College algebra graphs 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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TM

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
a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc.,
including, but not limited to, in any network or other electronic storage or transmission, or broadcast for

distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the
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
Vice President, Editor-in-Chief: Marty Lange
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Typeface: 10.5/12 Times Roman
Printer: R. R. Donnelley

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


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Brief Contents
Preface vi
Index of Applications

CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER
CHAPTER

R
1
2
3
4
5
6
7
8
9

xxxii

A Review of Basic Concepts and Skills 1
Relations, Functions, and Graphs 85
More on Functions 187
Quadratic Functions and Operations on Functions

281

Polynomial and Rational Functions 381
Exponential and Logarithmic Functions 479
Systems of Equations and Inequalities

575

Matrices and Matrix Applications 637
Analytic Geometry and the Conic Sections 707
Additional Topics in Algebra 761

Appendix I

The Language, Notation, and Numbers of Mathematics

Appendix II

Geometry Review with Unit Conversions

Appendix III

More on Synthetic Division

Appendix IV

More on Matrices A-30

Appendix V

Deriving the Equation of a Conic

Appendix VI

Proof Positive—A Selection of Proofs from College Algebra

A-14

A-28
A-32

Student Answer Appendix (SE only)

A-34

SA-1

Instructor Answer Appendix (AIE only)
Index

A-1

IA-1

I-1

iii


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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 mathematics 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.

iv


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About the Cover
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.

From the Authors

nges. From the
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the end we believe you’ll see this
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accent uates the
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strong fundam
use in all areas of their
solv ing acumen that studen ts will
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org
Gra phs and Models text as an idea
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—John Coburn and J.D. Her dlick
tics.
tea ching and lear ning of mathema

v

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Making Connections . . .
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 with tools 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:

vi

▶

Making Connections Visually, Symbolically,
Numerically, and Verbally

▶

Better Student Preparedness Through
Superior Course Management

▶

Increased Student Engagement

▶

Solid Skill Development

▶

Strong Mathematical Connections


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▶

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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 concepts 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.

vii


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Making Connections . . .
Visually, Symbolically, Numerically, and Verbally
, the concre te and numer ic
“It is widely known that for studen ts to grow stronger algebr aically
entations. In this transit ion
experiences from their past must give way to more symbolic repres

visual connections and verbal
from numer ic, to symbolic, to algebr aic thinking, the importance of
of rich concep ts or subtle ideas,
connections is too often overlooked. To reach a deep understanding
concep t or idea using the terms
studen ts must develop the ability to menta lly “see” and discuss the
seeing the connections that
and names needed to describ e it accurately. Only then can they begin
. A large part of this involves
exist between each new concep t, and concep ts that are already known
they’re able to see functio ns like
helping our studen ts to begin thinking visuall y, to a point where
ical attribu tes that immediately
f(x) = x2 – 4x as only one of a large family of functions, with graph
, solutions to inequa lities, the
lead to a discussion of maximums and minimums, end-behavior, zeroes
t. And while it’s important for
nature of the roots, and the application of these attribu tes in contex
, and that the intersection of
students to see that zeroes are x-intercepts and x-intercepts are zeroes
g these graphs, these should not
two graphs provides a simulta neous solution to the equations formin
tions, investigations, connections,
remain the sole focus of the tool. Graphing calculators allow explora
and we should use the technology
and visualizations far beyond what’s possible with paper and pencil,
more true-to-life equations,
to aid the development of these menta l-visua l skills, in addition to solving
ns involving real data, domain
engaging more applications, and explor ing the more substa ntial questio

tables, and other questions of
and range, anticipated graphical behavior, additional uses of lists and
be successful in these endeavors.”
interest. We believe this text offers instructors the tools they need to
—The Authors

EXAMPLE 1

ᮣ

“I think there is a good balance between technology

Solving a Logarithmic Equation

and paper/pencil techniques. I particularly like how
the technology portion does not take the place of
paper/pencil, but instead supplements it. I think a
lot of departments will like that.

Solve for x and check your answer: log x ϩ log 1x ϩ 32 ϭ 1.
ᮢ

ᮢ

Algebraic Solution
log x ϩ log 1x ϩ 32 ϭ 1
log 3x 1x ϩ 32 4 ϭ 1
x2 ϩ 3x ϭ 101
x2 ϩ 3x Ϫ 10 ϭ 0
1x ϩ 52 1x Ϫ 22 ϭ 0

x ϭ Ϫ5 or x ϭ 2

original equation
product property
exponential form,
distribute x
set equal to 0
factor
result

Graphical Solution

Using the intersection-ofgraphs method, we enter
Y1 ϭ log X ϩ log1X ϩ 32
and Y2 ϭ 1. From the domain
we know x 7 0, indicating
the solution will occur in QI.
After graphing both functions
using the window shown, the
intersection method shows
the only solution is x ϭ 2.

3

”

0

—Daniel Brock, Arkansas State University-Beebe

5

▶ Graphical Examples show students how

Ϫ3

Check: The “solution” x ϭ Ϫ5 is outside the domain and is ignored. For x ϭ 2,
log x ϩ log1x ϩ 32 ϭ 1 original equation
log 2 ϩ log12 ϩ 32 ϭ 1 substitute 2 for x
log 2 ϩ log 5 ϭ 1 simplify
log12 # 52 ϭ 1 product property
log 10 ϭ 1 Property I

the calculator can be used to supplement
their understanding of a problem.
EXAMPLE 1A

You could also use a calculator to verify log 2 ϩ log 5 ϭ 1 directly.
roug
g 14
Now try Exercises 7 through

Precalculus: Graphs and Models textbook
the best approach ever to the teaching of
Precalculus with the inclusion of graphing
calculator.

”

viii

—Alvio Dominguez, Miami-Dade
College-Wolfson

Solving an Equation Graphically
1
Solve the equation 21x Ϫ 32 ϩ 7 ϭ x Ϫ 2 using
2
a graphing calculator.

ᮣ

Solution

“I have certainly found the Coburn/Herdlick’s

ᮣ

ᮣ

Begin by entering the left-hand expression as Y1
and the right-hand expression as Y2 (Figure 1.74).
To find points of intersection, press 2nd TRACE
(CALC) and select option 5:intersect, which
automatically places you on the graphing
window, and asks you to identify the
“First curve?.” As discussed, pressing
three times in succession will identify each
graph, bypass the “Guess?” option, then
find and display the point of intersection

(Figure 1.75). Here the point of intersection
Ϫ10
is (Ϫ2, Ϫ3), showing the solution to this
equation is x ϭ Ϫ2 (for which both
expressions equal Ϫ3). This can be verified
by direct substitution or by using the
TABLE feature.
ENTER

Figure 1.74

Figure 1.75
10

10

Ϫ10


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▶ Calculator Explanations incorporate the

calculator without sacrificing.

Figure 3.2
Most graphing calculators are programmed to work
with imaginary and complex numbers, though for

some models the calculator must be placed in complex number mode. After pressing the MODE key
(located to the right of the 2nd option key), the
screen shown in Figure 3.2 appears and we use
the arrow keys to access “a ϩ bi” and active this
mode (by pressing ). Once active, we can validate
our previo
previous statements about imaginary numbers
(Figure
gure 3.3),
3.3 as well as verify our previous calculations like those in Examples 3(a), 3(d),
andd 4(a) (F
an
(Figure 3.4). Note the imaginary unit i is the 2nd option for the decimal point.
ENTER

“The technology (graphing calculator) explanations
and illustrations are superb. The level of detail is
valuable; even an experienced user (myself)
learned some new techniques and “tricks” in
reading through the text. The text frequently
references use of the calculator—yet without
sacrificing rigor or mathematical integrity.

Figure 3.3

Figure 3.4

”

—Light Bryant, Arizona Western College

Figure 4.4A

To help illustrate the Intermediate Value Theorem, many graphing calculators
offer a useful feature called split screen viewing, that enables us to view a table of
values and the graph of a function at the same time. To illustrate, enter the function
y ϭ x3 Ϫ 9x ϩ 6 (from Example 6) as Y1 on the Y= screen, then set the viewing
window as shown in Figure 4.4. Set your table in AUTO mode with ¢Tbl ϭ 1, then
press the MODE key (see Figure 4.4A) and notice the second-to-last entry on this screen
reads: Full for full screen viewing, Horiz for splitting the screen horizontally with the
graph above a reduced home screen, and G-T, which represents Graph-Table and
splits the screen vertically. In the G-T mode, the graph appears on the left and the
table of values on the right. Navigate the cursor to the G-T mode and press . Pressing the GRAPH key at this point should give you a screen similar to Figure 4.5. Scrolling
downward shows the function also changes sign between x ϭ 2 and x ϭ 3. For more
on this idea, see Exercises 31 and 32.
werful yet simple
As a final note, while the intermediate value theorem is a powerful
tool, it must be used with care. For example, given p1x2 ϭ Ϫx4 ϩ 10x2 Ϫ 5, p1Ϫ12 7 0
lly,
and p112 7 0, seeming to indicate that no zeroes exist in the intervall (Ϫ1, 1). Actual
Actually,
there are two zeroes, as seen in Figure 4.6.
ENTER

in every section. I have been using TI calculators
for 15 years and I learned a few new tricks while
reading this book.

Figure 4.6

Figure 4.5

“The authors give very good uses of the calculator

25

”

B. You’ve just seen how
we can use the intermediate
value theorem to identify
intervals containing a
polynomial zero

5

Ϫ5

—George Hurlburt, Corning Community College

Ϫ10

▶ Technology Applications show

students how technology can be
used to help apply lessons from
the classroom to real life.

“I think that the graphing examples, explanations,

and problems are perfect for the average college
algebra student who has never touched a graphing
calculator. . . . . I think this book would be great to
actually have in front of the students.

”

—Dale Duke, Oklahoma City Community College

Use Newton’s law of cooling to complete Exercises 75
and 76: T(x) ϭ TR ϩ (T0 Ϫ TR)ekx.

75. Cold party drinks: Janae was late getting ready for
the party, and the liters of soft drinks she bought
were still at room temperature (73°F) with guests
due to arrive in 15 min. If she puts these in her
freezer at Ϫ10°F, will the drinks be cold enough
(35°F) for her guests? Assume k Ϸ Ϫ0.031.
76. Warm party drinks: Newton’s law of cooling
applies equally well if the “cooling is negative,”
meaning the object is taken from a colder medium
and placed in a warmer one. If a can of soft drink is
taken from a 35°F cooler and placed in a room
where the temperature is 75°F, how long will it take
the drink to warm to 65°F? Assume k Ϸ Ϫ0.031.
Photochromat
Photochromatic sunglasses: Sunglasses that darken in
sunlight
sunl
unligh

ghtt (p
(photo
(photochromatic sunglasses) contain millions of
mole
m
ole
lecules
le
lec
cu of a substance known as silver halide. The
molecules
molecules
m
mo
o
are ttransparent indoors in the absence of
ultraviolent (U
(UV) light. Outdoors, UV light from the sun
causes the mol
molecules to change shape, darkening the
lenses in respo
response to the intensity of the UV light. For
certain lenses, the function T1x2 ϭ 0.85x models the
transparency of the lenses (as a percentage) based on a
UV index x. Fi
Find the transparency (to the nearest
percent), if the lenses are exposed to
77
li ht with a UV index of 7 (a high exposure).
77. sunlight

78. sunlight with a UV index of 5.5 (a moderate
exposure)

80. Use a trial-and-error process and a graphing
calculator to determine the UV index when the
lenses are 50% transparent.
Modeling inflation: Assuming the rate of inflation is 5%
per year, the predicted price of an item can be modeled
by the function P1t2 ϭ P0 11.052 t, where P0 represents the
initial price of the item and t is in years. Use this
information to solve Exercises 81 and 82.
81. What will the price of a new car be in the year
2015, if it cost $20,000 in the year 2010?
82. What will the price of a gallon of milk be in the
year 2015, if it cost $3.95 in the year 2010? Round
to the nearest cent.
Modeling radioactive decay: The half-life of a
radioactive substance is the time required for half an
initial amount of the substance to disappear through
decay. The amount of the substance
remaining is given
t
by the formula Q1t2 ϭ Q0 1 12 2 h, where h is the half-life,
t represents the elapsed time, and Q(t) represents the
amount that remains (t and h must have the same unit
of time). Use this information to solve Exercises 83
and 84.
83. Some isotopes of the substance known as thorium
have a half-life of only 8 min. (a) If 64 grams are
initially present, how many grams (g) of the

substance remain after 24 min? (b) How many
minutes until only 1 gram (g) of the substance
remains?

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Connect Math Hosted by ALEKS Corporation is an exciting, new assignment and
assessment platform combining the strengths of McGraw-Hill Higher Education and ALEKS Corporation.
Connect Math Hosted by ALEKS is the first platform on the market to combine an artificially-intelligent,
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Connect Math Hosted by ALEKS Corporation is the culmination of a one-of-a-kind market development
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Connect Math Hosted by ALEKS Corporation is built by Math educators for Math educators!

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Modern Student Homepage
▶ This homepage provides a dashboard for
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(Assignments include HW, quizzes, and tests.)

▶ Students can access their assignments through
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Modern, intuitive,
and simple interface.

2

You want a way to identify the strengths and weaknesses of your class at
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Integrated ALEKS® Assessment
▶ This artificially-intelligent (AI), diagnostic
assessment identifies precisely what a student
knows and is ready to learn next.
▶ Detailed assessment reports provide instructors
with specific information about where students
are struggling most.
▶ This AI-driven assessmentt is
the only one of its kind
in an online homework
platform.

Recommended to be used
as the first assignment in
any course.

ALEKS is a registered trademark of ALEKS Corporation.

x

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Built by Math Educators
for Math Educators
3

Y
Your
students want an assignment page that is easy to use and includes
llots of extra help resources.

Efficient Assignment Navigation
▶ Students have access to immediate
feedback and help while working
through assignments.
▶ Students have direct access
ess
to a media-rich eBook forr easy
referencing.
▶ Students can view detailed,
ed,
step-by-step solutions
written by instructors
who teach the course,
providing a unique solution
on

to each and every exercise.
e

4

Students can easily monitor
and track their progress on a
given assignment.

Y want a more intuitive and efficient assignment creation process
You
because of your busy schedule.
b

Assignment Creation Process
▶ Instructors can select textbookspecific questions organized by
chapter, section, and objective.
▶ Drag-and-drop functionality makes
creating an assignment quick and
easy.
▶ Instructors can preview their
assignments for efficient editing.

TM

www.connectmath.com

xi

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5

/Volume/204/MHDQ268/wea25324_disk1of1/0073525324/wea25324_pagefiles

Your students want an interactive eBook with rich functionality
integrated into the product.

Integrated Media-Rich eBook
▶ A Web-optimized eBook is seamlessly
integrated within ConnectPlus Math Hosted
by ALEKS Corp for ease of use.
▶ Students can access videos, images, and
other media in context within each chapter
or subject area to enhance their learning
experience.
▶ Students can highlight, take notes, or even
access shared instructor highlights/notes to
learn the course material.
▶ The integrated eBook provides students
with a cost-saving alternative to traditional
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6

You want a flexible gradebook that is easy to use.

Flexible Instructor Gradebook
▶ Based on instructor feedback,

Connect Math Hosted by ALEKS
Corp’s straightforward design creates
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▶ Assignment types are color-coded for
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the flexibility to import and export
additional grades.
Instructors have the ability
to drop grades as well as
assign extra credit.

xii


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Built by Math Educators
for Math Educators
7

Y want algorithmic content that was developed by math faculty to
You
ensure the content is pedagogically sound and accurate.
e

Digital Content Development Story

The development of McGraw-Hill’s Connect Math Hosted by ALEKS Corp. content involved
collaboration between McGraw-Hill, experienced instructors, and ALEKS, a company known for its
high-quality digital content. The result of this process, outlined below, is accurate content created
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1. McGraw-Hill selected experienced instructors to work as Digital Contributors.
2. The Digital Contributors selected the textbook exercises to be included in the algorithmic
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3. The Digital Contributors created detailed, stepped-out solutions for use in the Guided
Solution and Show Me features.
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Connect Math Hosted by ALEKS Corp.
Built by Math Educators for Math Educators
Lead Digital
Contributors

Tim Chappell
Metropolitan Community
College, Penn Valley

Digital Contributors
Al Bluman, Community College of
Allegheny County
John Coburn, St. Louis Community
College, Florissant Valley
Vanessa Coffelt, Blinn College

Donna Gerken, Miami-Dade College
Kimberly Graham
J.D. Herdlick, St. Louis Community
College, Meramec

Jeremy Coffelt
Blinn College

Nancy Ikeda
Fullerton College

Vickie Flanders, Baton Rouge
Community College
Nic LaHue, Metropolitan Community
College, Penn Valley
Nicole Lloyd, Lansing Community College
Jackie Miller, The Ohio State University
Anne Marie Mosher, St. Louis
Community College, Florissant Valley
Reva Narasimhan, Kean University
David Ray, University of Tennessee,
Martin

Amy Naughten

Kristin Stoley, Blinn College
Stephen Toner, Victor Valley College
Paul Vroman, St. Louis Community
College, Florissant Valley
Michelle Whitmer, Lansing Community

College

www.connectmath.com

TM

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Better Student Preparedness . . .
College Algebra
Enhanced Course Coverage Enables Seamless
Integration with Textbooks and Syllabi
ALEKS College Algebra features hundreds of new course topics to provide comprehensive course coverage, and ALEKS
AI-2, the next generation intelligence engine to dramatically improve student learning outcomes. This enhanced ALEKS
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ALEKS is a Web-based program that uses artificial intelligence and adaptive questioning to assess precisely a student’s
knowledge in College Algebra and provide personalized instruction on the exact topics the student is most ready to learn.
By providing individualized assessment and learning, ALEKS helps students to master course content quickly and easily.

Topics Added For Comprehensive Coverage:
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The ALEKS Pie summarizes a student’s
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Robust Graphing Features:
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The ALEKS Graphing Calculator is
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▶

◀  Realistic Input Tools provide an adaptive,
open-response environment that avoids
multiple-choice questions and ensures
student mastery.

xiv

ALEKS is a registered trademark of ALEKS Corporation.


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. . .T hrough Superior Course Management
New Instructor Module Features
for College Algebra
Help Students Achieve Success While
Saving Instructor Time
ALEKS includes an Instructor Module with powerful, assignment-driven features and extensive content flexibility to
simplify course management so instructors spend less time with administrative tasks and more time directing student
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students’ responses on multipart questions
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Instructors can also manually adjust scores.

◀

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Integration Topic Coverage:
Instructors have access to ALL course topics
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For more information about ALEKS, please visit: www.aleks.com/highered/math
ALEKS is a registered trademark of ALEKS Corporation.

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Increased Student Engagement . . .
Through
g Meaningful
g Applications
pp
a con nec tion bet wee n the
req uires that student s exp erie nce

Ma king mat hematics mea ning ful
is the result of a pow erf ul
on the wor ld they live in. This text
act
imp
its
and
y,
stud
y
the
s
atic
ing close ties to the
mat hem
qua lity, and greates t inte res t, hav
est
high
the
of
s
tion
lica
app
vide
commit men t to pro
larly ma de an eff ort to sup ply
ed leve ls of diff icul ty. We par ticu
itor
mon

lly
efu
car
h
wit
and
les,
assignm ent s, and
exa mp
illus trations, incl uded as hom ework
lass
in-c
for
d
use
be
to
y
ntit
qua
these in suf ficient
ir sup ply premat urely. Ma ny
and test s, wit hou t exhaus ting the
zes
quiz
of
n
ctio
stru
con

the
in
d
cur ious, eve n
emp loye
nces, wit h oth ers com ing from a
erie
exp
rse
dive
own
our
of
n
bor
atics in
app lications wer e
nts of life, and to see the mat hem
eve
ay
ryd
eve
the
on
e
seiz
to
ch tools, wit h
visiona ry folly that ena bles one
tial libr ary of ref ere nce and resear

stan
sub
a
by
ted
por
sup
e
wer
se
the backgr oun d. The
Aut hor s
nts, and modern tren ds. —The
an eye toward hist ory, cur ren t eve

▶ Chapter Openers highlight Chapter Connections, an interesting

application exercise from the chapter, and provide a list of other
real-world connections to give context for students who wonder
how math relates to them.

“I think the book has very modern applications and quite a few
of them. The calculator instructions are very well done.”

CHAPTER CONNECTIONS

—Nezam Iraniparast, Western Kentucky University

More on Functions
CHAPTER OUTLINE

2.5 Piecewise-Defined Functions 245

power in watts and v is the wind velocity in
miles per hour. While the formula enables us
to predict the power generated for a given wind
speed, the graph offers a visual representation
of this relationship, where we note a rapid
growth in power output as the wind speed
increases. This application appears as
Exercise 107 in Section 2.2.

2.6 Variation: The Toolbox Functions in Action 259

Check out these other real-world connections:

2.1 Analyzing the Graph of a Function 188

▶ Examples throughout the text feature word problems, providing

students with a starting point for how to solve these types of
problems in their exercise sets.

Viewing a function in terms of an equation, a
table of values, and the related graph, often
brings a clearer understanding of the
relationships involved. For example, the power
generated by a wind turbine is often modeled
8v3
by the function P 1v2 ϭ

, where P is the
125

2.2 The Toolbox Functions and Transformations 202
2.3 Absolute Value Functions, Equations,
and Inequalities 218

2.4 Basic Rational Functions and Power Functions;
More on the Domain 230

ᮣ

ᮣ

ᮣ

“ The students always want to know ‘When am I ever going to have

ᮣ

Analyzing the Path of a Projectile
(Section 2.1, Exercise 57)
Altitude of the Jet Stream
(Section 2.3, Exercise 61)
Amusement Arcades
(Section 2.5, Exercise 42)
Volume of Phone Calls
(Section 2.6, Exercise 55)

to use algebra anyway?’ Now it will not be hard for them to see for

themselves some REAL ways.
—Sally Haas, Angelina College

187

”

EXAMPLE 2

▶ Application Exercises at the end of each section are the hallmark of

the Coburn series. Never contrived, always creative, and born out of
the author’s life and experiences, each application tells a story and
appeals to a variety of teaching styles, disciplines, backgrounds, and
interests. The authors have ensured that the applications reflect the
most common majors of college algebra students.

“ The amount of technology is great, as are the applications.
The quality of the applications is better than my current text.”
—Daniel Russow, Arizona Western College–Yuma

▶ Math
M th iin Action
A ti Applets,
A l t llocated
t d online,
li enable
bl students
t d t tto work
k

collaboratively as they manipulate applets that apply mathematical
concepts in real-world contexts.
xvi

ᮣ

Identifying Functions
Two relations named f and g are given; f is pointwise-defined (stated as a set of
ordered pairs), while g is given as a set of plotted points. Determine whether each
is a function.
f: 1Ϫ3, 02, 11, 42, 12, Ϫ52, 14, 22, 1Ϫ3, Ϫ22, 13, 62, 10, Ϫ12, (4, Ϫ5), and (6, 1)

Solution

ᮣ

The relation f is not a function, since Ϫ3 is paired
with two different outputs: 1Ϫ3, 02 and 1Ϫ3, Ϫ22 .
The relation g shown in the figure is a function.
Each input corresponds to exactly one output,
otherwise one point would be directly above the
other and have the same first coordinate.

g

5

y
(0, 5)

(Ϫ4, 2)
(3, 1)

(Ϫ2, 1)
Ϫ5

5

x

(4, Ϫ1)
(Ϫ1, Ϫ3)
Ϫ5

Now try Exercises 11 through 18 ᮣ


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Through Timely Examples
mp les that set the sta ge
to overstate the imp ortance of exa
t
icul
diff
be
ld

wou
it
,
tics
ma
the
was too
In ma
falt ered due to an exa mp le that
e
hav
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nce
erie
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cat
edu
for lear ning. No t a few
a car efu l and
a dist rac ting result. In this ser ies,
had
or
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uen
seq
of
out
fit,

r
on
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ely and clea r, wit h a direct foc us
tim
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tha
les
mp
exa
ct
sele
to
deliber ate eff ort was ma de
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e, they wer e fur the r designed to
sibl
pos
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Eve
d.
han
at
l
the concep t or skil
e. As a tra ined educat or

gro undwor k for concep ts to com
the
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and
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idea
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ore it’s ever asked, and a tim ely
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way in this regard, ma king each
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can
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en
Wh
.
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ate
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ant
n
eve
cal,
nex t logi
. —The Aut hor s
it was just sup posed to be that way
in unn otic ed incr ements, as tho ugh

“ The authors have succeeded with numerous

calculator examples with easy-to-use instructions
to follow along. I truly enjoy seeing plenty of
calculator examples throughout the text!!

▶ Side by side graphical and algebraic solutions illustrate the

difference between problem-solving methods, emphasize the
connections between algebraic and graphical information,
and enable students to understand why one method might be
preferable to another for any given problem.

”

—David Bosworth, Huchinson Community College

▶ Titles have been added to examples to

highlight relevant learning objectives
and reinforce the importance of speaking
mathematically using vocabulary.

EXAMPLE 8

ᮣ

Analytical Solution

ᮣ

Solve the inequality Ϫx2 ϩ 6x Յ 9.

▶ Annotations located to the right of the

solution sequence help the student recognize
which property or procedure is being
applied.
▶ “Now Try” boxes immediately following

examples guide students to specific matched
exercises at the end of the section, helping
them identify exactly which homework
problems coincide with each discussed
concept.

Solving a Quadratic Inequality

WORTHY OF NOTE

Begin by writing the inequality in standard form: Ϫx2 ϩ 6x Ϫ 9 Յ 0. Note this is
equivalent to g1x2 Յ 0 for g1x2 ϭ Ϫx2 ϩ 6x Ϫ 9. Since a 6 0, the graph of g will
open downward. The factored form is g1x2 ϭ Ϫ1x Ϫ 32 2, showing 3 is a zero and a
repeated root. Using the x-axis, we plot the point (3, 0) and visualize a parabola
opening downward through this point.
Figure 3.29 shows the graph is below the x-axis (outputs are negative) for all
values of x except x ϭ 3. But since this is a less than or equal to inequality, the
solution is x ʦ ‫ޒ‬.

Since x ϭ 3 was a zero of
multiplicity 2, the graph “bounced
off” the x-axis at this point, with no
change of sign for g. The graph is
entirely below the x-axis, except at
the vertex (3, 0).

Graphical Solution

Figure 3.29
Ϫ1

0

1

2

3

4

5

6

7

x

aϽ0
ᮣ

The complete graph of g shown in Figure 3.30 confirms the analytical solution
(using the zeroes method). For the intervals of the domain shown in red:
1Ϫq, 32 ´ 13, q2 , the graph of g is below the x-axis 3 g1x2 6 04 . The point (3, 0)

is on the x-axis 3 g132 ϭ 04 . As with the analytical solution, the solution to this
“less than or equal to” inequality is all real numbers. A calculator check of the
original inequality is shown in Figure 3.31.
Figure 3.30

Figure 3.31

y

10

2

Ϫ2

6

x
Ϫ2

8

g(x)

“The modeling and regression

examples in this text are excellent,
and the instructions for using the
graphing calculator to investigate
these types of problems are great.

Ϫ3
Ϫ8

Now try Exercises 121 thro
through 132

”

ᮣ

—Allison Sutton, Austin
Community College

“ The examples support the exercises which is very
important. The chapter is very well written and is
easy to read and understand.

”

—Joseph Lloyd Harris, Gulf Coast Community College

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Solid Skill Development . . .

Through
g Exercises
s. The exe rcise
sup por t of eac h sec tion’s ma in idea
in
es
rcis
exe
of
lth
wea
a
d
ude
We hav e incl
t for wea ker stu den ts,
e, in an eff ort to pro vide sup por
car
at
gre
h
wit
ed
uct
str
con
e
of
wer
set s

n fur the r. The qua ntit y and qua lity
eve
ch
rea
to
ts
den
stu
ed
anc
adv
whi le cha llenging mo re
ies to guide
eff ort s, and num ero us opp ort unit
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tea
a
for
t
por
sup
ong
str
exe rcis es off ers
ing idea s.
to illus tra te imp ortant pro blem solv
and
ions
ulat

calc
t
icul
diff
h
oug
stu den ts thr
—The Aut hor s

Mid-Chapter Checks

MID-CHAPTER CHECK

Mid-Chapter Checks provide students with a good stopping
place to assess their knowledge before moving on to the
second half of the chapter.

1. Determine whether the following function is even,
ͿxͿ
odd, or neither. f 1x2 ϭ x2 ϩ
4x

4. Write the equation of the function that has the same
graph of f 1x2 ϭ 2x, shifted left 4 units and up 2
units.
5. For the graph given, (a) identify
the function family, (b) describe
or identify the end-behavior,
inflection point, and x- and
y-intercepts, (c) determine

the domain and range and

2. Use a graphing calculator to find the maximum and
minimum values of
f 1x2 ϭ Ϫ1.91x4 Ϫ 2.3x3 ϩ 2.2x Ϫ 5.12 . Round to
the nearest hundredth.
3 Use interval notation to identify the interval(s)

End-of-Section Exercise Sets

Exercise 5
y
5

f(x)

Ϫ5

5 x

2.2 EXERCISES

▶ Concepts and Vocabulary exercises to help students

ᮣ

CONCEPTS AND VOCABULARY

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.

recall and retain important terms.

1. After a vertical
, points on the graph are
farther from the x-axis. After a vertical
,
points on the graph are closer to the x-axis.

2. Transformations that change only the location of a
graph and not its shape or form, include
and
.

3. The vertex of h1x2 ϭ 31x ϩ 52 Ϫ 9 is at
and the graph opens
.

4. The inflection point of f 1x2 ϭ Ϫ21x Ϫ 42 3 ϩ 11 is
at
and the end-behavior is
,
.

2

5 Gi

▶ Developing Your Skills exercises to provide

ᮣ

practice of relevant concepts just learned with
increasing levels of difficulty.

th

h f

lf

ti

f ( ) di

/

/E l i

h th

7. f 1x2 ϭ x2 ϩ 4x
5

2

ϩ 3i

15. r 1x2 ϭ Ϫ3 14 Ϫ x ϩ 3 16. f 1x2 ϭ 2 1x ϩ 1 Ϫ 4
5

y

5

Ϫ5

r(x)

8. g1x2 ϭ Ϫx2 ϩ 2x

y

5

5 x

Ϫ5

5

5 x

Ϫ5

f(x)

5 x

Ϫ5

Ϫ5

y

y

Ϫ5

5 x

17. g1x2 ϭ 2 14 Ϫ x

formulas and applications bring forward some interesting
ideas and problems that are more in depth. These would
help hold the students’ interest in the topic.

hift f f 1 2

DEVELOPING YOUR SKILLS

By carefully inspecting each graph given, (a) identify the
function family; (b) describe or identify the end-behavior,
vertex, intervals where the function is increasing or
decreasing, maximum or minimum value(s) and x- and
y-intercepts; and (c) determine the domain and range.
Assume required features have integer values.

“ The sections in the assignments headed working with

6 Di

18. h1x2 ϭ Ϫ2 1x ϩ 1 ϩ 4

y

5

y

g(x)
h(x)

”

Ϫ5

Ϫ5

—Sherri Rankin, Huchinson Community College

ᮣ

WORKING WITH FORMULAS

61. Discriminant of the reduced cubic x3 ؉ px ؉ q ‫ ؍‬0: D ‫ ؍‬؊14p3 ؉ 27q2 2
The discriminant of a cubic equation is less well known than that of the quadratic, but serves the same purpose.
The discriminant of the reduced cubic is given by the formula shown, where p is the linear coefficient and q is
the constant term. If D 7 0, there will be three real and distinct roots. If D ϭ 0, there are still three real roots, but
one is a repeated root (multiplicity two). If D 6 0, there are one real and two complex roots. Suppose we wish to
study the family of cubic equations where q ϭ p ϩ 1.

a. Verify the resulting discriminant is D ϭ Ϫ14p3 ϩ 27p2 ϩ 54p ϩ 272.
b. Determine the values of p and q for which this family of equations has a repeated real root. In other words,
solve the equation Ϫ14p3 ϩ 27p2 ϩ 54p ϩ 272 ϭ 0 using the rational zeroes theorem and synthetic division
to write D in completely factored form.

▶ Working with Formulas exercises to demonstrate

contextual applications of well-known formulas.
▶ Extending the Concept exercises that require

ᮣ

communication of topics, synthesis of related
concepts, and the use of higher-order thinking
skills.

EXTENDING THE CONCEPT

59. Use the general solutions from the quadratic formula
to show that the average value of the x-intercepts is
Ϫb
. Explain/Discuss why the result is valid even if
2a
the roots are complex.
Ϫb ϩ 2b2 Ϫ 4ac

▶ Maintaining Your Skills exercises that address

ᮣ

skills from previous sections to help students
retain previously learning knowledge.

Ϫb Ϫ 2b2 Ϫ 4ac

62. Referring to Exercise 39, discuss the nature (real or
complex, rational or irrational) and number of
zeroes (0, 1, or 2) given by the vertex/intercept
formula if (a) a and k have like signs, (b) a and k
k
have unlike signs, (c) k is zero, (d) the ratio Ϫ
a
is positive and a perfect square and (e) the

MAINTAINING YOUR SKILLS

37. (1.3) Is the graph shown here, the graph of a
function? Discuss why or why not.

38. (R.2/R.3) Determine
the area of the figure
shown
1A ϭ LW, A ϭ ␲r2 2.

18 cm
24 cm

39. (1.5) Solve for r: A ϭ P ϩ Prt

“The exercise sets are plentiful. I like having many to

choose from when assigning homework. When there
are only one or two exercises of a particular type, it’s
hard for the students to get the practice they need.

”

—Sarah Jackson, Pratt Community College

xviii

40

S l

f

(if
if

ibl )

“ There seems to be a good selection of easy, moderate,
and difficult problems in the exercises.”

—Ed Gallo, Sinclair Community College


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End-of-Chapter Review Material
Exercises located at the end of the chapter provide students
with the tools they need to prepare for a quiz or test. Each
chapter features the following:
▶

Making Connections matching exercises are groups of
problems where students must identify graphs based on
an equation or description. This feature helps students
make the connection between graphical and algebraic
information while it enhances students’ ability to read
and interpret graphical data.

“ Not only was the algebra rigorously treated, but it

was reinforced throughout the chapters with the MidChapter Check and the Chapter Review and Tests.

”

—Mark Crawford, Waubonsee Community College

MAKING CONNECTIONS
Making
M
ki Connections:
C
ti
G

Graphically,
hi ll Symbolically,
S b li ll Numerically,
N
i ll and
d Verbally
V b ll
Eight graphs (a) through (h) are given. Match the characteristics shown in 1 through 16 to one of the eight graphs.
y

(a)

5 x

Ϫ5

▶

Chapter Summary and Concept Reviews that present
key concepts with corresponding exercises by section in
a format easily
y used by
y students.

5 x

“ The problem sets are really magnificent. I deeply enjoy

”

▶

Practice Tests that give students the opportunity to check
their knowledge and prepare for classroom quizzes, tests,
and other assessments.

▶

Cumulative Reviews that are presented at the end of
each chapter help students retain previously learned
skills and concepts by revisiting important ideas from
earlier chapters (starting with Chapter 2).

▶

Graphing Calculator icons appear next to exercises
where important concepts can be supported by the
use of graphing technology.

5 x

Ϫ5

y

(h)

5

5 x

Ϫ5

Ϫ5

Ϫ5

5 x

Ϫ5

Ϫ5

y

(g)

5

Ϫ5

5 x

Ϫ5

5

Ϫ5

y

(f)

5

y

(d)

5

Ϫ5

y

(e)

5 x

Ϫ5

y

(c)

5

Ϫ5

Ϫ5

and appreciate the many problems that incorporate
telescopes, astronomy, reflector design, nuclear cooling
tower profiles, charged particle trajectories, and
other such examples from science, technology, and
engineering.
—Light Bryant, Arizona Western College

y

(b)

5

5

5 x

Ϫ5

Ϫ5

1
1. ____ y ϭ x ϩ 1
3

9. ____ f 1Ϫ32 ϭ 4, f 112 ϭ 0

2. ____ y ϭ Ϫx ϩ 1

10. ____ f 1Ϫ42 ϭ 3, f 142 ϭ 3

SUMMARY AND CONCEPT REVIEW
SECTION 1.1
SE

Rectangular Coordinates; Graphing Circles and Other Relations

KEY CONCEPTS
KE
• A relation is a collection of ordered pairs (x, y) and can be stated as a set or in equation form.
• As a set of ordered pairs, we say the relation is pointwise-defined. The domain of the relation is the set of all first
coordinates, and the range is the set of all corresponding second coordinates.
• A relation can be expressed in mapping notation x S y, indicating an element from the domain is mapped to
(corresponds to or is associated with) an element from the range.
• The graph of a relation in equation form is the set of all ordered pairs (x, y) that satisfy the equation. We plot a
sufficient number of points and connect them with a straight line or smooth curve, depending on the pattern
formed.
• The x- and y-variables of linear equations and their graphs have implied exponents of 1.
• With a relation entered on the Y= screen, a graphing calculator can provide a table of ordered pairs and the
related graph.
x1 ϩ x2 y1 ϩ y2
,
b.
• The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is a
2
2

• The distance between the points (x1, y1) and (x2, y2) is d ϭ 21x2 Ϫ x1 2 2 ϩ 1y2 Ϫ y1 2 2.
• The equation of a circle centered at (h, k) with radius r is 1x Ϫ h2 2 ϩ 1y Ϫ k2 2 ϭ r2.

EXERCISES
1. Represent the relation in mapping notation, then state the domain and range.
51Ϫ7, 32, 1Ϫ4, Ϫ22, 15, 12, 1Ϫ7, 02, 13, Ϫ22, 10, 826
2

“The authors give very good uses of the calculator in every

section. I have been using TI calculators for 15 years and
I learned a few new tricks while reading this book.

”

—George Hurlburt, Corning Community College

Homework Selection Guide
A list of suggested homework exercises has been provided for each section of the text (Annotated Instructor’s Edition only).
This feature may prove especially useful for departments that encourage consistency among many sections, or those having a
large adjunct population. The feature was also designed as a convenience to instructors, enabling them to develop an inventory
of exercises that is more in tune with the course as they like to teach it. The guide provides prescreened and preselected
p
assignments at four different levels: Core, Standard, Extended, and In Depth.
8
10
• Core: These assignments go right to the heart of the material,
HOMEWORK SELECTION GUIDE
offering a minimal selection of exercises that cover the primary
concepts and solution strategies of the section, along with a
small selection of the best applications.
• Standard: The assignments at this level include the Core exercises, while providing for additional practice without excessive drill.
A wider assortment of the possible variations on a theme are included, as well as a greater variety of applications.

• Extended: Assignments from the Extended category expand on the Standard exercises to include more applications, as well
as some conceptual or theory-based questions. Exercises may include selected items from the Concepts and Vocabulary,
Working with Formulas, and the Extending the Concept categories of the exercise sets.
• In Depth: The In Depth assignments represent a more comprehensive look at the material from each section, while
attempting to keep the assignment manageable for students. These include a selection of the most popular and highest-quality
exercises from each category of the exercise set, with an additional emphasis on Maintaining Your Skills.
Additional answers can be found in the Instructor Answer Appendix.

Core: 7–91 every other odd, 95–101 odd (26 Exercises)
Standard: 1–4, 7–83 every other odd, 85–92 all, 95–101 odd (36 Exercises)

Ϫ13
3

Ϫ19
2

Extended: 1–4, 7–31 every other odd, 35–38 all, 39–79 every other odd,
85–92 all, 95–101 odd, 106, 109 (39 Exercises)
In Depth: 1–4, 7–31 every other odd, 35–38 all, 39–83 every other odd,
85–92 all, 95, 96, 98, 99, 100, 101, 105, 106, 109 (44 Exercises)

xix


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Strong Mathmatical Connections . . .

Through a Conversational Writing Style
featur es of a mathematics text,
While examples and applications are arguably the most promin ent
togeth er. It may be true that
it’s the readability and writing style of the author s that bind them
when looking for an example
some studen ts don’t read the text, and that others open the text only
for those studen ts that do (read
similar to the exercise they’re working on. But when they do and
g concep ts in a form and
the text), it’s important they have a text that “speak s to them,” relatin
style of this text will help draw
at a level they understand and can relate to. We feel the writing
and bringing them back a second
studen ts in and keep their interest, becoming a positiv e experience
begin to see the true value
and third time, until it becomes habitual. At this point studen ts might
g with any other form of
of their text, as it becomes a resour ce for learning on equal footin
—The Authors
direction.
supplementa l instruction. This text represents our best effort s in this

Conversational Writing Style
John and J.D.’s experience in the classroom and their strong connections
to how students comprehend the material are evident in their writing
style. They use a conversational 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. The effort they have put into the writing is representative of

John Coburn’s unofficial mantra: “If you want more students to reach
the top, you gotta put a few more rungs on the ladder.”

“Coburn strikes a good balance between

providing all of the important information
necessary for a certain topic without going
too deep.

”

—Barry Monk, Macon State College

“I think the authors have done an excellent job

of interweaving the formal explanations with the
‘plain talk’ descriptions, illustrating with
meaningful examples and applications.

”

—Ken Gamber, Hutchinson Community College

Through Student Involvement
How do you design a student-friendly textbook? We decided to get students involved by hosting
two separate focus groups. During these sessions we asked students to advise us on how they use their books, what
pedagogical elements are useful, which elements are distracting and not useful,
as well as general feedback on page layout. During this process there were
times when we thought, “Now why hasn’t anyone ever thought of that before?”
Clearly these student focus groups were invaluable. Taking direct student

feedback and incorporating what is feasible and doesn’t detract from instructor
use of the text is the best way to design a truly student-friendly text. The next
two pages will highlight what we learned from students so you can see for
yourself how their feedback played an important role in the development of the
Coburn/Herdlick series.

xx


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5.2

Students said that Learning Objectives
should clearly define the goals of each
section.

Exponential Functions

LEARNING OBJECTIVES
In Section 5.2 you will see
how we can:

A. Evaluate an exponential

Demographics is the statistical study of human populations. In this section, we introduce the family of exponential functions, which are widely used to model population
growth or decline with additional applications in science, engineering, and many other
fields. As with other functions, we begin with a study of the graph and its characteristics.

function

A. Evaluating Exponential Functions

B. Graph general
exponential functions

C. Graph base-e
exponential functions
D. Solve exponential
equations and
applications

In the boomtowns of the old west, it was not
uncommon for a town to double in size every year
(at least for a time) as the lure of gold drew more
and more people westward. When this type of
growth is modeled using mathematics, exponents
play a lead role. Suppose the town of Goldsboro
h d 1000 id t h
ld
fi t di
d

Examples are “boxed” so students can
clearly see where they begin and end.
Examples are called out in the margins
so they are easy for students to spot.

EXAMPLE 4

ᮣ

Graphing Exponential
xponential Functions Using Transformations
i transformations
f
i
Graph F1x2 ϭ 2xxϪ11 ϩ 2 using
off
the basic function f 1x2 ϭ 2x (not by simply plotting
points). Clearly state what transformations are
applied.

Solution

ᮣ

Students asked for Check Points
throughout each section to alert them
when a specific learning objective has
been covered and to reinforce the use
of correct mathematical terms.

(1, 3)
yϭ2

Ϫ4

4

x

To help sketch a more accurate graph, the point (3, 6) can be used: F132 ϭ 6.
Now try Exercises 15 through 30

ᮣ

Students told us they liked when the
examples were linked to the exercises.

Described by students as one of the
most useful features in a math text,
Caution Boxes signal a student to stop
and take note in order to avoid mistakes
in problem solving.

CAUTION

ᮣ

S d
Students
told
ld us that
h the
h color
l red
d should

h ld
only be used for things that are really
important. Also, anything significant
should be included in the body of the
text; marginal readings imply optional.

Because students spend a lot of time in
the exercise section of a text, they said
that a white background is hard on their
eyes . . . so we used a soft, off-white color
for the background.

(0, 2.5)

(3, 6)

F102 ϭ 2102Ϫ1 ϩ 2
ϭ 2Ϫ1 ϩ 2
1
ϭ ϩ2
2
ϭ 2.5
B. You’ve just seen how
we can graph general
exponential functions

Students said having a lot of icons was
confusing. The graphing calculator is the
only icon used in the exercise sets; no
unnecessary icons are used.

The graph of F is that of the basic function
f 1x2 ϭ 2x with a horizontal shift 1 unit right and
a vertical shift 2 units up. With this in mind the
horizontal asymptote also shifts from y ϭ 0 to
y ϭ 2 and (0, 1) shifts to (1, 3). The y-intercept of
F is at (0, 2.5):

y
F(x) = 2x is shifted
1 unit right
2 units up

For equations like those in Example 1, be careful not to treat the absolute value bars as
simple grouping symbols. The equation Ϫ51x Ϫ 72 ϩ 2 ϭ Ϫ13 has only the solution
x ϭ 10, and “misses” the second solution since it yields x Ϫ 7 ϭ 3 in simplified form.
The equation Ϫ5Ϳx Ϫ 7Ϳ ϩ 2 ϭ Ϫ13 simplifies to Ϳx Ϫ 7Ϳ ϭ 3 and there are actually two
solutions. Also note that Ϫ5Ϳx Ϫ 7Ϳ ͿϪ5x ϩ 35Ϳ!

Students told us that directions should be
in bold so they are easily distinguishable
from the problems.

ᮣ

APPLICATIONS

Use the information given to build a linear equation
model, then use the equation to respond. For exercises
71 to 74, develop both an algebraic and a graphical

solution.

71. Business depreciation: A business purchases a
copier for $8500 and anticipates it will depreciate
in value $1250 per year.
a. What is the copier’s value after 4 yr of use?
b. How many years will it take for this copier’s
value to decrease to $2250?
72. Baseball card value: After purchasing an
autographed baseball card for $85, its value
increases by $1.50 per year.
a. What is the card’s value 7 yr after purchase?
b. How many years will it take for this card’s
value to reach $100?

74. Gas mileage: When empty, a large dump-truck
gets about 15 mi per gallon. It is estimated that for
each 3 tons of cargo it hauls, gas mileage decreases
by 34 mi per gallon.
a. If 10 tons of cargo is being carried, what is the
truck’s mileage?
b. If the truck’s mileage is down to 10 mi per
gallon, how much weight is it carrying?
75. Parallel/nonparallel roads: Aberville is 38 mi
north and 12 mi west of Boschertown, with a
straight “farm and machinery” road (FM 1960)
connecting the two cities. In the next county,
Crownsburg is 30 mi north and 9.5 mi west of
Dower, and these cities are likewise connected by a
straight road (FM 830). If the two roads continued

indefinitely in both directions, would they intersect
at some point?

xxi


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Coburn’s Precalculus Series
College Algebra: Graphs & Models, First Edition
A Review of Basic Concepts and Skills ◆ Functions and Graphs ◆ Relations; More on
Functions ◆ Quadratic Functions and Operations on Functions ◆ Polynomial and
Rational Functions ◆ Exponential and Logarithmic Functions ◆ Systems of Equations
and Inequalities ◆ Matrices and Matrix Applications ◆ Analytic Geometry and the
Conic Sections ◆ Additional Topics in Algebra

Precalculus: Graphs & Models, First Edition
Functions and Graphs ◆ Relations; More on Functions ◆ Quadratic Functions and
Operations on Functions ◆ Polynomial and Rational Functions ◆ Exponential and
Logarithmic Functions ◆ Introduction to Trigonometry ◆ trigonometric Identities,
Inverses, and Equations ◆ Applications of Trigonometry ◆ Systems of Equations and
Inequalities; Matrices ◆ Analytic Geometry; Polar and parametric Equations ◆ Sequences,
Series, Counting, and Probability ◆ Bridges to Calculus—An Introduction to Limits

College Algebra
Second Edition
Review ◆ Equations and Inequalities ◆ Relations,
Functions, and Graphs ◆ Polynomial and Rational

Functions ◆ Exponential and Logarithmic Functions
◆ Systems of Equations and Inequalities ◆ Matrices
◆ Geometry and Conic Sections ◆ Additional
Topics in Algebra
MHID 0-07-351941-3, ISBN 978-0-07-351941-8

College Algebra Essentials
Second Edition
Review ◆ Equations and Inequalities ◆ Relations,
Functions, and Graphs ◆ Polynomial and Rational
Functions ◆ Exponential and Logarithmic Functions
◆ Systems of Equations and Inequalities
MHID 0-07-351968-5, ISBN 978-0-07-351968-5

Algebra and Trigonometry
Second Edition
Review ◆ Equations and Inequalities ◆ Relations,
Functions, and Graphs ◆ Polynomial and Rational
Functions ◆ Exponential and Logarithmic Functions
◆ Trigonometric Functions ◆ Trigonometric
Identities, Inverses, and Equations ◆ Applications
of Trigonometry ◆ Systems of Equations and Inequalities ◆ Matrices
◆ Geometry and Conic Sections ◆ Additional Topics in Algebra
MHID 0-07-351952-9, ISBN 978-0-07-351952-4

xxii

Precalculus
Second Edition
Equations and Inequalities ◆ Relations,

Functions, and Graphs ◆ Polynomial and
Rational Functions ◆ Exponential and
Logarithmic Functions ◆ Trigonometric
Functions ◆ Trigonometric Identities,
Inverses, and Equations ◆ Applications of Trigonometry
◆ Systems of Equations and Inequalities ◆ Matrices ◆ Geometry
and Conic Sections ◆ Additional Topics in Algebra ◆ Limits
MHID 0-07-351942-1, ISBN 978-0-07-351942-5

Trigonometry
Second Edition
Introduction to Trigonometry ◆ Right
Triangles and Static Trigonometry
◆ Radian Measure and Dynamic
Trigonometry ◆ Trigonometric Graphs
and Models ◆ Trigonometric Identities
◆ Inverse Functions and Trigonometric Equations
◆ Applications of Trigonometry ◆ Trigonometric
Connections to Algebra
MHID 0-07-351948-0, ISBN 978-0-07-351948-7