Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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Graphs and Models
John W. Coburn
St. Louis Community College at Florissant Valley
J.D. Herdlick
St. Louis Community College at Meramec-Kirkwood
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COLLEGE ALGEBRA: GRAPHS AND MODELS
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the
Americas, New York, NY 10020. Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
No part of this publication may be reproduced or distributed in any form or by any means, or stored in
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United States.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 DOW/DOW 1 0 9 8 7 6 5 4 3 2 1
ISBN 978–0–07–351954–8
MHID 0–07–351954–5
ISBN 978–0–07–723057–9 (Annotated Instructor’s Edition)
MHID 0–07–723057–4
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Typeface: 10.5/12 Times Roman
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All credits appearing on page or at the end of the book are considered to be an extension of the copyright page.
Library of Congress Cataloging-in-Publication Data
Coburn, John W.
College algebra : graphs and models / John W. Coburn, J.D. Herdlick.
p. cm.
Includes index.
ISBN 978–0–07–351954–8 — ISBN 0–07–351954–5 (hard copy : alk. paper) 1. Algebra—
Textbooks. 2. Algebra—Graphic methods—Textbooks. I. Herdlick, John D. II. Title.
QA154.3.C5953 2012
512.9—dc22
2010035347
www.mhhe.com
TM
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Preface vi
Index of Applications xxxii
CHAPTER R A Review of Basic Concepts and Skills 1
CHAPTER 1 Relations, Functions, and Graphs 85
CHAPTER 2 More on Functions 187
CHAPTER 3 Quadratic Functions and Operations on Functions 281
CHAPTER 4 Polynomial and Rational Functions 381
CHAPTER 5 Exponential and Logarithmic Functions 479
CHAPTER 6 Systems of Equations and Inequalities 575
CHAPTER 7 Matrices and Matrix Applications 637
CHAPTER 8 Analytic Geometry and the Conic Sections 707
CHAPTER 9 Additional Topics in Algebra 761
Appendix I The Language, Notation, and Numbers of Mathematics A-1
Appendix II Geometry Review with Unit Conversions A-14
Appendix III More on Synthetic Division A-28
Appendix IV More on Matrices A-30
Appendix V Deriving the Equation of a Conic A-32
Appendix VI Proof Positive—A Selection of Proofs from College Algebra A-34
Student Answer Appendix (SE only) SA-1
Instructor Answer Appendix (AIE only) IA-1
Index I-1
Brief Contents
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About the Authors
John Coburn John Coburn grew up in the Hawaiian Islands, the seventh of sixteen children. He
received his Associate of Arts degree in 1977 from Windward Community College,
where he graduated with honors. In 1979 he earned a Bachelor’s Degree in Education
from the University of Hawaii. After working in the business world for a number
of years, he returned to teaching, accepting a position in high school mathematics
where he was recognized as Teacher of the Year (1987). Soon afterward, the decision
was made to seek a Master's Degree, which he received two years later from the
University of Oklahoma. John is now a full professor at the Florissant Valley campus
of St. Louis Community College. During his tenure there he has received numerous
nominations as an outstanding teacher by the local chapter of Phi Theta Kappa,
two nominations to Who’s Who Among America’s Teachers, and was recognized
as Post Secondary Teacher of the Year in 2004 by the Mathematics Educators of
Greater St. Louis (MEGSL). He has made numerous presentations and local, state,
and national conferences on a wide variety of topics and maintains memberships
in several mathematics organizations. Some of John’s other interests include body
surfing, snorkeling, and beach combing whenever he gets the chance. He is also
an avid gamer, enjoying numerous board, card, and party games. His other loves
include his family, music, athletics, composition, and the wild outdoors.
J.D. Herdlick J.D. Herdlick was born and raised in St. Louis, Missouri, very near the Mississippi
river. In 1992, he received his bachelor’s degree in mathematics from Santa Clara
University (Santa Clara, California). After completing his master’s in mathemat-
ics at Washington University (St. Louis, Missouri) in 1994, he felt called to serve
as both a campus minister and an aid worker for a number of years in the United
States and Honduras. He later returned to education and spent one year teaching
high school mathematics, followed by an appointment at Washington University
as visiting lecturer, a position he held until 2006. Simultaneously teaching as an
adjunct professor at the Meramec campus of St. Louis Community College, he
eventually joined the department full time in 2001. While at Santa Clara University,
he became a member of the honorary societies Phi Beta Kappa, Pi Mu Epsilon, and
Sigma Xi under the tutelage of David Logothetti, Gerald Alexanderson, and Paul
Halmos. In addition to the Dean’s Award for Teaching Excellence at Washington
University, J.D. has received numerous awards and accolades for his teaching at St.
Louis Community College. Outside of the office and classroom, he is likely to be
found in the water, on the water, and sometimes above the water, as a passionate
wakeboarder and kiteboarder. It is here, in the water and wind, that he finds his
inspiration for writing. J.D. and his family currently split their time between the
United States and Argentina.
Dedication
With boundless gratitude, we dedicate this work to the special people in
our lives. To our children, whom we hope were joyfully oblivious to the
time, sacrifice, and perseverance required; and to our wives, who were
well acquainted with every minute of it.
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From the Authors
In the last two decades, mathematics education has seen some enormous changes. From the
introduction of graphing calculators and the advent of the Internet, to online homework and
visual supplements we could only dream about decades ago, the changes have been unrelenting.
Together, John Coburn and J.D. Herdlick share a combined 40 years of experience teaching
college algebra with graphing calculators and other technologies, and have developed a wealth of
firsthand experience related to the endeavor.
In the Coburn/Herdlick Graphs and Models text, we have combined the conversational style and
the wealth of applications that our texts are known for, with this depth of experience. As one of
our primary goals, we set out to help students think visually, to a point where they see functions
like f(x) = x
2
– 4x as one of a family of graphs, with attributes that immediately lead to a
discussion of maximums and minimums, end-behavior, zeroes, solutions to inequalities, the nature of
the roots, and the application of these attributes in context—instead of merely an equation that
must be solved by factoring or by interpreting a graph on the screen of a calculator. And while
graphing calculators may relieve some computational drudgery, we believe our text offers much
more than a simple side-by-side comparison of algebraic methods versus graphical methods, with
the calculator playing a more significant role than simply checking answers to work done manually.
Graphing calculators are used to work and investigate far beyond what’s possible with paper and
pencil, with the technology used to solve more true-to-life equations, engage more applications, and
explore more substantial questions of interest. In the end we believe you’ll see this text is built on
strong fundamentals, yet one that offers a visual and dynamic excursion that accentuates the
organizational planning and problem solving acumen that students will use in all areas of their
lives. To this end we offer the Coburn/Herdlick Graphs and Models text as an ideal tool for the
teaching and learning of mathematics. —John Coburn and J.D. Herdlick
Most coral reefs in the world are 7000–9000 years old, but new reefs
can fully develop in as few as 20 years. In addition to being home to over 4000
species of tropical or reef fish, coral reefs are immensely beneficial to humans and
must be carefully preserved. They buffer coastal regions from strong waves and
storms, provide millions of people with food and jobs, and prompt advances in
modern medicine.
Similar to the ancient reefs, a course in College Algebra is based on thousands
of years of mathematical curiosity, insight, and wisdom. In this one short course,
we study a wealth of important concepts that have taken centuries to mature. Just as
the variety of fish in the sea rely on the coral reefs to survive, students in a College
Algebra course rely on mastery of this bedrock of concepts to successfully pursue
more advanced courses, as well as their career goals.
About the Cover
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College Algebra tends to be a challenging course for many students.
They may not see the connections that College Algebra has to their life or why it
is so critical that they succeed in this course. Others may enter into this course
underprepared or improperly placed and with very little motivation.
Instructors are faced with several challenges as well. They are given the task of
improving pass rates and student retention while ensuring the students are adequately
prepared for more advanced courses, as a College Algebra course attracts a very diverse
audience, with a wide variety of career goals and a large range of prerequisite skills.
The goal of this textbook series is to provide both students and instructors withtools to
address these challenges, so that both can experience greater success in College Algebra.
For instance, the comprehensive exercise sets have a range of difficulty that provides
very strong support for weaker students, while advanced students are challenged to
reach even further. The rest of this preface further explains the tools that John Coburn,
J.D. Herdlick, and McGraw-Hill have developed and how they can be used to connect
students to College Algebra and connect instructors to their students.
The Coburn/Herdlick College Algebra Series provides you with
strong tools to achieve better outcomes in your College Algebra
course as follows:
▶
Making Connections Visually, Symbolically,
Numerically, and Verbally
▶
Better Student Preparedness Through
Superior Course Management
▶
Increased Student Engagement
▶
Solid Skill Development
▶
Strong Mathematical Connections
Making Connections . . .
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▶
Making Connections Visually, Symbolically, Numerically, and Verbally
In writing their Graphs and Models series, the Coburn/Herdlick team took great care to help students
think visually by relating a basic graph to an algebraic equation at every opportunity. This empowers
students to see the “Why?” behind many algebraic rules and properties, and offers solid preparation for
the connections they’ll need to make in future courses which often depend on these visual skills.
▶
Better Student Preparedness Through Superior Course Management
McGraw-Hill is proud to offer instructors a choice of course management options to accompany Coburn/
Herdlick. If you prefer to assign text-specific problems in a brand new, robust online homework system
that contains stepped out and guided solutions for all questions, Connect Math Hosted by ALEKS may
be for you. Or perhaps you prefer the diagnostic nature and artificial intelligence engine that is the
driving force behind our ALEKS 360 Course product, a true online learning environment, which has
been expanded to contain hundreds of new College Algebra & Precalculus topics. We encourage you to
take a closer look at each product on preface pages x through xiii and to consult your McGraw-Hill sales
representative to setup a demonstration.
▶
Increased Student Engagement
There are many texts that claim they “engage” students, but only the Coburn Series has carefully
studied and implemented features and options that make it truly possible. From the on-line support,
to the textbook design and a wealth of quality applications, students will remain engaged throughout
their studies.
▶
Solid Skill Development
The Coburn/Herdlick series intentionally relates the examples to the exercise sets so there is a strong
connection between what students are learning while working through the examples in each section and
the homework exercises that they complete. This development of strong mechanical skills is followed
closely by a careful development of problem solving skills, with the use of interesting and engaging
applications that have been carefully chosen with regard to difficulty and the skills currently under study.
There is also an abundance of exercise types to choose from to ensure that homework challenges
a wide variety of skills. Furthermore, John and J.D. reconnect students to earlier chapter material with
Mid-Chapter Checks; students have praised these exercises for helping them understand what key con-
cepts require additional practice.
▶
Strong Mathematical Connections
John Coburn and J.D. Herdlick’s experience in the classroom and their strong connections to how
students comprehend the material are evident in their writing style. This is demonstrated by the way they
provide a tight weave from topic to topic and foster an environment that doesn’t just focus on procedures
but illustrates the big picture, which is something that so often is sacrificed in this course. Moreover,
they employ a clear and supportive writing style, providing the students with a tool they can depend on
when the teacher is not available, when they miss a day of class, or simply when working on their own.
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