*For courses in college algebra.*

**Visualize. Interact. Succeed.**

The **Graphs and Models** series by Bittinger, Beecher, Ellenbogen, and Penna is known for helping students “see the math” through its focus on visualization and technology. These texts continue to maintain the features that have helped students succeed for years: focus on functions, visual emphasis, side-by-side algebraic and graphical solutions, and real-data applications. With the **Sixth Edition**, visualization is taken to a new level with technology, and students find even more ongoing review.

**Also available with MyMathLab**

MyMathLab^{®} is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and pursue a personalized study plan that helps them absorb course material and understand difficult concepts.

New **Guided Visualizations** in MyMathLab help students allow for hands-on manipulation to gain understanding of difficult concepts. References to **28 Just-In-Time review topics** are placed throughout the text and MyMathLab to help students right when they need it most, and new **Cumulative Review Assignments **and **Skill Maintenance Quizzes** are pre-made and assignable in MyMathLab to help students connect concepts and maintain skills throughout the course. Plus, new **Video Assessment Exercises** and a new **Video Notebook **further enhance the MyMathLab course and resources available.

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**Marvin Bittinger** has been teaching math at the university level for more than thirty-eight years. Since 1968, he has been employed at Indiana University—Purdue University Indianapolis, and is now professor emeritus of mathematics education. Professor Bittinger has authored over 190 publications on topics ranging from basic mathematics to algebra and trigonometry to applied calculus. He received his BA in mathematics from Manchester College and his PhD in mathematics education from Purdue University. Special honors include Distinguished Visiting Professor at the United States Air Force Academy and his election to the Manchester College Board of Trustees from 1992 to 1999. His hobbies include hiking in Utah, baseball, golf, and bowling. Professor Bittinger has also had the privilege of speaking at many mathematics conventions, most recently giving a lecture entitled "Baseball and Mathematics." In addition, he also has an interest in philosophy and theology, in particular, apologetics. Professor Bittinger currently lives in Carmel, Indiana, with his wife, Elaine. He has two grown and married sons, Lowell and Chris, and four granddaughters.

**Judy Beecher** has an undergraduate degree in mathematics from Indiana University and a graduate degree in mathematics from Purdue University. She has taught at both the high school and college levels with many years of developmental math and precalculus teaching experience at Indiana University—Purdue University Indianapolis. In addition to her career in textbook publishing, she spends time traveling, enjoying her grandchildren, and promoting charity projects for a children's camp.

**David Ellenbogen** has taught math at the college level for twenty-two years, spending most of that time in the Massachusetts and Vermont community college systems, where he has served on both curriculum and developmental math committees. He has also taught at St. Michael's College and The University of Vermont. Professor Ellenbogen has been active in the American Mathematical Association of Two Year Colleges since 1985, having served on its Developmental Mathematics Committee and as a delegate, and has been a member of the Mathematical Association of America since 1979. He has authored dozens of publications on topics ranging from prealgebra to calculus and has delivered lectures at numerous conferences on the use of language in mathematics. Professor Ellenbogen received his BA in mathematics from Bates College and his MA in community college mathematics education from The University of Massachusetts at Amherst. A co-founder of the Colchester Vermont Recycling Program, Professor Ellenbogen has a deep love for the environment and the outdoors, especially in his home state of Vermont. In his spare time, he enjoys playing keyboard in the band Soularium, volunteering as a community mentor, hiking, biking, and skiing. He has two sons, Monroe and Zack.

**Judy Penna **received her undergraduate degree in mathematics from Kansas State University and her graduate degree in mathematics from the University of Illinois. Since then, she has taught at Indiana University—Purdue University Indianapolis and at Butler University, and continues to focus on writing quality textbooks for undergraduate mathematics students. In her free time she likes to travel, read, knit, and spend time with her children.

Preface

To the Student

**Just-in-Time Review**

1. Real Numbers

2. Properties of Real Numbers

3. Order on the Number Line

4. Absolute Value

5. Operations with Real Numbers

6. Interval Notation

7. Integers as Exponents

8. Scientific Notation

9. Order of Operations

10. Introduction to Polynomials

11. Add and Subtract Polynomials

12. Multiply Polynomials

13. Special Products of Binomials

14. Factor Polynomials; The FOIL Method

15. Factor Polynomials: The ac-Method

16. Special Factorizations

17. Equation-Solving Principles

18. Inequality-Solving Principles

19. The Principle of Zero Products

20. The Principle of Square Roots

21. Simplify Rational Expressions

22. Multiply and Divide Rational Expressions

23. Add and Subtract Rational Expressions

24. Simplify Complex Rational Expressions

25. Simplify Radical Expressions

26. Rationalize Denominators

27. Rational Exponents

28. The Pythagorean Theorem

**1. Graphs, Functions, and Models**

1.1. Introduction to Graphing

Visualizing the Graph

1.2. Functions and Graphs

1.3. Linear Functions, Slope, and Applications

Visualizing the Graph

Mid-Chapter Mixed Review

1.4. Equations of Lines and Modeling

1.5. Linear Equations, Functions, Zeros, and Applications

1.6. Solving Linear Inequalities

Summary and Review

Review Exercises

Chapter Test

**2. More on Functions**

2.1. Increasing, Decreasing, and Piecewise Functions; Applications

2.2. The Algebra of Functions

2.3. The Composition of Functions

Mid-Chapter Mixed Review

2.4. Symmetry

2.5. Transformations

Visualizing the Graph

2.6. Variation and Applications

Summary and Review

Review Exercises

Chapter Test

**3. Quadratic Functions and Equations; Inequalities**

3.1. The Complex Numbers

3.2. Quadratic Equations, Functions, Zeros, and Models

3.3. Analyzing Graphs of Quadratic Functions

Visualizing the Graph

Mid-Chapter Mixed Review

3.4. Solving Rational Equations and Radical Equations

3.5. Solving Equations and Inequalities with Absolute Value

Summary and Review

Review Exercises

Chapter Test

**4. Polynomial Functions and Rational Functions**

4.1. Polynomial Functions and Modeling

4.2. Graphing Polynomial Functions

Visualizing the Graph

4.3. Polynomial Division; The Remainder Theorem and the Factor Theorem

Mid-Chapter Mixed Review

4.4. Theorems about Zeros of Polynomial Functions

4.5. Rational Functions

Visualizing the Graph

4.6. Polynomial Inequalities and Rational Inequalities

Study Guide

Review Exercises

Chapter Test

**5. Exponential Functions and Logarithmic Functions**

5.1. Inverse Functions

5.2. Exponential Functions and Graphs

5.3. Logarithmic Functions and Graphs

Visualizing the Graph

Mid-Chapter Mixed Review

5.4. Properties of Logarithmic Functions

5.5. Solving Exponential Equations and Logarithmic Equations

5.6. Applications and Models: Growth and Decay; Compound Interest 364

Study Guide

Review Exercises

Chapter Test

**6. Systems of Equations and Matrices**

6.1. Systems of Equations in Two Variables

Visualizing the Graph

6.2. Systems of Equations in Three Variables

6.3. Matrices and Systems of Equations

6.4. Matrix Operations

Mid-Chapter Mixed Review

6.5. Inverses of Matrices

6.6. Determinants and Cramer’s Rule

6.7. Systems of Inequalities and Linear Programming

6.8. Partial Fractions

Study Guide

Review Exercises

Chapter Test

**7. Conic Sections**

7.1. The Parabola

7.2. The Circle and the Ellipse

Mid-Chapter Mixed Review

7.3. The Hyperbola

7.4. Nonlinear Systems of Equations and Inequalities

Visualizing the Graph

Study Guide

Review Exercises

Chapter Test

**8. Sequences, Series, and Combinatorics**

8.1. Sequences and Series

8.2. Arithmetic Sequences and Series

8.3. Geometric Sequences and Series

Visualizing the Graph

8.4. Mathematical Induction

Mid-Chapter Mixed Review

8.5. Combinatorics: Permutations

8.6. Combinatorics: Combinations

8.7. The Binomial Theorem

8.8. Probability

Study Guide

Review Exercises

Chapter Test

Photo Credits

Answers

Additional Instructor’s Answers

Index

Index of Applications