Precalculus
An Investigation of Functions
Edition 1.3
David Lippman
Melonie Rasmussen
This book is also available to read free online at
http://www.opentextbookstore.com/precalc/
If you want a printed copy, buying from the bookstore is cheaper than printing yourself.
ii
Copyright © 2012 David Lippman and Melonie Rasmussen
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Attribution. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
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• Notice — For any reuse or distribution, you must make clear to others the license terms of
this work. The best way to do this is with a link to this web page:
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In addition to these rights, we give explicit permission to remix small portions of this
book (less than 10% cumulative) into works that are CC-BY, CC-BY-SA-NC, or GFDL
licensed.
Selected exercises were remixed from Precalculus by D.H. Collingwood and K.D.
Prince, originally licensed under the GNU Free Document License, with permission from
the authors.
Cover Photo by David Lippman, of artwork by
John Rogers
Lituus, 2010
Dichromatic glass and aluminum
Washington State Arts Commission in partnership with Pierce College
This is the fourth official version of Edition 1. It contains typo corrections and language clarification, but is page number and problem set number equivalent to the original
Edition 1.
i
About the Authors
David Lippman received his master’s degree in mathematics from
Western Washington University and has been teaching at Pierce
College since Fall 2000.
Melonie Rasmussen also received her
master’s degree in mathematics from
Western Washington University and has
been teaching at Pierce College since Fall
2002. Prior to this Melonie taught for the
Puyallup School district for 6 years after
receiving her teaching credentials from Pacific Lutheran
University.
We have both been long time advocates of open learning, open materials, and basically
any idea that will reduce the cost of education for students. It started by supporting the
college’s calculator rental program, and running a book loan scholarship program.
Eventually the frustration with the escalating costs of commercial text books and the
online homework systems that charged for access led them to take action.
First, David developed IMathAS, open source online math homework software that runs
WAMAP.org and MyOpenMath.com. Through this platform, we became integral parts
of a vibrant sharing and learning community of teachers from around Washington State
that support and contribute to WAMAP. Our pioneering efforts, supported by dozens of
other dedicated faculty and financial support from the Transition Math Project, have led
to a system used by thousands of students every quarter, saving hundreds of thousands of
dollars over comparable commercial offerings.
David continued further and wrote his first open textbook, Math in Society, a math for
liberal arts majors book, after being frustrated by students having to pay $100+ for a
textbook for a terminal course. Together, frustrated by both cost and the style of
commercial texts, we began writing PreCalculus: An Investigation of Functions in 2010.
ii
Acknowledgements
We would like to thank the following for their generous support and feedback.
• The community of WAMAP users and developers for creating a majority of the
homework content used in our online homework sets.
• Pierce College students in our Fall 2010 - Summer 2011 Math 141 and Math 142
classes for helping correct typos, identifying videos related to the homework, and
being our willing test subjects.
• The Open Course Library Project for providing the support needed to produce a
full course package for these courses.
• Mike Kenyon, Chris Willett, Tophe Anderson, and Vauhn Foster-Grahler for
reviewing the course and giving feedback and suggestions.
• Our Pierce College colleagues for providing their suggestions.
• Tophe Anderson, James Gray, and Lawrence Morales for their feedback and
suggestions in content and examples.
• Jeff Eldridge for extensive proofreading and suggestions for clarification.
• James Sousa for developing videos associated with the online homework.
• Kevin Dimond for his work on indexing the book and creating PowerPoint slides.
• Faculty at Green River Community College and the Maricopa College District for
their feedback and suggestions.
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Preface
Over the years, when reviewing books we found that many had been mainstreamed by
the publishers in an effort to appeal to everyone, leaving them with very little character.
There were only a handful of books that had the conceptual and application driven focus
we liked, and most of those were lacking in other aspects we cared about, like providing
students sufficient examples and practice of basic skills. The largest frustration, however,
was the never ending escalation of cost and being forced into new editions every three
years. We began researching open textbooks, however the ability for those books to be
adapted, remixed, or printed were often limited by the types of licenses, or didn’t
approach the material the way we wanted.
This book is available online for free, in both Word and PDF format. You are free to
change the wording, add materials and sections or take them away. We welcome
feedback, comments and suggestions for future development at
precalc@opentextbookstore.com. Additionally, if you add a section, chapter or problems,
we would love to hear from you and possibly add your materials so everyone can benefit.
In writing this book, our focus was on the story of functions. We begin with function
notation, a basic toolkit of functions, and the basic operation with functions: composition
and transformation. Building from these basic functions, as each new family of functions
is introduced we explore the important features of the function: its graph, domain and
range, intercepts, and asymptotes. The exploration then moves to evaluating and solving
equations involving the function, finding inverses, and culminates with modeling using
the function.
The "rule of four" is integrated throughout - looking at the functions verbally,
graphically, numerically, as well as algebraically. We feel that using the “rule of four”
gives students the tools they need to approach new problems from various angles. Often
the “story problems of life” do not always come packaged in a neat equation. Being able
to think critically, see the parts and build a table or graph a trend, helps us change the
words into meaningful and measurable functions that model the world around us.
There is nothing we hate more than a chapter on exponential equations that begins
"Exponential functions are functions that have the form f(x)=ax." As each family of
functions is introduced, we motivate the topic by looking at how the function arises from
life scenarios or from modeling. Also, we feel it is important that precalculus be the
bridge in level of thinking between algebra and calculus. In algebra, it is common to see
numerous examples with very similar homework exercises, encouraging the student
to mimic the examples. Precalculus provides a link that takes students from the basic
plug & chug of formulaic calculations towards building an understanding that equations
and formulas have deeper meaning and purpose. While you will find examples and
similar exercises for the basic skills in this book, you will also find examples of multistep
problem solving along with exercises in multistep problem solving. Often times these
exercises will not exactly mimic the exercises, forcing the students to employ their
critical thinking skills and apply the skills they've learned to new situations. By
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developing students’ critical thinking and problem solving skills this course prepares
students for the rigors of Calculus.
While we followed a fairly standard ordering of material in the first half of the book, we
took some liberties in the trig portion of the book. It is our opinion that there is no need
to separate unit circle trig from triangle trig, and instead integrated them in the first
chapter. Identities are introduced in the first chapter, and revisited throughout. Likewise,
solving is introduced in the second chapter and revisited more extensively in the third
chapter. As with the first part of the book, an emphasis is placed on motivating the
concepts and on modeling and interpretation.
Supplements
During Spring 2010, the Washington Open Course Library (OCL) project was announced
with the goal of creating open courseware for the 81 highest enrolled community college
courses with a price cap on course materials of $30. We were chosen to work on
precalculus for this project, and that helped drive us to complete our book, and allowed
us to create supplemental materials.
A course package is available that contains the following features:
• Suggested syllabus
• Day by day course guide
• Instructor guide with lecture outlines and examples
• Additional online resources, with links to other textbooks, videos, and other
resources
• Discussion forums
• Diagnostic review
• Online homework for each section (algorithmically generated, free response)
• A list of videos related to the online homework
• Printable class worksheets, activities, and handouts
• Chapter review problems
• Sample quizzes
• Sample chapter exams
The course shell was built for the IMathAS online homework platform, and is available
for Washington State faculty at www.wamap.org and mirrored for others at
The course shell was designed to follow Quality Matters (QM) guidelines, but has not yet
been formally reviewed.
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How To Be Successful In This Course
This is not a high school math course, although for some of you the content may seem
familiar. There are key differences to what you will learn here, how quickly you will be
required to learn it and how much work will be required of you.
You will no longer be shown a technique and be asked to mimic it repetitively as the only
way to prove learning. Not only will you be required to master the technique, but you
will also be required to extend that knowledge to new situations and build bridges
between the material at hand and the next topic, making the course highly cumulative.
As a rule of thumb, for each hour you spend in class, you should expect this course will
require an average of 2 hours of out-of-class focused study. This means that some of you
with a stronger background in mathematics may take less, but if you have a weaker
background or any math anxiety it will take you more.
Notice how this is the equivalent of having a part time job, and if you are taking a
fulltime load of courses as many college students do, this equates to more than a full time
job. If you must work, raise a family and take a full load of courses all at the same time,
we recommend that you get a head start & get organized as soon as possible. We also
recommend that you spread out your learning into daily chunks and avoid trying to cram
or learn material quickly before an exam.
To be prepared, read through the material before it is covered in class and note or
highlight the material that is new or confusing. The instructor’s lecture and activities
should not be the first exposure to the material. As you read, test your understanding
with the Try it Now problems in the book. If you can’t figure one out, try again after
class, and ask for help if you still can’t get it.
As soon as possible after the class session recap the day’s lecture or activities into a
meaningful format to provide a third exposure to the material. You could summarize
your notes into a list of key points, or reread your notes and try to work examples done in
class without referring back to your notes. Next, begin any assigned homework. The
next day, if the instructor provides the opportunity to clarify topics or ask questions, do
not be afraid to ask. If you are afraid to ask, then you are not getting your money’s
worth! If the instructor does not provide this opportunity, be prepared to go to a tutoring
center or build a peer study group. Put in quality effort and time and you can get quality
results.
Lastly, if you feel like you do not understand a topic. Don’t wait, ASK FOR HELP!
ASK: Ask a teacher or tutor, Search for ancillaries, Keep a detailed list of questions FOR: Find additional resources, Organize the material, Research other learning options HELP: Have a support network, Examine your weaknesses, List specific examples & Practice Best of luck learning! We hope you like the course & love the price.
David & Melonie
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Table of Contents
Chapter 1: Functions ........................................................................................................ 1
Section 1.1 Functions and Function Notation ................................................................. 1
Section 1.2 Domain and Range ..................................................................................... 21
Section 1.3 Rates of Change and Behavior of Graphs .................................................. 34
Section 1.4 Composition of Functions .......................................................................... 49
Section 1.5 Transformation of Functions ..................................................................... 61
Section 1.6 Inverse Functions ....................................................................................... 90
Chapter 2: Linear Functions.......................................................................................... 99
Section 2.1 Linear Functions ........................................................................................ 99
Section 2.2 Graphs of Linear Functions ..................................................................... 111
Section 2.3 Modeling with Linear Functions .............................................................. 126
Section 2.4 Fitting Linear Models to Data .................................................................. 138
Section 2.5 Absolute Value Functions ........................................................................ 146
Chapter 3: Polynomial and Rational Functions ......................................................... 155
Section 3.1 Power Functions & Polynomial Functions .............................................. 155
Section 3.2 Quadratic Functions ................................................................................. 163
Section 3.3 Graphs of Polynomial Functions ............................................................. 176
Section 3.4 Rational Functions ................................................................................... 188
Section 3.5 Inverses and Radical Functions ............................................................... 206
Chapter 4: Exponential and Logarithmic Functions ................................................. 215
Section 4.1 Exponential Functions ............................................................................. 215
Section 4.2 Graphs of Exponential Functions ............................................................ 232
Section 4.3 Logarithmic Functions ............................................................................. 242
Section 4.4 Logarithmic Properties ............................................................................ 253
Section 4.5 Graphs of Logarithmic Functions ............................................................ 262
Section 4.6 Exponential and Logarithmic Models ...................................................... 270
Section 4.7 Fitting Exponentials to Data .................................................................... 289
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Chapter 5: Trigonometric Functions of Angles ......................................................... 297
Section 5.1 Circles ...................................................................................................... 297
Section 5.2 Angles ...................................................................................................... 307
Section 5.3 Points on Circles using Sine and Cosine ................................................. 321
Section 5.4 The Other Trigonometric Functions ........................................................ 333
Section 5.5 Right Triangle Trigonometry ................................................................... 343
Chapter 6: Periodic Functions ..................................................................................... 353
Section 6.1 Sinusoidal Graphs .................................................................................... 353
Section 6.2 Graphs of the Other Trig Functions ......................................................... 369
Section 6.3 Inverse Trig Functions ............................................................................. 379
Section 6.4 Solving Trig Equations ............................................................................ 387
Section 6.5 Modeling with Trigonometric Equations ................................................. 397
Chapter 7: Trigonometric Equations and Identities ................................................. 409
Section 7.1 Solving Trigonometric Equations with Identities .................................... 409
Section 7.2 Addition and Subtraction Identities ......................................................... 417
Section 7.3 Double Angle Identities ........................................................................... 431
Section 7.4 Modeling Changing Amplitude and Midline ........................................... 442
Chapter 8: Further Applications of Trigonometry.................................................... 451
Section 8.1 Non-right Triangles: Law of Sines and Cosines ...................................... 451
Section 8.2 Polar Coordinates ..................................................................................... 467
Section 8.3 Polar Form of Complex Numbers ............................................................ 480
Section 8.4 Vectors ..................................................................................................... 491
Section 8.5 Parametric Equations ............................................................................... 504
Solutions to Selected Exercises .................................................................................... 519
Chapter 1 ..................................................................................................................... 519
Chapter 2 ..................................................................................................................... 526
Chapter 3 ..................................................................................................................... 530
Chapter 4 ..................................................................................................................... 534
Chapter 5 ..................................................................................................................... 539
Chapter 6 ..................................................................................................................... 542
Chapter 7 ..................................................................................................................... 546
Chapter 8 ..................................................................................................................... 549
Index ............................................................................................................................... 555
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Section 1.1 Functions and Function Notation 1
Chapter 1: Functions
Section 1.1 Functions and Function Notation
What is a Function?
The na