Algebra 2

Algebra 2

Common Core

Randall I. Charles
Basia Hall

Dan Kennedy
Allan E. Bellman
Sadie Chavis Bragg
William G. Handlin
Stuart J. Murphy

Grant Wiggins

Boston, Massachusetts • Chandler, Arizona • Glenview, Illinois • Hoboken, New Jersey

Acknowledgments appear on page 1160, which constitutes an extension of this copyright page.

Copyright © 2015 Pearson Education, Inc., or its affiliates. All Rights Reserved. Printed in the United States of

America. This publication is protected by copyright, and permission should be obtained from the publisher prior to
any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic,
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Pearson, Prentice Hall, Pearson Prentice Hall, and MathXL are trademarks, in the U.S. and/or other countries, of
Pearson Education, Inc., or its affiliates.

Common Core State Standards: © 2010 National Governors Association Center for Best Practices and Council of
Chief State School Officers. All rights reserved.

UNDERSTANDING BY DESIGN® and UbD™ are trademarks of the Association for Supervision and Curriculum
Development (ASCD), and are used under license.

SAT® is a trademark of the College Entrance Examination Board. ACT® is a trademark owned by ACT, Inc. Use of
the trademarks implies no relationship, sponsorship, endorsement, sale, or promotion on the part of Pearson Education,
Inc., or its affiliates.

ISBN-13: 978-0-13-328116-3
ISBN-10: 0-13-328116-7
4 5 6 7 8 9 10  V057  18 17 16 15 14

Contents in Brief

Welcome to Pearson Algebra 2 Common Core Edition student book. Throughout this textbook,
you will find content that has been developed to cover many of the High School Standards for
Mathematical Content and all of the Standards for Mathematical Practice. The End-of-Course
Assessment provides students with practice with all of the Standards for Mathematical Content
listed on pages xx to xxiii.

Using Your Book for Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
Entry-Level Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl

CC Content Focus: Seeing Structure in Expressions
Chapter 1 Expressions, Equations, and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 Functions, Equations, and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 3 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

CC Content Focus: Interpreting and Building Functions
Chapter 4 Quadratic Functions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Chapter 5 Polynomials and Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Chapter 6 Radical Functions and Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Chapter 7 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Chapter 8 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
Chapter 9 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

CC Content Focus: Expressing Geometric Properties with Equations
Chapter 10 Quadratic Relations and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

CC Content Focus: Rules of Probability, Interpreting Data, and Making Inferences
Chapter 11 Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
Chapter 12 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761

CC Content Focus: Trigonometric Functions
Chapter 13 Periodic Functions and Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825
Chapter 14 Trigonometric Identities and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901
End-of-Course Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964

Skills Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985
Visual Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994
Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160

Contents iii

Series Authors

Randall I. Charles, Ph.D.,  is Professor Emeritus in the Department of Mathematics and Computer
Science at San Jose State University, San Jose, California. He began his career as a high school
mathematics teacher, and he was a mathematics supervisor for five years. Dr. Charles has been
a member of several NCTM committees and is the former Vice President of the National Council
of Supervisors of Mathematics. Much of his writing and research has been in the area of problem
solving. He has authored more than 90 mathematics textbooks for kindergarten through college.

Dan Kennedy, Ph.D.,  is a classroom teacher and the Lupton Distinguished Professor of Mathematics at
the Baylor School in Chattanooga, Tennessee. A frequent speaker at professional meetings on the
subject of mathematics education reform, Dr. Kennedy has conducted more than 50 workshops and
institutes for high school teachers. He is coauthor of textbooks in calculus and precalculus, and from
1990 to 1994, he chaired the College Board’s AP Calculus Development Committee. He is a 1992
Tandy Technology Scholar and a 1995 Presidential Award winner.

Basia Hall  currently serves as Manager of Instructional Programs for the Houston Independent
School District. With 33 years of teaching experience, Ms. Hall has served as a department chair,
instructional supervisor, school improvement facilitator, and professional development trainer. She
has developed curricula for Algebra 1, Geometry, and Algebra 2 and co-developed the Texas
state mathematics standards. A 1992 Presidential Awardee, Ms. Hall is past president of the Texas
Association of Supervisors of Mathematics and is a state representative for the National Council of
Supervisors of Mathematics (NCSM).

Consulting Authors

Stuart J. Murphy  is a visual learning author and consultant. He is a champion of helping students
develop learning skills so they become more successful students. He is the author of MathStart, a
series of children’s books that presents mathematical concepts in the context of stories and I See
I Learn, a Pre-Kindergarten and Kindergarten learning initiative that focuses on social and emotional
skills. A graduate of the Rhode Island School of Design, he has worked extensively in educational
publishing and has been on the authorship teams of a number of elementary and high school
mathematics programs. He is a frequent presenter at meetings of the National Council of Teachers
of Mathematics, the International Reading Association, and other professional organizations.

Grant Wiggins, Ed.D.,  is the President of Authentic Education in Hopewell, New Jersey. He
earned his B.A. from St. John’s College in Annapolis and his Ed.D. from Harvard University.
Dr. Wiggins consults with schools, districts, and state education departments on a variety of reform
matters; organizes conferences and workshops; and develops print materials and Web resources
on curricular change. He is perhaps best known for being the coauthor, with Jay McTighe, of
Understanding by Design and The Understanding by Design Handbook1, the award-winning and
highly successful materials on curriculum published by ASCD. His work has been supported by the
Pew Charitable Trusts, the Geraldine R. Dodge Foundation, and the National Science Foundation.

1 ASCD, publisher of “The Understanding by Design Handbook” coauthored by Grant Wiggins and registered owner
of the trademark “Understanding by Design,” has not authorized or sponsored this work and is in no way affiliated with
Pearson or its products.

iv

Program Authors v

Algebra 1 and Algebra 2

Allan E. Bellman, Ph.D.,  is an Associate Professor of Mathematics Education at the University of
Mississippi. He previously taught at the University of California, Davis for 12 years and in public
school in Montgomery County, Maryland for 31. He has been an instructor for both the Woodrow
Wilson National Fellowship Foundation and the Texas Instruments’ T3 program. Dr. Bellman has
a expertise in the use of technology in education and assessment-driven instruction, and speaks
frequently on these topics. He was a 1992 Tandy Technology Scholar and has twice been listed in
Who’s Who Among America’s Teachers.

Sadie Chavis Bragg, Ed.D.,  is Senior Vice President of Academic Affairs and professor of
mathematics at the Borough of Manhattan Community College of the City University of New York.
She is a past president of the American Mathematical Association of Two-Year Colleges (AMATYC).
In recognition of her service to the field of mathematics locally, statewide, nationally, and
internationally, she was awarded AMATYC’s most prestigious award, the Mathematics Excellence
Award for 2010. Dr. Bragg has coauthored more than 50 mathematics textbooks for kindergarten
through college.

William G. Handlin, Sr.,  is a classroom teacher and Department Chair of Mathematics and former
Department Chair of Technology Applications at Spring Woods High School in Houston, Texas.
Awarded Life Membership in the Texas Congress of Parents and Teachers for his contributions to the
well-being of children, Mr. Handlin is also a frequent workshop and seminar leader in professional
meetings.

Geometry

Laurie E. Bass  is a classroom teacher at the 9–12 division of the Ethical Culture Fieldston School in
Riverdale, New York. A classroom teacher for more than 30 years, Ms. Bass has a wide base of
teaching experience, ranging from Grade 6 through Advanced Placement Calculus. She was the
recipient of a 2000 Honorable Mention for the Radio Shack National Teacher Awards. She has
been a contributing writer for a number of publications, including software-based activities for the
Algebra 1 classroom. Among her areas of special interest are cooperative learning for high school
students and geometry exploration on the computer. Ms. Bass is a frequent presenter at local,
regional, and national conferences.

Art Johnson, Ed.D.,  is a professor of mathematics education at Boston University. He is a
mathematics educator with 32 years of public school teaching experience, a frequent speaker and
workshop leader, and the recipient of a number of awards: the Tandy Prize for Teaching Excellence,
the Presidential Award for Excellence in Mathematics Teaching, and New Hampshire Teacher of the
Year. He was also profiled by the Disney Corporation in the American Teacher of the Year Program.
Dr. Johnson has contributed 18 articles to NCTM journals and has authored over 50 books on
various aspects of mathematics.

 

Reviewers National

Tammy Baumann Sharon Liston Robert Thomas, Ph.D.
K-12 Mathematics Coordinator Mathematics Department Chair Mathematics Teacher
School District of the City Moore Public Schools Yuma Union High School
Oklahoma City, Oklahoma
of Erie District #70
Erie, Pennsylvania Ann Marie Palmeri Monahan Yuma, Arizona
Mathematics Supervisor
Sandy Cowgill Bayonne Public Schools Linda Ussery
Mathematics Department Chair Bayonne, New Jersey Mathematics Consultant
Muncie Central High School Alabama Department of
Muncie, Indiana Indika Morris
Mathematics Department Chair Education
Sheryl Ezze Queen Creek School District Tuscumbia, Alabama
Mathematics Chairperson Queen Creek, Arizona
DeWitt High School Denise Vizzini
Lansing, Michigan Jennifer Petersen Mathematics Teacher
K-12 Mathematics Curriculum Clarksburg High School
Dennis Griebel Montgomery County,
Mathematics Coordinator Facilitator
Cherry Creek School District Springfield Public Schools Maryland
Aurora, Colorado Springfield, Missouri
Marcia White
Bill Harrington Tammy Popp Mathematics Specialist
Secondary Mathematics Mathematics Teacher Academic Operations,
Mehlville School District
Coordinator St. Louis, Missouri Technology and Innovations
State College School District Memphis City Schools
State College, Pennsylvania Mickey Porter Memphis, Tennessee
Mathematics Teacher
Michael Herzog Dayton Public Schools Merrie Wolf
Mathematics Teacher Dayton, Ohio Mathematics Department Chair
Tucson Small School Project Tulsa Public Schools
Tucson, Arizona Steven Sachs Tulsa, Oklahoma
Mathematics Department Chair
Camilla Horton Lawrence North High School
Secondary Instruction Support Indianapolis, Indiana
Memphis School District
Memphis, Tennessee John Staley
Secondary Mathematics
Gary Kubina
Mathematics Consultant Coordinator
Mobile County School System Office of Mathematics, PK-12
Mobile, Alabama Baltimore, Maryland

vi

From the Authors

Welcome

Math is a powerful tool with far-reaching applications throughout your life. We have designed
a unique and engaging program that will enable you to tap into the power of mathematics
and mathematical reasoning. This award-winning program has been developed to align fully
to the Common Core State Standards.

Developing mathematical skills and problem-solving strategies is an ongoing process—a
journey both inside and outside the classroom. This course is designed to help make sense
of the mathematics you encounter in and out of class each day and to help you develop
mathematical proficiency.

You will learn important mathematical principles. You will also learn how the principles are
connected to one another and to what you already know. You will learn to solve problems and
learn the reasoning that lies behind your solutions. You will also develop the key mathematical
practices of the Common Core State Standards.

Each chapter begins with the “big ideas” of the chapter and some essential questions that you
will learn to answer. Through this question-and-answer process you will develop your ability to
analyze problems independently and solve them in different applications.

Your skills and confidence will increase through practice and review. Work through the problems
so you understand the concepts and methods presented and the thinking behind them. Then do
the exercises. Ask yourself how new concepts relate to old ones. Make the connections!

Everyone needs help sometimes. You will find that this program has built-in opportunities, both
in this text and online, to get help whenever you need it.

This course will also help you succeed on the tests you take in class and on other tests like
the SAT, ACT, and state exams. The practice exercises in each lesson will prepare you for the
format and content of such tests. No surprises!

The problem-solving and reasoning habits and skills you develop in this program will serve you
in all your studies and in your daily life. They will prepare you for future success not only as a
student, but also as a member of a changing technological society.

Best wishes,

  vii

PowerAlgebra.com

Welcome to Algebra 2. Pearson Algebra 2 Common Core
Edition is part of a blended digital and print environment for
the study of high school mathematics. Take some time to look
through the features of our mathematics program, starting with

PowerAlgebra.com, the site of the

digital features of the program.

Hi, I’m Darius. My
friends and I will
be showing you the
great features of
Pearson Algebra 2
Common Core Edition
program.

On each Chapter Opener, you will

find a listing of the online features of the
program. Look for these buttons throughout
the lessons.

viii 

The Common Core State Standards have

a similar organizing structure. They

Big Ideas begin with Conceptual Categories,

We start with Big Ideas. Each chapter is such as Algebra or Functions.

organized around Big Ideas that convey the Within each category are
key mathematics concepts you will be studying
in the program. Take a look at the Big Ideas on CommondoCmoarines and clusters. ds
pages xxiv and xxv.
State Standar

The Big Ideas are organizing ideas for all In the Chapter Review at the end of the

of the lessons in the program. At the beginning of chapter, you’ll find an answer to the Essential
Question for each Big Idea. We’ll also remind
each chapter, we’ll tell you which Big Ideas you’ll you of the lesson(s) where you studied the
concepts that support the Big Ideas.
be studying. We’ll also present an Essential
Question for each Big Idea.

Using Your Book for Success ix

Exploring Concepts

The lessons offer many opportunities to explore concepts
in different contexts and through different media.

Hi, I’m Serena. I never
have to power down when
I am in math class now.

For each chapter, there is a Common Here’s another cool feature. Each lesson opens
Core Performance Task that you
with a Solve It, a problem that helps
will work on throughout the chapter. See
you connect what you know to an important
pages xii and xiii for more information. concept in the lesson. Do you notice how
the Solve It frame looks like it comes from a
computer? That’s because all of the Solve Its

can be found at PowerAlgebra.com.

x 

DevelThe Standards for Mathematical Practice ciency
describe processes, practices, and habits

of mind of mathematically proficient
students. Many of the features in
Pearson Algebra 2 Common Core

opingEdpMitirooanficthiheenlepcmyy oinau mtdieacvtahel.lopProfi

Want to do some more exploring? Try the Try a Concept Byte! In a

Math Tools at PowerAlgebra.com. Concept Byte, you might explore
technology, do a hands-on activity,
Click on this icon to access these tools: or try a challenging extension.
Graphing Utility, Number Line, Algebra Tiles, The text in the top right corner
and 2D and 3D Geometric Constructor. With of the first page of a lesson
the Math Tools, you can continue to explore or Concept Byte tells you
the concepts presented in the lesson.
the Standards for

Mathematical Content
and the Standards for
Mathematical Practice that

you will be studying.

Using Your Book for Success xi

Solving Problems

Pearson Algebra 2 Common Core Edition includes many opportunities
to build on and strengthen your problem-solving abilities. In each
chapter, you’ll work through a multi-part Performance Task.

Hi, I’m Maya. These Common
Core Performance Tasks will
help you become a proficient
problem solver.

On the Chapter Opener, you’ll be
introduced to the chapter Performance
Task. You’ll start to make sense of the

problem and think about solution plans.

xii 

Proficient Problem Solvers make sense

of problem situations, develop workable
Developin m
solution plans, model the problem Solving

situation with mathematics,

g and communicate their

Proficthieinnkicnyg clearly. Proble

with

Throughout the chapter, you will Apply In the Pull It All Together at
What You’ve Learned to solve
the end of the chapter, you will use the
problems that relate to the Performance Task. concepts and skills presented throughout
You’ll be asked to reason quantitatively and the chapter to solve the Performance
model with mathematics. Task. Then you’ll have another Task to

solve On Your Own.

Using Your Book for Success xiii

Thinking Mathematically

Mathematical reasoning is the key to making sense of math and
solving problems. Throughout the program you’ll learn strategies to
develop mathematical reasoning habits.

Hello, I’m Tyler.
These Think-Write
and Know-Need-Plan
boxes help me plan
my work.

The worked-out problems include Other worked-out problems model a
call-outs that reveal the strategies and
reasoning behind the solution. The problem-solving plan that includes the

Think-Write problems model the steps of stating what you Know,
identifying what you Need, and
thinking behind each step of a solution. developing a Plan.

Also, look for the boxes labeled Plan
and Think.

xiv 

The Standards for Mathematical Practice

emphasize sense-making, reasoning, and

critical reasoning. Many features in

Stand Pearson Algebra 2 Common Core
Edition provide opportunities for
you to develop these skills ctice
ards
and dispositions. Pra

for Mathematical

A Take Note box highlights key concepts Part of thinking mathematically is

in a lesson. You can use these boxes to review making sense of the concepts that are

concepts throughout the year. being presented. The Essential
Understandings help you build

a framework for the Big Ideas.

Using Your Book for Success xv

M Practice Makes Perfect
OL
L Ask any professional and you’ll be told that the one requirement
FO for becoming an expert is practice, practice, practice. Pearson
Algebra 2 Common Core Edition offers rich and varied exercises
to help you become proficient with the mathematics.

Hello, I’m Anya. I can leave
my book at school and still get
my homework done. All of the
lessons are at PowerAlgebra.com

Want more practice? Look for this icon athX in
R SCHO

your book. Check out all of the opportunities in

MathXL® for School. Your teacher can

assign you some practice exercises or you can
choose some on your own. And you’ll know right
away if you got the right answer!

xvi 

Acing the Test AssessinAll of these opportunities for practiceandards
help you prepare for assessments
Doing well on tests, whether they are chapter throughout the year, including the
tests or state assessments, depends on a deep assessments to measure your
understanding of math concepts, fluency with proficiency with the Common
calculations and computations, and strong Core State Standards.
problem-solving abilities.
g the Common Core State St

At the end of the chapter, you’ll find a Quick Review In the Cumulative Standards Review
at the end of the chapter, you’ll also
of the concepts in the chapter and a few examples and
exercises so you can check your skill at solving problems find Tips for Success,
related to the concepts.
reminders to help with problem
solving. We include problems of all
different formats and types so you
can feel comfortable with any test
item on your state assessment.

Using Your Book for Success xvii

Standards for Mathematical Practice

The Common Core State Standards are made of two separate,
but equally important sets of standards:

• Standards for Mathematical Content
• Standards for Mathematical Practice

The Math Content Standards are grade-specific, while the
Math Practices Standards are the same from Kindergarten
through High School. The Math Practices describe qualities
and habits of mind that strong mathematical thinkers exhibit.

The eight Standards for Mathematical Practice, numbered
1 through 8, can be put into the four groups shown on this page
and the next. Included with the statement of each standard is a
description of what the Math Practice means for you.

Making Sense of and Solving Problems

1. Make sense of problems and persevere in solving them.
When you make sense of problems, you can explain the meaning of the problem, and you are able to find
an entry point to its solution and plan a solution pathway. You can look at a problem and analyze givens,
constraints, relationships, and goals. You can think of similar problems or can break the problem into
easier-to-solve problems. You are able to track your progress as you work through the solution and check
your answer using a different method. As you work through your solution, you frequently check whether
the results you are getting make sense.

6.  Attend to precision.
You attend to precision when you communicate clearly and precisely the approach you used to solve a
problem, and you also understand the approaches that your classmates used. You identify the meaning
of symbols that you use, you specify units of measure, and you include labels on the axes of graphs.
Your answers are expressed with the appropriate degree of accuracy. You are able to give clear, concise
definitions of math terms.

xviii

Reasoning and Communicating xix

2. Reason abstractly and quantitatively.
As a strong math thinker and problem solver, you are able to make sense of quantities in problem
situations. You can both represent a problem situation using symbols or equations and explain what
the symbols or equations represent in relationship to the problem situation. As you represent a situation
symbolically or mathematically, you can explain the meaning of the quantities.

3. Construct viable arguments and critique the reasoning of others.
You are able to communicate clearly and convincingly about your solutions to problems. You can build
sound mathematical arguments, drawing on definitions, assumptions, or established solutions. You
can develop and explore conjectures about mathematical situations. You make use of examples and
counterexamples to support your arguments and justify your conclusions. You respond clearly and
logically to the positions and conclusions of your classmates, and are able to compare two arguments,
identifying any flaws in logic or reasoning that the arguments may contain. You can ask useful questions
to clarify or improve the argument of a classmate.

Representing and Connecting

4. Model with mathematics.
As a strong math thinker, you are able to use mathematics to represent a problem situation and can
make connections between a real-world problem situation and mathematics. You see the applicability
of mathematics to everyday problems. You can explain how geometry can be used to solve a carpentry
problem or algebra to solve a proportional relationship problem. You can define and map relationships
among quantities in a problem, using appropriate tools to do so. You are able to analyze the
relationships and draw conclusions.

5. Use appropriate tools strategically.
As you develop models to match a given problem situation, you are able to strategize about which
tools would be most helpful to use to solve the problem. You consider all tools, from paper and
pencil to protractors and rulers, to calculators and software applications. You can articulate the
appropriateness of different tools and recognize which would best serve your needs for a given
problem. You are especially insightful about technology tools and use them in ways that deepen or
extend your understanding of concepts. You also make use of mental tools, such as estimation, to
determine the reasonableness of a solution.

Seeing Structure and Generalizing

7. Look for and make use of structure.
You are able to go beyond simply solving problems, to see the structure of the mathematics in
these problems, and to generalize mathematical principles from this structure. You are able to
see complicated expressions or equations as single objects, or a being composed of many parts.

8. Look for and express regularity in repeated reasoning.
You notice when calculations are repeated and can uncover both general methods and shortcuts for
solving similar problems. You continually evaluate the reasonableness of your solutions as you solve
problems arising in daily life.

Using Your Book for Success

Standards for Mathematical Content

Algebra 2

Number and Quantity

Hi, I’m Max. Here The Real Number System
is a list of many of
the Common Core Extend the properties of exponents to rational exponents
State Standards N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of
that you will study integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
this year. Mastering N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
these topics will
help you be ready Quantities
for your state
assessment. Reason quantitatively and use units to solve problems
N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

The Complex Number System

Perform arithmetic operations with complex numbers

N.CN.A.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with
a and b real.

N-CN.A.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and
multiply complex numbers.

Use complex numbers in polynomial identities and equations.
N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

Algebra

Seeing Structure in Expressions

Interpret the structure of expressions
A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems
A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression.
A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions.
A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the
formula to solve problems.

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomial

A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p (x) and a number a, the remainder on division by
x - a is p (a), so p (a) = 0 if and only if (x - a) is a factor of p (x).

A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough
graph of the function defined by the polynomial.

Use polynomial identities to solve problems

A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

Rewrite rational expressions

A-APR.D.6 Rewrite simple rational expressions in different forms; write a (x)>b (x) in the form q (x) + r (x)>b (x), where
a (x), b (x), q (x), and r (x) are polynomials with the degree of r (x) less than the degree of b (x), using
inspection, long division, or, for the more complicated examples, a computer algebra system.

xx

Creating Equations

Create equations that describe numbers or relationships
A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising

from linear and quadratic functions, and simple rational and exponential functions.
Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.

A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous
solutions may arise.

Solve equations and Inequalities in one variable

A-REI.B.4 Solve quadratic equations in one variable.
A-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square,

the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a { bi for real numbers a and b.
Solve systems of equations

A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear
equations in two variables.

A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically
and graphically.

Represent and solve equations and inequalities graphically

A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f (x) and y = g (x)
intersect are the solutions of the equation f (x) = g (x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find successive approximations. Include cases
where f (x) and/or g (x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Functions

Interpreting Functions

Understand the concept of a function and use function notation

F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the
integers.

Interpret functions that arise in applications in terms of the context

F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in
terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a
specified interval. Estimate the rate of change from a graph.

Analyze functions using different representations

F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.

F-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end
behavior.

F-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.

F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different
properties of the function.

Using Your Book for Success xxi

Look at the domain F-IF.C.8b Write a function defined by an expression in different but equivalent forms to reveal and explain
titles and cluster F-IF.C.9 different properties of the function. Use the properties of exponents to interpret expressions for
descriptions in bold to exponential functions.
get a good idea of the Compare properties of two functions each represented in a different way (algebraically, graphically,
topics you’ll study this numerically in tables, or by verbal descriptions)
year.
Building Functions

Build a function that models a relationship between two quantities

F-BF.A.1 Write a function that describes a relationship between two quantities.

F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-BF.A.1b Combine standard function types using arithmetic operations.

F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them
to model situations, and translate between the two forms.

Build new functions from existing functions

F-BF.B.3 Identify the effect on the graph of replacing f (x) by f (x) + k, k f (x), f (kx), and f (x + k) for
F-BF.B.4 specific values of k (both positive and negative); find the value of k given the graphs. Experiment
with cases and illustrate an explanation of the effects on the graph using technology. Include
recognizing even a nd odd functions from their graphs and algebraic expressions for them.
Find inverse functions.

F-BF.B.4a Solve an equation of the form f (x) = c for a simple function f that has an inverse and write an
expression for the inverse.

Linear and Exponential Models

Construct and compare linear and exponential models and solve problems

F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a
graph, a description of a relationship, or two input-output pairs (include reading these from a table).

F-LE.A.4 For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are
numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret expressions for functions in terms of the situation they model

F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.

Trigonometric Functions

Extend the domain of trigonometric functions using the unit circle

F-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the
angle.

F-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions
to all real numbers, interpreted as radian measures of angles traversed counterclockwise around
the unit circle.

Model periodic phenomena with trigonometric functions

F-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency,
and midline.

Prove and apply trigonometric identities

F-TF.C.8 Prove the Pythagorean identity sin2 (u) + cos2 (u) = 1 and use it to find sinu, cosu, or tanu
given sinu, cosu, or tanu and the quadrant of the angle.

Geometry

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section
G.GPE.2 Derive the equation of a parabola given a focus and directrix.

xxii

Statistics and Probability

Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on a single count or measurement variable

S-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate
population percentages. Recognize that there are data sets for which such a procedure is not appropriate.
Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

S-ID.A.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are
related.

S-ID.A.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use
given functions or choose a function suggested by the context. Emphasize linear and exponential models.

Making Inferences and Justifying Conclusions

Understand and evaluate random processes underlying statistical experiments

S-IC.A.1 Understand statistics as a process for making inferences to be made about population parameters based
on a random sample from that population.

S-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using
simulation.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

S-IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies;
explain how randomization relates to each.

S-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error
through the use of simulation models for random sampling.

S-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if
differences between parameters are significant.

S-IC.B.6 Evaluate reports based on data.

Conditional Probability and the Rules of Probability

Understand independence and conditional probability and use them to interpret data

S-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of
the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S-CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is
the product of their probabilities, and use this characterization to determine if they are independent.

S-CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A
and B as saying that the conditional probability of A given B is the same as the probability of A, and the
conditional probability of B given A is the same as the probability of B.

S-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each
object being classified. Use the two-way table as a sample space to decide if events are independent and
to approximate conditional probabilities.

S-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language
and everyday situations.

Use the rules of probability to compute probabilities of compound events in a uniform probability model

S-CP.B.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and
interpret the answer in terms of the model.

S-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the
model.

Using Your Book for Success xxiii

Stay connected! BIGideas
These Big Ideas will
help you understand These Big Ideas are the organizing ideas for the study of important areas of mathematics: algebra,
how the math you geometry, and statistics.
study in high school
fits together. Algebra

Properties
• In the transition from arithmetic to algebra, attention shifts from arithmetic operations

(addition, subtraction, multiplication, and division) to use of the properties of these
operations.
• All of the facts of arithmetic and algebra follow from certain properties.
Variable
• Quantities are used to form expressions, equations, and inequalities.
• An expression refers to a quantity but does not make a statement about it. An equation (or
an inequality) is a statement about the quantities it mentions.
• Using variables in place of numbers in equations (or inequalities) allows the statement of
relationships among numbers that are unknown or unspecified.
Equivalence
• A single quantity may be represented by many different expressions.
• The facts about a quantity may be expressed by many different equations (or inequalities).
Solving Equations & Inequalities
• Solving an equation is the process of rewriting the equation to make what it says about its
variable(s) as simple as possible.
• Properties of numbers and equality can be used to transform an equation (or inequality) into
equivalent, simpler equations (or inequalities) in order to find solutions.
• Useful information about equations and inequalities (including solutions) can be found by
analyzing graphs or tables.
• The numbers and types of solutions vary predictably, based on the type of equation.
Proportionality
• Two quantities are proportional if they have the same ratio in each instance where they are
measured together.
• Two quantities are inversely proportional if they have the same product in each instance
where they are measured together.
Function
• A function is a relationship between variables in which each value of the input variable is
associated with a unique value of the output variable.
• Functions can be represented in a variety of ways, such as graphs, tables, equations, or
words. Each representation is particularly useful in certain situations.
• Some important families of functions are developed through transformations of the simplest
form of the function.
• New functions can be made from other functions by applying arithmetic operations or by
applying one function to the output of another.
Modeling
• Many real-world mathematical problems can be represented algebraically. These
representations can lead to algebraic solutions.
• A function that models a real-world situation can be used to make estimates or predictions
about future occurrences.

xxiv

Statistics and Probability xxv

Data Collection and Analysis
• Sampling techniques are used to gather data from real-world situations. If the data are

representative of the larger population, inferences can be made about that population.
• Biased sampling techniques yield data unlikely to be representative of the larger population.
• Sets of numerical data are described using measures of central tendency and dispersion.

Data Representation
• The most appropriate data representations depend on the type of data—quantitative or

qualitative, and univariate or bivariate.
• Line plots, box plots, and histograms are different ways to show distribution of data over a

possible range of values.
Probability
• Probability expresses the likelihood that a particular event will occur.
• Data can be used to calculate an experimental probability, and mathematical properties can be
used to determine a theoretical probability.
• Either experimental or theoretical probability can be used to make predictions or decisions about
future events.
• Various counting methods can be used to develop theoretical probabilities.

Geometry

Visualization
• Visualization can help you see the relationships between two figures and connect properties of

real objects with two-dimensional drawings of these objects.
Transformations
• Transformations are mathematical functions that model relationships with figures.
• Transformations may be described geometrically or by coordinates.
• Symmetries of figures may be defined and classified by transformations.
Measurement
• Some attributes of geometric figures, such as length, area, volume, and angle measure, are
measurable. Units are used to describe these attributes.
Reasoning & Proof
• Definitions establish meanings and remove possible misunderstanding.
• Other truths are more complex and difficult to see. It is often possible to verify complex truths
by reasoning from simpler ones using deductive reasoning.
Similarity
• Two geometric figures are similar when corresponding lengths are proportional and
corresponding angles are congruent.
• Areas of similar figures are proportional to the squares of their corresponding lengths.
• Volumes of similar figures are proportional to the cubes of their corresponding lengths.
Coordinate Geometry
• A coordinate system on a line is a number line on which points are labeled, corresponding to the
real numbers.
• A coordinate system in a plane is formed by two perpendicular number lines, called the x - and
y-axes, and the quadrants they form. The coordinate plane can be used to graph many functions.
• It is possible to verify some complex truths using deductive reasoning in combination with the
distance, midpoint, and slope formulas.

Using Your Book for Success

1 Expressions, Equations,
and Inequalities

Get Ready! 1

Common Core Performance Task 3
4
1-1 Patterns and Expressions 11
1-2 Properties of Real Numbers 18
1-3 Algebraic Expressions
25
Mid-Chapter Quiz 26
33
1-4 Solving Equations 41
1-5 Solving Inequalities
1-6 Absolute Value Equations and Inequalities 49
50
Assessment and Test Prep 53
54
Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Algebra Chapters 1 & 2 Functions
Seeing Structure in Expressions Interpreting Functions
Interpret the structure of expressions Interpret functions that arise in applications in terms of the context
Creating Equations Analyze functions using different representations
Create equations that describe numbers or relationships
Building Functions
xxvi Contents Build a function that models a relationship between two quantities

2 Functions, Equations,
and Graphs

Get Ready! 57

Common Core Performance Task 59

2-1 Relations and Functions 60
2-2 Direct Variation 68
2-3 Linear Functions and Slope-Intercept Form 74
2-4 More About Linear Equations 81

Mid-Chapter Quiz 89

Concept Byte:  Piecewise Functions 90
2-5 Using Linear Models 92
2-6 Families of Functions 99
2-7 Absolute Value Functions and Graphs 107
2-8 Two-Variable Inequalities 114

Assessment and Test Prep 121
122
Pull It All Together 127
Chapter Review 128
Chapter Test
Cumulative Standards Review

Visual See It! Reasoning Try It! Practice Do It!

YNAM NLINE athX
TIVITI MEWO R SCHO

Essential Understanding 60
Think-Write 108
Know S Need S Plan 20
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Virtual Nerd™ 2 Practice by Example 22
Solve It! 11
Connecting BIG IDEAS 50 Think About a Plan 105

Error Analysis/Reasoning 78

Contents xxvii

3 Linear Systems

Get Ready! 131

Common Core Performance Task 133
134
3-1 Solving Systems Using Tables and Graphs 142
3-2 Solving Systems Algebraically 149
3-3 Systems of Inequalities
156
Mid-Chapter Quiz 157

3-4 Linear Programming 163
Concept Byte TECHNOLOGY: 
164
Linear Programming 166
Concept Byte ACTIVITY:  174

Graphs in Three Dimensions 182
3-5 Systems With Three Variables 183
3-6 Solving Systems Using Matrices 187
188
Assessment and Test Prep

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Number and QuantityChapters 3 & 4 Algebra
The Complex Number System Seeing Structure in Expressions
Perform arithmetic operations with complex numbers Interpret the structure of expressions
Use complex numbers in polynomial identities and equations
Functions Arithmetic with Polynomials and Rational Expressions
Interpreting Functions Understand the relationship between zeros and factors of polynomials
Interpret functions that arise in applications in terms of the context
Analyze functions using different representations Creating Equations
Building Functions Create equations that describe numbers or relationships
Build new functions from existing functions
Reasoning with Equations and Inequalities
xxviii Contents Solve systems of equations
Represent and solve equations and inequalities graphically

4 Quadratic Functions

and Equations 191

Get Ready! 193
194
Common Core Performance Task 202
209
4-1 Quadratic Functions and Transformations 215
4-2 Standard Form of a Quadratic Function 216
4-3 Modeling With Quadratic Functions
Concept Byte:  Identifying Quadratic Data 224
4-4 Factoring Quadratic Expressions 225
226
Mid-Chapter Quiz 232
233
Algebra Review:  Square Roots and Radicals 240
4-5 Quadratic Equations 248
Concept Byte:  Writing Equations From Roots 256
4-6 Completing the Square 258
4-7 The Quadratic Formula
4-8 Complex Numbers 265
Concept Byte:  Quadratic Inequalities
4-9 Quadratic Systems 266
Concept Byte EXTENSION:  267
273
Powers of Complex Numbers 274

Assessment and Test Prep

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Visual See It! Reasoning Try It! Practice Do It!

YNAM NLINE athX
TIVITI MEWO R SCHO

Essential Understanding 226
Think-Write 144
Know S Need S Plan 205
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Virtual Nerd™ 132 Practice by Example 221
Solve It! 166
Connecting BIG IDEAS 267 Think About a Plan 199

Error Analysis/Reasoning 152

Contents xxix

5 Polynomials and
Polynomial Functions

Get Ready! 277

Common Core Performance Task 279
280
5-1 Polynomial Functions 288
5-2 Polynomials, Linear Factors, and Zeros 296
5-3 Solving Polynomial Equations 303
5-4 Dividing Polynomials
311
Mid-Chapter Quiz 312

5-5 Theorems About Roots of Polynomial Equations 318
Concept Byte EXTENSION:  319

Using Polynomial Identities 325
5-6 The Fundamental Theorem of Algebra 326
Concept Byte ACTIVITY:  331
339
Graphing Polynomials Using Zeros
5-7 The Binomial Theorem 346
5-8 Polynomial Models in the Real World 347
5-9 Transforming Polynomial Functions 353
354
Assessment and Test Prep

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Number and QuantityChapters 5 & 6 Algebra
The Complex Number System Seeing Structure in Expressions
Use complex numbers in polynomial identities and equations Interpret the structure of expressions
Functions
Interpreting Functions Creating Equations
Interpret functions that arise in applications in terms of the context Create equations that describe numbers or relationships
Analyze functions using different representations
Building Functions Arithmetic with Polynomials and Rational Expressions
Build a function that models a relationship between two quantities Understand the relationship between zeros and factors of polynomials
Build new functions from existing functions Use polynomial identities to solve problems

xxx Contents

6 Radical Functions
and Rational Exponents

Get Ready! 357

Common Core Performance Task 359

Concept Byte REVIEW: Properties of Exponents 360
6-1 Roots and Radical Expressions 361
6-2 Multiplying and Dividing Radical Expressions 367
6-3 Binomial Radical Expressions 374
6-4 Rational Exponents 381

Mid-Chapter Quiz 389

6-5 Solving Square Root and Other Radical Equations 390
6-6 Function Operations 398
6-7 Inverse Relations and Functions 405
Concept Byte TECHNOLOGY:  Graphing Inverses 413
6-8 Graphing Radical Functions 414

Assessment and Test Prep 421
422
Pull It All Together 427
Chapter Review 428
Chapter Test
Cumulative Standards Review

Visual See It! Reasoning Try It! Practice Do It!

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TIVITI MEWO R SCHO

Essential Understanding 288
Think-Write 300
Know S Need S Plan 391
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Virtual Nerd™ 358 Practice by Example 322
Solve It! 296
Connecting BIG IDEAS 422 Think About a Plan 402

Error Analysis/Reasoning 379

Contents xxxi

7 Exponential and
Logarithmic Functions

Get Ready! 431

Common Core Performance Task 433

7-1 Exploring Exponential Models 434

7-2 Properties of Exponential Functions 442

7-3 Logarithmic Functions as Inverses 451

Concept Byte TECHNOLOGY:  Fitting Curves to Data 459

Mid-Chapter Quiz 461

7-4 Properties of Logarithms 462

7-5 Exponential and Logarithmic Equations 469

Concept Byte TECHNOLOGY:  Using Logarithms for

Exponential Models 477

7-6 Natural Logarithms 478

Concept Byte EXTENSION: 

Exponential and Logarithmic Inequalities 484

Assessment and Test Prep 486
487
Pull It All Together 491
Chapter Review 492
Chapter Test
Cumulative Standards Review

AlgebraChapters 7 & 8 Functions
Seeing Structure in Expressions Interpreting Functions
Interpret the structure of expressions Analyze functions using different representations
Creating Equations Building Functions
Create equations that describe numbers or relationships Build a function that models a relationship between two quantities
Arithmetic with Polynomials and Rational Expressions Build new functions from existing functions
Rewrite rational expressions
Reasoning with Equations and Inequalities Linear and Exponential Models
Represent and solve equations and inequalities graphically Construct and compare linear and exponential models and solve problems

xxxii Contents

8 Rational Functions

Get Ready! 495

Common Core Performance Task 497
498
8-1 Inverse Variation
Concept Byte TECHNOLOGY:  506
507
Graphing Rational Functions 515
8-2 The Reciprocal Function Family 524
8-3 Rational Functions and Their Graphs
Concept Byte TECHNOLOGY:  Oblique Asymptotes 526
527
Mid-Chapter Quiz 534
542
8-4 Rational Expressions 549
8-5 Adding and Subtracting Rational Expressions 550
8-6 Solving Rational Equations
Concept Byte:  Systems With Rational Equations 552
Concept Byte TECHNOLOGY:  Rational Inequalities 553
557
Assessment and Test Prep 558

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Visual See It! Reasoning Try It! Practice Do It!

YNAM NLINE athX
TIVITI MEWO R SCHO

Essential Understanding 451
Think-Write 529
Know S Need S Plan 445
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Virtual Nerd™ 496 Practice by Example 473
Solve It! 434
Connecting BIG IDEAS 553 Think About a Plan 546

Error Analysis/Reasoning 522

Contents xxxiii

9 Sequences and Series

Get Ready! 561

Common Core Performance Task 563
564
9-1 Mathematical Patterns 572
9-2 Arithmetic Sequences
Concept Byte EXTENSION:  578
579
The Fibonacci Sequence 580
587
Mid-Chapter Quiz
594
9-3 Geometric Sequences 595
9-4 Arithmetic Series
Concept Byte  602
603
Geometry and Infinite Series 607
9-5 Geometric Series 608

Assessment and Test Prep

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Chapters 9 & 10 Algebra Geometry
Seeing Structure in Expressions Expressing Geometric Properties with Equations

Write expressions in equivalent forms to solve problems Translate between the geometric description and the equation
for a conic section
Functions
Interpreting Functions

Understand the concept of a function and use function notation

Analyze functions using different representations

xxxiv Contents

10 Quadratic Relations

and Conic Sections 611

Get Ready! 613
614
Common Core Performance Task
621
10-1 Exploring Conic Sections 622
Concept Byte TECHNOLOGY:  630

Graphing Conic Sections 637
10-2 Parabolas 638
10-3 Circles 645
653
Mid-Chapter Quiz 661

10-4 Ellipses 662
10-5 Hyperbolas 663
10-6 Translating Conic Sections 667
Concept Byte:  Solving Quadratic Systems 668

Assessment and Test Prep

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Visual See It! Reasoning Try It! Practice Do It!

YNAM NLINE athX
TIVITI MEWO R SCHO

Essential Understanding 614
Think-Write 566
Know S Need S Plan 631
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Virtual Nerd™ 612 Practice by Example 599
Solve It! 564
Connecting BIG IDEAS 663 Think About a Plan 585

Error Analysis/Reasoning 634

Contents xxxv

11 Probability and Statistics

Get Ready! 671

Common Core Performance Task 673
674
11-1 Permutations and Combinations 681
11-2 Probability 688
11-3 Probability of Multiple Events 694
Concept Byte ACTIVITY:  Probability Distributions 696
11-4 Conditional Probability 703
11-5 Probability Models
710
Mid-Chapter Quiz 711
719
11-6 Analyzing Data 725
11-7 Standard Deviation 731
11-8 Samples and Surveys 739
11-9 Binomial Distributions
11-10 Normal Distributions 746
Concept Byte ACTIVITY: 
748
Margin of Error
Concept Byte ACTIVITY:  750
751
Drawing Conclusions from Samples 757
758
Assessment and Test Prep

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Number and QuantityChapters 11 & 12 Making Inferences and Justifying Conclusions
Vector and Matrix Quantities Understand and evaluate random processes underlying statistical experiments
Represent and model with vector quantities Make inferences and justify conclusions from sample surveys, experiments,
Perform operations on vectors
Perform operations on matrices and use matrices in applications and observational studies
Statistics and Probability
Interpreting Categorical and Quantitative Data Conditional Probability and the Rules of Probability
Summarize, represent, and interpret data on a single count or Understand independence and conditional probability and use them to

measurement variable interpret data
Use the rules of probability to compute probabilities of compound events in
xxxvi Contents
a uniform probability model
Use Probability to Make Decisions
Use probability to evaluate outcomes of decisions

12 Matrices

Get Ready! 761

Common Core Performance Task 763

12-1 Adding and Subtracting Matrices 764

Concept Byte TECHNOLOGY: 

Working With Matrices 771

12-2 Matrix Multiplication 772

Concept Byte:  Networks 780

12-3 Determinants and Inverses 782

Mid-Chapter Quiz 791

12-4 Inverse Matrices and Systems 792
12-5 Geometric Transformations 801
12-6 Vectors 809

Assessment and Test Prep 816
817
Pull It All Together 821
Chapter Review 822
Chapter Test
Cumulative Standards Review

Visual See It! Reasoning Try It! Practice Do It!

YNAM NLINE athX
TIVITI MEWO R SCHO

Essential Understanding 688
Think-Write 786
Know S Need S Plan 732
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Virtual Nerd™ 672 Practice by Example 788
Solve It! 696
Connecting BIG IDEAS 817 Think About a Plan 692

Error Analysis/Reasoning 723

Contents xxxvii

13 Periodic Functions and
Trigonometry

Get Ready! 825

Common Core Performance Task 827
828
13-1 Exploring Periodic Data 835
Geometry Review:  Special Right Triangles 836
13-2 Angles and the Unit Circle 843
Concept Byte Activity: Measuring Radians 844
13-3 Radian Measure 851
13-4 The Sine Function
859
Mid-Chapter Quiz
860
Concept Byte TECHNOLOGY:  861
Graphing Trigonometric Functions 868
875
13-5 The Cosine Function 883
13-6 The Tangent Function
13-7 Translating Sine and Cosine Functions 891
13-8 Reciprocal Trigonometric Functions 892
897
Assessment and Test Prep 898

Pull It All Together
Chapter Review
Chapter Test
Cumulative Standards Review

Chapters 13 & 14 Functions Geometry
Interpreting Functions Similarity, Right Triangles, and Trigonometry
Interpret functions that arise in applications in terms of the context
Analyze functions using different representations Define trigonometric ratios and solve problems involving right triangles

Trigonometric Functions Apply trigonometry to general triangles
Extend the domain of trigonometric functions using the unit circle
Model periodic phenomena with trigonometric functions
Prove and apply trigonometric identities

xxxviii Contents

14 Trigonometric Identities
and Equations

Get Ready! 901

Common Core Performance Task 903
904
14-1 Trigonometric Identities 911
14-2 Solving Trigonometric Equations Using Inverses 919
14-3 Right Triangles and Trigonometric Ratios
927
Mid-Chapter Quiz 928
935
14-4 Area and the Law of Sines 936
Concept Byte:  The Ambiguous Case 943
14-5 The Law of Cosines 951
14-6 Angle Identities
14-7 Double-Angle and Half-Angle Identities 958
959
Assessment and Test Prep 963
964
Pull It All Together
Chapter Review
Chapter Test
End-of-Course Assessment

Visual See It! Reasoning Try It! Practice Do It!

YNAM NLINE athX
TIVITI MEWO R SCHO

Essential Understanding 911
Think-Write 884
Know S Need S Plan 839
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Virtual Nerd™ 902 Practice by Example 832
Solve It! 851
Connecting BIG IDEAS 892 Think About a Plan 925

Error Analysis/Reasoning 880

Contents xxxix

Entry-Level
Assessment

Multiple Choice 6. Solve 2(1 - 2w) = 4w + 18. 8
-4 16
Read each question. Then write the letter of the correct
-2
answer on your paper.

1. Let A = 51, 2, 3, 46 be a set in the universe 7. Which of the following lines is perpendicular to the
U = 51, 2, 3, 4, 5, 6, 7, 86. What is the complement
line 3x + y = 2?
of A?
y = 3x + 4
{2, 3} {1, 2, 3, 4}
y = 1 x - 2
{5, 6, 7, 8} {2, 3, 7, 8} 3

2. Solve x2 + 2x - 3 = 0 by factoring. y = -3x + 3
x = -3 and x = 1
# y = - 1 x + 1
3

x = -1 and x = 3 8. If y = 1, then (x + 5) y = x + 5. Which property

x=0 supports this statement?

Inverse Property of Multiplication

x = -3 and x = 0 Identity Property of Multiplication

3. Simplify 3a2b3 - 12a4b 3 + 6a4b 2 Associative Property of Addition
3a2b
.

b2 - 4a2b 2 + 2a2b Commutative Property of Addition

a2b - 4a2b 2 + 2a2b 9. y

3b2 - 12a2b + 6b2 4

3ab2 - 4a2b + 2ab2 4 O x
4 4
4. Which relation is not a function?
{(1, -5), (2, 4), (1, -4)} Which inequality does the graph represent?
{(1, -5), (2, 4), (3, -3)}
{(1, -5), (2, 4), (3, 2)} y 6 2x - 4
{(1, -5), (2, 4), (3, -4)}
y 7 -4x + 2
5. In the diagram, m and n are parallel.
y 7 2x - 4

132 m y 6 -4x + 2

1 0. The area of a trapezoid is A = 1 h(b1 + b2). Solve
2
n for b1.
(x  12) b1 2A - b2
= h

What is the value of x? 120 b1 = 2A - h
36 144 b1 = 2hAb-2 b2

60 b1 = 2A - b2

xl Entry-Level Assessment

1 1. Let <AB> be parallel to <CD>, with A( -2, 3), B(1, 4), 17. A rectangular photograph is being enlarged to poster
and C(1, 2). Which of the following could be the
size by making both the length and width six times
coordinates of point D?
as large as the original. How many times as large

(4, 1) ( -2, 3) as the area of the original photograph is the area of

( -2, -1) (4, 3) the poster? 12
61
12. Solve 3 Ú 4g - 5 Ú -1.
-32 … g … 2 -4 … g … 8 6 36
- 1 … g … 43 1 … g … 2
18. A rectangle has a length of 2x + 3 and a width of
13. Which is not a solution of 5(2x + 4) Ú 2(x + 34)? x - 4. Find the area of the rectangle.

48 6 2x 2 - 12

8 3 2x 2 - 8x

1 4. Factor 6x 2 - 216. 2x 2 - 5x - 12
6(x - 6)(x + 6)
(6x - 36)(6x + 36) 2x 2 - 11x - 12
6(x - 6)
6(x - 36)(x + 6) 19. What is the y-intercept of the line that passes through

the points ( -4, 4) and (2, -5)? 3
2
-2

- 3 2
2

15. Mike and Jane leave their home on bikes traveling 2 0. Which of the following is equivalent to 12 (16 - 4)?

in opposite directions on a straight road. Mike rides 112 - 4 112 - 8

5 mi/h faster than Jane. After 4 h they are 124 mi apart. 213 - 212 213 - 412

At what rate does Mike ride his bike? 2 1. Which of the following represents the system shown in
the graph?
5 mi/h 18 mi/h

13 mi/h 31 mi/h y

1 6. What is the point-slope form for the equation of the 4
line in the graph?
2
y
x
2 x O4
2
2 O
2 y = x - 3 y 6 x - 3
ex Ú 5 ex = 5

y - 2 = 23(x + 2) y … x - 3 y 7 x - 3
ex 7 5 ex … 5
y - 2 = 12(x + 2)
1 2 2. Which of the following equations represents the line
y - 2 = - 2 (x + 2)
that is parallel to the line y = 5x + 2 and that passes
- = - 2 + through the point (1, -3)?
y 2 3 (x 2) 1
5
y = -5x + 2 y = x -8

y = 5x + 8 y = 5x - 8

Entry-Level Assessment xli

23. Which equation represents a line that would be 2 8. What is the solution to 2n + 8 = n + 7 ?
1 3 2
perpendicular to a second line with a slope of 5 ? -9 5

y = -5x + 2 -1 13

y = - 1 x + 3 2 9. Solve the system of equations below.
5

y = 5x - 2 e 34xx + y = -7
- y = - 14
5y + x = 2

2 4. △ABC is similar to △DEF . ( -3, 2) ( -3, -2)

A (3, 2) no solution

kD 30. Which of the following is equivalent to x2 2x - 12 24 ?
x - 2x -
2 1
B 6 ft C + 4 x2 - 2

6 ft 1 2 2x - 3
+ x- 6
x

E 9 ft F 31. What is (are) the solution(s) of the graphed function

What is the value of k? when the value of the function is 0?
3 ft
y
4 ft
6 ft 4 2 O x
9 ft 2 4

25. Solve the equation using the Quadratic Formula.
6x 2 - 10x + 3 = 0
5 { 17
6
-1 and 2
3 { 15 1 and -2 2
6 2.2

-2 and 5 3 2. Which of the following is true?
185 6 9
3 and 2 8 6 162 16 7 16
4 3 5 25 54

26. A rectangle in the coordinate plane has vertices (3, 2), 1121 6 1144
(8, 2), (3, 6), and (8, 6). Which of the following sets
of vertices describes a rectangle that is congruent to 3 3. A firefighter leans a 30-ft ladder against a building
this one?
in order to reach a window that is 24 ft high. How far

(3, -2),(3, -8), (5, -8), (5, -2) away from the building is the base of the ladder?

( -2, -4), ( -2, -8), (3, -8), (3, -4) 18 ft 24 ft

(0, 0), (5, 0), (5, 5), (0, 5) 20 ft 30 ft

( -3, 2), (1, 2), (1, 6), ( -3, 6) 3 4. What is the number of x-intercepts of the parabola
with equation y = 6x2 - 4x - 3?
27. Simplify the expression below.
( - 6y-4)5 0 2

7776y 20 - 7776y 20 1 3

7776 - 7776
y 20 y 20

xlii Entry-Level Assessment

Get Ready! CHAPTER

Skills Adding Rational Numbers 1
Handbook,
Find each sum.
page 973

1. 6 + (-6) 2. -8 + 6 3. 5.31 + (-7.40) 4. -1.95 + 10
7. 652 + 4130 8. - 156 + 513
( ) 5. 734 + -812 6. -231 + 341

Skills Subtracting Rational Numbers
Handbook,
Find each difference.
page 973

9. -28 - 14 10. 61 - (-11) 11. -16 - ( -25) 12. -6.2 - 3.6
15. 223 - 731
( ) 13. -523 - -231 14. -214 - 341 16. 5 - 13
2 4

Skills Multiplying and Dividing Rational Numbers
Hpaagnedb9o7o3k,
# #Find each product or quotient. 2 3 , 5
17. -3 7 18. -2.1 ( -3.5) 19. - 3 , 4 20. - 8 8

Skills Using the Order of Operations
Hpaagnedb9o7o5k,
Simplify each expression.
# # 21. 8 (-3) + 4
22. 3 4 - 8 , 2 23. 1 , 22 - 0.54 + 1.26
# 24. 9 , (-3) - 2 25. 5(3 5 - 4)
26. 1 - (1 - 5)2 , ( -8)
# # 27. Reasoning  Why don’t the expressions 3 + 52 3 , 15 and (3 + 52) 3 , 15
yield the same answer?

Looking Ahead Vocabulary

28. The current of a river flows north at a constant rate. What is the constant in the
mathematical expression 3x + 5y + 3?

29. Before signing a contract, you must review the terms and conditions of the contract.
How many terms are there in the surface area formula below?
2(/w + wh + h/)

30. Engineers evaluate the efficiency of the memory and speed of a computer. What
does evaluate mean in mathematics?

31. Smiling is a facial expression of happiness or contentment. In math, what is the
expression that represents the quotient of 3 and 3 less than a number?

Chapter 1  Expressions, Equations, and Inequalities 1

CHAPTER Expressions, Equations,

1 and Inequalities

Download videos VIDEO Chapter Preview 1 Variable
connecting math Essential Question  How do variables
to your world.. 1-1 Patterns and Expressions help you model real-world situations?
1-2 Properties of Real Numbers
Interactive! ICYNAM 1-3 Algebraic Expressions 2 Properties
Vary numbers, ACT I V I TI 1-4 Solving Equations Essential Question  How can you use
graphs, and figures D 1-5 Solving Inequalities the properties of real numbers to simplify
to explore math ES 1-6 Absolute Value Equations and algebraic expressions?
concepts..
Inequalities 3 Solving Equations and Inequalities
Essential Question  How do you solve an
The online equation or inequality?
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Seeing Structure in Expressions
Spanish English/Spanish Vocabulary Audio Online: • Creating Equations

English Spanish

absolute value, p. 41 valor absoluto

Online access algebraic expression, p. 5 expresión algebraica
to stepped-out
problems aligned compound inequality, p. 36 desigualdad compuesta
to Common Core
Get and view like terms, p. 21 términos semejantes
your assignments
online. NLINE literal equation, p. 29 ecuación literal
ME WO
O term, p. 20 término
RK
HO variable, p. 5 variable

Extra practice
and review
online

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Where’s My Car?

Cody leaves his friend Mia’s house and drives along the road shown in the
diagram below. Somewhere between Mia’s house and the restaurant, Cody’s car
runs out of gas.

Mia’s Gas Restaurant
House Station

9 mi 11 mi

Cody has an empty gas can in his car, but he does not want to leave the car
unattended. Cody calls Mia, who drives to Cody’s car to pick up his gas can. She
then drives to the gas station that is located on the same road. After Mia fills the
gas can, she drives back to Cody’s car. She gives Cody the gas can, and then drives
to the restaurant along the same road to meet another friend for lunch. When she
reaches the restaurant, Mia has driven a total of 34 mi.

Task Description

Determine how far Cody is from Mia’s house when his car runs out of gas. Find all
possible distances.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll draw diagrams to help you make sense of the problem. (MP 1)

• You’ll assign a variable to an unknown distance and use your variable to write
expressions that represent other distances. (MP 2)

• You’ll model the problem situation with an equation. (MP 4)

Chapter 1  Expressions, Equations, and Inequalities 3

1-1 Patterns and CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Expressions
AM-ASFSSE.9B1.32 .ACh-SoSoEse.2a.n3d  Cphrodouseceaannd epqroudivuaclenatnform of
Objective To identify and describe patterns eaqnueivxaplreensstifoonrmtoorfevaenaelxapnrdesesxiopnlatino prervoepaelrtaiensdoefxtphlaein
pqruoapnetrityierseopfretsheenqteudanbtyittyhreeperxepsreenssteiodnb. y the expression.

MP 1, MP 2, MP 3, MP 7

You are playing a video game. You reach a locked gate. The lock is a
square with 9 sections. You can make a key by placing a red or yellow
block in each section. Near the gate is a carving of a pattern of squares.

Look for a pattern
to find a shortcut.

MATHEMATICAL

PRACTICES

1st 2nd 3rd

Lesson The key to the gate is the eighth image in the pattern. Draw the key to
the gate. How do you know it will work?
Vocabulary
• constant In the Solve It, you identified and used a geometric pattern. In this lesson, you will identify
• variable quantity patterns in pictures, tables, and graphs and describe them using numbers and variables.
• variable
• numerical Essential Understanding  ​You can represent some patterns using diagrams,
words, numbers, or algebraic expressions.
expression
• algebraic

expression

Problem 1 Identifying a Pattern

How can you identify Look at the figures from left to right. What is the pattern?
a pattern? What would the next figure in the pattern look like?
Look for the same type
of change between The pattern shows regular polygons with the number of
consecutive figures. sides increasing by one.

The last figure shown above has six sides, so the next figure would have seven sides.
This is a heptagon: .

Got It? 1. Look at the figures from left to right.
What is the pattern? Draw the next
figure in the pattern.

4 Chapter 1  Expressions, Equations, and Inequalities

A mathematical quantity is anything that can be measured or counted. The value of
the quantity is its measure or the number of items that are counted. Quantities whose
values do not change are called constants. In other situations, the value of a quantity
can change. Quantities whose values change or vary are called variable quantities.

Key Concepts  Variables and Expressions

Definition Examples

A variable is a symbol, usually a letter, that n x
represents one or more numbers.

A numerical expression is a mathematical 3+5 (8 - 2) + 5

phrase that contains numbers and operation symbols.

An algebraic expression is a mathematical phrase 3n + 5 (8x - 2) + 5n
that contains one or more variables.

Tables are a convenient way to organize data and discover patterns. They work much
like an “input/output” machine: a machine that takes one value as an input, processes
it, and gives a value as an output. A process column in the table provides a way to
understand what happens to the input values.

Problem 2 Expressing a Pattern with Algebra

Use a pattern to answer each question.
A How many toothpicks are in the 20th figure?
use a table. look for a pattern that relates the

figure number to the number of toothpicks.

Figure Number Process Number of To get the output,
(Input) Column Toothpicks (Output) multiply the input by 4.
1
2 1(4) 4
3 2(4) 8
3(4) 12

…
…
…

n■ ■

Pattern: Multiply the figure number by 4 to get the number of toothpicks. So, there
are 20(4) = 80 toothpicks in the 20th figure.

W hat would the B What is an expression that describes the number of toothpicks in the nth figure?
pt hroecnetshs rloowok? like for Use the pattern from part (a). There are 4n toothpicks in the nth figure.

multiply the figure Got It? 2. How many tiles are in the 25th figure in this pattern?
n umber, n, by 4.

Show a table of values with a process column.

Lesson 1-1  Patterns and Expressions 5

Problem 3 Using a Graph Capacity (gal) Tank Sizes

Aquarium  You want to set up an aquarium and need to 10
determine what size tank to buy. The graph shows 8
tank sizes using a rule that relates the capacity of the 6
tank to the combined lengths of the fish it can hold. 4
2
If you want five 2-in. platys, four 1-in. guppies, and a 00 2 4 6 8 10
3-in. loach, which is the smallest capacity tank you
can buy: 15-gallon, 20-gallon, or 25-gallon? Use a Combined Length of Fish (in.)
table to find the answer.

How can you use the Choose some points on the graph. (0, 2), (5, 7), (10, 12)
given graph?
You can use the graph Make a table using the input and Process
to make a table and output values shown in the ordered Input Column Output
find a pattern relating pairs.
capacity and combined Find a pattern in the process
fish length. column. Each output is 2 more than
the corresponding input.
0 0؉2 2
You want 5 platys, 4 guppies, and
1 loach. So, you will have a total of 5 5؉2 7
17 in. of fish. Find the output when
the input is 17. 10 10 ؉ 2 12

output = input + 2
= 17 + 2
= 19

Write the answer in words. You need to buy the 20-gal tank.

Got It? 3. The graph at the right shows the total cost of platys at the Total Cost ($) Cost of Platys
aquarium shop. Use a table to answer the questions.
6
a. How much do six platys cost? 4
b. How much do ten platys cost? 2
c. Reasoning  Why is the graph in Problem 3 a line 00 2 4 6

while the graph at the right is a set of points? Number of Platys

6 Chapter 1  Expressions, Equations, and Inequalities

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Describe a rule for each pattern.
1. 35, 70, 105, 140, c 5. Explain the strategy you use to identify a pattern.
2.
6. Compare and Contrast  ​How are tables of values like
Make a table to represent each pattern. Use a process pictorial representations? How are they different?
column.
3. 2, 4, 6, 8,  c 7. Error Analysis  Y​ our friend looks for a pattern in the
4. table below and claims that the output equals the
input divided by 2. Is your friend correct? Explain.

Input 3 6.8 8 10 25
Output 2 3.4 4 5 12.5

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Describe each pattern using words. Draw the next likely figure in each pattern. See Problem 1.

8. 9.

10. 11.

Make a table with a process column to represent each pattern. write an See Problem 2.
expression for the number of tiles or circles in the nth figure.

12. 13.

14. 15.

Lesson 1-1  Patterns and Expressions 7

Copy and complete each table. Include a process column.

16. Input Output 17. Input Output 18. Input Output
Ϫ3
10 13 1 Ϫ6
21 24 2 Ϫ9
32 35 3 Ϫ12
43 46 4 ■
5■ 5■ 5
6■ 6■ 6 ■

…
…

…
…

…
…

n■ n■ n■

Identify a pattern by making a table. Include a process column. See Problem 3.

19. 6 20. 6 21. 6

Output 44 Output Output 4

22 2

00 1 2 3 4 5 6 00 1 2 3 4 5 6 00 1 2 3 4 5 6
Input Input Input

The graph shows the number of bottles of water needed for students Bottles of Water Class Field Trip
going on a field trip. 6

22. How many bottles of water are needed if 5 students attend? 4
2
23. How many bottles of water are needed if 20 students attend? 00 1 2 3 4
Number of Students
24. How many bottles of water are needed if n students attend?

B Apply Identify a pattern and find the next three numbers
in the pattern.

25. 6, 12, 18, 24, c 26. 1, 4, 3, 6, 5, c

27. 3, 6, 10, 15, c 28. 2, 6, 10, 14, c

29. 1, 3, 9, 27, 81, c 30. 4, 20, 100, 500, c

31. Identify a pattern and draw the next three figures in the pattern.

32. Open-Ended  Write a rule so that for every input, the output is an
even number.

8 Chapter 1  Expressions, Equations, and Inequalities