (1) 


847  (7) 

A Binomial Expansion Pattern 











rth Term of a Binomial Expansion 




854  (7) 

Proof by Mathematical Induction 





Generalized Principle of Mathematical Induction 



Proof of the Binomial Theorem 



Reviewing Basic Concepts (Sections 11.4 and 11.5) 


861  (1) 


861  (9) 

Fundamental Principle of Counting 







Distinguishing between Permutations and Combinations 




870  (20) 



Complements and Venn Diagrams 









Reviewing Basic Concepts (Sections 11.6 and 11.7) 


881  (1) 


881  (4) 

Chapter 11 Review Exercises 


885  (2) 


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) 



Terminology for Polynomials 



Adding and Subtracting Polynomials 






897  (7) 

Factoring Out the Greatest Common Factor 







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 





Review of Negative and Rational Exponents 


911  (6) 

Negative Exponents and the Quotient Rule 






917  (7) 









Rationalizing Denominators 




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  