Intermediate algebra 2e conecting concepts throught applications 1

Brief ContentsChApTErChApTErChApTErChApTErChApTErChApTErChApTErChApTErChApTErChApTErAppEndIxAppEndIxAppEndIxAppEndIxAppEndIx linear FunCTionS 1 SySTemS oF linear equaTionS anD inequaliTi

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Second Edition Intermediate Algebra

Connecting Concepts Through Applications

Mark ClarkPalomar CollegeCynthia AnfinsonPalomar College

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our lives together and to our children Will and rosemary

MC

To my husband Fred and son Sean, thank you for your love

and support

CA

extra thanks go to Jim and mary eninger for opening up their

family cabin for me to spend many a day and night writing

also to Dedad and mimi, who built the cabin in the 1940s

MC

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Intermediate Algebra: Connecting Concepts

Through Applications

Second Edition

Mark Clark, Cynthia Anfinson

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Brief Contents

ChApTErChApTErChApTErChApTErChApTErChApTErChApTErChApTErChApTErChApTErAppEndIxAppEndIxAppEndIxAppEndIxAppEndIx

linear FunCTionS 1 SySTemS oF linear equaTionS anD inequaliTieS 149

exPonenTS, PolynomialS, anD FunCTionS 253 quaDraTiC FunCTionS 331

exPonenTial FunCTionS 469 logariThmiC FunCTionS 547 raTional FunCTionS 621 raDiCal FunCTionS 691 ConiC SeCTionS 769 SequenCeS anD SerieS ( ONLINE ONLY ) 815 BaSiC algeBra revieW a-1

maTriCeS B-1 uSing The graPhing CalCulaTor C-1 anSWerS To PraCTiCe ProBlemS D-1 anSWerS To SeleCTeD exerCiSeS e-1

1 2 3 4 5 6 7 8 9 10 A B C D E

iii

index i-1 unit Conversions reF-2 geometric Formulas reF-3 equation Solving Toolbox reF-4 expression Simplifying Toolbox reF-5 modeling Toolbox reF-6

Factoring Toolbox reF-6

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1.1 Solving Linear Equations 2

Linear equations • Applications • Literal equations

1.2 Fundamentals of Graphing and Slope 19

Introduction to Graphing Equations • Linear Equations in Two Variables • The Meaning of Slope

in an Application • Graphing Lines Using Slope and Intercept

1.3 Intercepts and Graphing 41

The General Form of Lines • Intercepts and Their Meaning • Graphing Lines Using Intercepts • Horizontal and Vertical Lines

1.4 Finding Equations of Lines 58

Equations of Lines • Parallel and Perpendicular Lines • Interpreting the Characteristics of

a Line: A Review

1.5 Functions and Function Notation 74

Relations and Functions • Vertical line test • Function Notation • Domain and Range of Functions

1.6 Using Data to Create Scatterplots 95

Using Data to Create Scatterplots • Adjusting Data • Graphical Models • Domain and Range of a Model • Applications

1.7 Finding Linear Models 114

Using a Calculator to Create Scatterplots • Linear Models • Applications

Chapter 1 Summary 133Chapter 1 Review Exercises 142Chapter 1 Test 145

Chapter 1 Projects 146

Systems of Linear Equations and Inequalities 149

2.1 Systems of Linear Equations 150

Definition of Systems • Graphical and Numerical Solutions • Types of Systems • Applications

2.2 Solving Systems of Equations Using the Substitution Method 166

Substitution Method • Consistent and Inconsistent Systems • Applications

2.3 Solving Systems of Equations Using the Elimination Method 179

Elimination Method • Applications of Systems

Contents

iv

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2.4 Solving Linear Inequalities 189

Introduction to Inequalities • Solving Inequalities • Systems as Inequalities • Solving Inequalities Numerically and Graphically • Solving Compound Inequalities • Applications

2.5 Absolute Value Equations and Inequalities 205

Chapter 2 Projects 248Cumulative Review Chapters 1-2 250

3.1 Rules for Exponents 254

Rules for Exponents • Negative Exponents and Zero as an Exponent • Using Scientific Notation • Rational exponents • Applications

3.5 Special Factoring Techniques 312

Perfect Square Trinomials • Difference of Squares • Difference and Sum of Cubes • Multistep Factorizations • Trinomials in Quadratic Form

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4.2 Graphing Quadratic Functions from Vertex Form 345

Axis of Symmetry • Vertex Form • Graphing Quadratic Functions from Vertex Form • Domainand Range • Applications

4.3 Finding Quadratic Models 363

Quadratic Models • Domain and Range • Applications

4.4 Solving Quadratic Equations by the Square Root Property and

Completing the Square 378

Solving from Vertex Form • Square Root Property • Using the Pythagorean Theorem • TheDistance Formula • Completing the Square • Converting to Vertex Form • Graphing from

Vertex Form with x-Intercepts • Applications

4.5 Solving Equations by Factoring 399

The Product Property of Zero • Solving by Factoring • Finding a Quadratic Function from the Graph • Solving Nonlinear Polynomial Inequalities in One Variable • Applications

4.6 Solving Quadratic Equations by Using the Quadratic Formula 417

Solving by the Quadratic Formula • Determining Which Algebraic Method to Use When Solving

a Quadratic Equation • Solving Systems of Equations Involving Quadratic Functions

4.7 Graphing Quadratic Functions from Standard Form 430

Graphing from Standard Form • Graphing Quadratic Inequalities in Two Variables • Applications

5.1 Exponential Functions: Patterns of Growth and Decay 470

Exploring Exponential Growth and Decay • Recognizing Exponential Patterns • Applications

5.2 Solving Equations Using Exponent Rules 489

Recap of the Rules for Exponents • Solving Power Equations • Solving Exponential Equations

by Inspection • Identifying Exponential Equations and Power Equations • Applications

5.3 Graphing Exponential Functions 498

Exploring Graphs of Exponentials • Domain and Range of Exponential

Functions • Exponentials of the Form f 1x2 5 a#b x 1 c

5.4 Finding Exponential Models 510

Exponential Functions • Exponential Models • Domain and Range for Exponential Models • Applications

5.5 Exponential Growth and Decay Rates and Compounding Interest 523

Exponential Growth and Decay Rates • Compounding Interest • Growth Rates and Exponential Functions • Applications

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6.1 Functions and Their Inverses 548

Introduction to Inverse Functions • One-to-One Functions • Applications

6.5 Solving Exponential Equations 584

Solving Exponential Equations • Compounding Interest • Applications

6.6 Solving Logarithmic Equations 596

Solving Logarithmic Equations • Applications

7.1 Rational Functions and Variation 622

Rational Functions • Direct and Inverse Variation • Domain of a Rational Function • Applications • Vertical Asymptotes and Holes in Graphs

7.2 Simplifying Rational Expressions 638

Simplifying Rational Expressions • Long Division of Polynomials • Synthetic Division • Relationship between Division and Factoring

7.3 Multiplying and Dividing Rational Expressions 651

Multiplying Rational Expressions • Dividing Rational Expressions

7.4 Adding and Subtracting Rational Expressions 657

Least Common Denominator • Adding Rational Expressions • Subtracting Rational Expressions • Simplifying Complex Fractions

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8.1 Radical Functions 692

Relationships between Radicals and Rational Exponents • Radical Functions

That Model Data • Square Roots and Higher Roots • Simplifying Radicals • Applications

8.2 Graphing Radical Functions 705

Domain and Range of Radical Functions • Graphing Radical Functions • Odd and Even Indexes

8.3 Adding and Subtracting Radicals 715

Adding and Subtracting radicals

8.4 Multiplying and Dividing Radicals 718

Multiplying Radicals • Dividing Radicals and Rationalizing the Denominator • Conjugates

8.5 Solving Radical Equations 729

Solving Radical Equations • Solving Radical Equations Involving More Than One Square

Root • Extraneous Solution(s) • Solving Radical Equations Involving Higher-Order Roots • Applications

8.6 Complex Numbers 741

Definition of Imaginary and Complex Numbers • Operations with Complex Numbers • Solving

Equations with Complex Solutions

9.1 Parabolas and Circles 770

Introduction to Conic Sections • Revisiting Parabolas • A Geometric

Approach to Parabolas • Circles • Applications

9.2 Ellipses and Hyperbolas 791

Ellipses • Hyperbolas • Recognizing the Equations for Conic Sections • Applications

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Index I-1 Unit Conversions REF-2 Geometric Formulas REF-3 Equation Solving Toolbox REF-4 Expression Simplifying Toolbox REF-5 Modeling Toolbox REF-6

to an Appropriate Window • Zero, Minimum, Maximum, and Intersect Features • Regression

Answers to Practice Problems D-1 Answers to Selected Exercises E-1

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In teaching the Intermediate Algebra course, we wanted to help our students apply traditional mathematical skills in real-world contexts This text is our attempt to address the challenge of supplying applications that students relate to and lay a strong math foundation This text is the result of many years of using this approach in our class-rooms, refined with the help of many of you who shared your thoughtful feedback with

us throughout our extensive revision process

Our application-driven approach is designed to engage students while they master

algebraic concepts, critical thinking, and communication skills Our goal is simple:

to present mathematics as concepts in action rather than as a series of techniques to

memorize and to have students understand and connect with mathematics—what it means in real-world contexts—while developing a solid foundation in algebra.About The Authors

MARK CLARK

Mark Clark graduated from California State University, Long Beach, with a elor’s and Master’s in Mathematics He is a full-time Associate Professor at Palomar College and has taught there since 1996 He is committed to teaching his students through applications and using technology to help them both understand the mathe-matics in context and communicate their results clearly Intermediate algebra is one of his favorite courses to teach, and he continues to teach several sections of this course each year Mark also loves to share his passion for teaching concepts of developmental math through applications by giving workshops and talks to other instructors at local and national conferences

Bach-CINDY ANFINSON

Cindy Anfinson graduated from the University of California, San Diego, with a elor of Arts Degree in Mathematics and is a member of Phi Beta Kappa Under the Army Science and Technology Graduate Fellowship, she earned a Master of Science Degree in Applied Mathematics from Cornell University She is currently an As-sociate Professor of Mathematics at Palomar College At Palomar College, she has worked with the First-Year Experience and the Summer Bridge programs, she was the Mathematics Learning Center Director for a 3-year term, and has served on mul-tiple committees, including the Basic Skills Committee and the Student Success and Equity Council

Bach-preface

x

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Four Toolboxes are included throughout the new edition: the Equations Solving

Tool-box, the Factoring ToolTool-box, the Expression Simplifying ToolTool-box, and the Modeling

Toolbox These Toolboxes are integrated throughout the text with visual icons so that

students receive just-in-time help connecting them to the solving techniques and tools

used for different types of problems Each Toolbox emphasizes how these fundamental

tools are used throughout the course A quick reference of all four Toolboxes as well as

the Geometric Formulas and Unit Conversions appears at the back of the text The text

highlights the importance of using these from the start of the course

There is an increased emphasis on students identifying equation and function types

within solving, graphing, and modeling problems This helps students to review

previ-ous material and connect it to the current topics

Students are asked to provide reasons for each step they took in selected Solving

exercises This helps students to think critically about and explain the Solving process,

and it connects with the Toolboxes that have been integrated throughout the text

Vocabulary short-answer exercises have been added at the start of each exercise set

These help students learn the vocabulary of algebra and improve their communication

skills

The Exercise Sets have been updated with new data and current applications to help

students see connections between mathematics and the world in which they live

Within WebAssign, there is expanded problem coverage with an emphasis on conceptual

problems, full “WatchIt” coverage with closed-captioning, and expanded “MasterIt”

and “Expanded Problem” coverage with emphasis on conceptual problems

The Annotated Instructor’s Edition has been replaced with a comprehensive

Instruc-tor’s Manual Practical tips and classroom examples are provided on how to approach

and pace chapters as well as integrate features such as Concept Investigations into the

classroom For every student example in the student text, there is a different instructor

classroom example with accompanying answers that can be used for additional in-class

practice and/or homework

Activities that are hand-selected by the authors provide additional opportunities

for Instructors to get their students involved using active learning in the classroom

WebAssign suggestions integrated throughout the Instructor Manual also tie in the

digital aspects of the course so that no matter how instructors approach their class, they

feel supported

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For the Student

Online Student Solutions Manual

(ISBN: 978-1-337-61561-7)

Author: Scott Barnett

The Student Solutions Manual provides worked-out

solutions to all of the odd-numbered exercises in the text

STUDENT

www.webassign.com (Printed Access Card ISBN: 978-1-337-61566-2,

Online Access Code ISBN: 978-1-337-61565-5)

Prepare for class with confidence using WebAssign from

Cengage for Intermediate Algebra: Connecting Concepts

through Applications, 2e This online learning platform,

which includes an interactive ebook, fuels practice, so you

truly absorb what you learn—and are better prepared come

test time Videos and tutorials walk you through concepts

and deliver instant feedback and grading, so you always

know where you stand in class Focus your study time and

get extra practice where you need it most Study smarter

with WebAssign!

Ask your instructor today how you can get access

to WebAssign, or learn about self-study options at

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Online Complete Solutions Manual

(ISBN: 978-1-337-61560-0)Author: Scott BarnettThe Complete Solutions Manual provides worked-out solutions to all of the problems in the text

Instructor’s Companion Website

Everything you need for your course in one place! Access and download a comprehensive Instructor’s Manual that paces the chapters, includes WebAssign suggestions, and provides additional opportunities for in-class practice, homework, and activities In addition, you can find the online chapter, PowerPoint presentations, and more on the companion site This collection of book-specific lecture and class tools is available online via www.cengage.com/login

INSTRUCTOR

www.webassign.com/cengage (Printed Access Card ISBN: 978-1-337-61566-2, Online Access Code ISBN: 978-1-337-61565-5)

WebAssign from Cengage for Intermediate Algebra: necting Concepts through Applications, 2e is a fully custom-izable online solution, including an interactive ebook, for STEM disciplines that empowers you to help your students learn, not just do homework Insightful tools save you time and highlight exactly where your students are struggling Decide when and what type of help students can access while working on assignments—and incentivize indepen-dent work so help features aren’t abused Meanwhile, your students get an engaging experience, instant feedback and better outcomes A total win-win!

Con-To try a sample assignment, learn about LMS tion or connect with our digital course support visit www.webassign.com/cengage

integra-For the Instructor

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We would like to thank our reviewers and users for their many helpful suggestions

for improving the text In particular, we thank Karen Mifflin and Gina Hayes for their

suggestions and work on the solutions We are extremely grateful to Scott Barnett

for helping with the accuracy checking and solutions for this text We also thank the

editorial, production, and marketing staffs of Cengage, Frank Snyder, Michael Lepera,

Samantha Gomez, Alison Duncan, Pamela Polk, and Jaime Manz; for all of their help

and support during the development and production of this edition Thanks also to

Vernon Boes, Diane Beasley, Irene Morris, Leslie Lahr, and Lisa Torri for their work

on the design and art program, and to the Lumina Datamatics staff for their

copyedit-ing and proofreadcopyedit-ing expertise We especially want to thank Danielle Derbenti for

believing in us and mentoring our development as authors Our gratitude also goes to

Katy Gabel who had an amazing amount of patience with us throughout production

We truly appreciate all the hard work and efforts of the entire team

Mark ClarkCindy Anfinson

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T he U.S Census Bureau found that the average number of square feet of floor area in new one-family houses in 1980 was 1740 square feet In 2015, the average number of square feet of floor area increased

to 2687 square feet In this chapter, we will discuss how to use linear models to analyze trends in real-life data One of the chapter projects will ask you to investigate the costs associated with installing new flooring in a home.

1.1 Solving Linear Equations

1.2 Fundamentals of

Graphing and Slope

1.3 Intercepts and Graphing

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Variables are used to represent quantities that can change or are unknown In application problems, it is important to define variables so the reader knows what the variables

represent In traditional algebra problems, it is common to use the variables x and y In

application problems, variables often are given a letter that closely matches what the variable represents For example, if a problem is discussing the weekly pay of a student

tutor, then the variable P can represent the weekly pay in dollars.

Variables are also identified as input or output variables The input variable can be

thought of as the information that is being entered into the problem The output variable

is the information that results, or is produced by, the problem The input variable is

also called the independent variable The output variable is also called the dependent variable In traditional algebra problems, x is typically the input (or independent) variable The variable y is the output (or dependent) variable In application problems,

we need to think about the effect of the variable to determine whether it is the input

or output variable For example, suppose a student tutor is paid $10.00 per hour An equation that computes the tutor’s weekly pay is

P 5 10h where h 5 the number of hours worked per week and P 5 the weekly pay in dollars Here h is the input variable, as it is the information that must be entered into the problem

in order to compute the weekly pay P is the output variable, as it is the information that

is produced by the problem (that is, it is the output)

Equations can be used to represent many real-life situations One of the uses of algebra is to solve equations for an unknown quantity, or variable In this section, you will learn how to solve linear equations for a missing variable and how to write a complete solution A complete solution includes a sentence that gives the units (how

a quantity is measured, such as in dollars) and the meaning of the solution in that situation Providing these details demonstrates a clear understanding of the solution

Solving Linear Equations

LEARNING OBJECTIVES

Solve linear equations

Write complete answers to application problems

Determine whether a solution is reasonable

Solve literal equations

1.1

U-Haul charges $19.95 for the day and $0.79 per mile driven to rent a 10-foot truck The total cost to rent a 10-foot truck for the day can be represented by the equation

U 5 19.95 1 0.79m where U is the total cost in dollars to rent a 10-foot truck from U-Haul for the day and

m is the number of miles the truck is driven

a State the given variables and their definitions.

b Determine how much it will cost to rent a 10-foot truck from U-Haul and drive it

75 miles

c Determine the number of miles you can travel for a total cost of $175.00

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a m 5 the number of miles driven

U 5 the total cost in dollars to rent a 10-foot truck from U-Haul for the day

b Because the number of miles driven was given, replace the variable m in the

equation with the number 75 and solve for the missing variable U as follows:

U 5 19.95 1 0.79m

U 5 19.95 1 0.79(75)

U 5 19.95 1 59.25

U 5 79.20 This answer indicates that renting a 10-foot truck from U-Haul for the day and driving it 75 miles will cost $79.20

c Because the total cost of $175.00 is given in the statement, substitute 175.00 for the

variable U and solve for the missing variable m.

Since U-Haul would charge for a full mile for the 0.266, round down to 196 miles to stay

within the budget of $175 Check this answer by substituting m 5196 to be sure U will

equal $175.

U 5 19.95 1 0.79(196)

U 5 19.95 1 154.84

U 5 174.79 This answer indicates that for a cost of $175.00, you can rent a 10-foot truck from U-Haul for a day and drive it 196 miles.

PRACTICE PROBLEM FOR ExAMPLE 1

While you are on spring break in Fort Lauderdale, Florida, the cost, C, in dollars of

your taxi ride from the airport to your hotel can be represented by the equation

C 5 2.10 1 2.40m when the ride is m miles long.

a Define the variables.

b What will an 8-mile taxi ride cost?

c How many miles can you ride if you budget $35 for the taxi?

(Note: The answers to the Practice Problems are in Appendix D.)

Connecting the Concepts

What is the difference between the equal (5) symbol and the approximation (F) symbol?

In mathematics, we use these symbols and others to show

a relationship between two quantities or between two expressions

The equal sign (5) is used when two quantities or expressions are equal and exactly the same

The approximation symbol (<) is used to show that two quantities or expressions are approximately the same The approximation symbol will be used whenever a quantity is rounded.

Many problems in this book will investigate applications that involve money and business Defining some business terms will help in understanding the problems

and explaining the solutions Three main concepts in business are revenue, cost,

and profit.

Revenue is the total amount of money that is brought into a business through sales For example, if a pizza place sells 10 pizzas for $12 each, the revenue would

Connecting the Concepts

How are we going to round?

In general, we will round values

to at least one more decimal place than the given numbers in the problem

In some applications, rounding will be determined by what makes sense in the situation

When a specific rounding rule

is given in a problem, we will follow that rule.

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be 10 pizzas#$12 per pizza 5 $120 Revenue is often calculated as price times the quantity sold The revenue for a business cannot be a negative number.

Cost is defined as the amount of money paid out for expenses Expenses often are categorized in two ways: fixed costs and variable costs The same pizza place might have fixed costs such as rent, utilities, and salaries It would have variable costs of supplies and food ingredients depending on the number of pizzas made The cost for the business would be the fixed costs and the variable costs added together

The profit for a business is the revenue minus the cost If this pizza place had a cost

of $100 when making the 10 pizzas, they would have a profit of $120 2 $100 5 $20 Although a business cannot have a negative revenue, profit can be negative When profit is negative, it is sometimes called a loss

The break-even point of a business is the point at which the revenue from a product

is the same as the cost The break-even point also occurs when the profit is zero For

a company that is considering a new product and wants to know how many should

be produced or sold to start making a profit, the point at which profit changes from negative to positive is important to know

DEFINITIONS

Revenue The amount of money brought into a business through sales

Revenue is often calculated as

revenue 5 price#quantity sold

Cost The amount of money spent by a business to create and/or sell a product

Cost usually includes both fixed costs and variable costs Fixed costs are the same each month or year, and variable costs change depending on the number of items produced and/or sold

cost 5 fixed cost 1 variable cost or

cost 5 fixed cost 1 cost per item#quantity sold

Profit The amount of money left after all costs.

profit 5 revenue 2 cost

Break-even point A company breaks even when their revenue equals their

cost or when their profit is zero

revenue 5 cost profit 5 0

What’s That Mean?

Mathematical Verbs

Simplify: Use arithmetic and

basic algebra rules to make an

expression simpler We simplify

expressions.

Evaluate: Substitute any given

values for variables and simplify

the resulting expression or

equation.

Solve: Isolate the given variable

in an equation on one side of the

equal sign using the properties of

equality This will result in a value

that the isolated variable is equal

to We solve equations.

A small bicycle company produces high-tech bikes for international race teams The company has fixed costs of $54,500 per month for rent, salaries, and utilities For every bike they produce, it costs them $5750 in materials and other expenses related

to that bike The company can sell each bike for an average price of $13,995, but it can produce a maximum of only 20 bikes per month

a Write an equation for the monthly cost of producing b bikes.

b How much does it cost the bicycle company to produce 20 bikes in a month?

c Write an equation for the monthly revenue from selling b bikes.

d How much revenue will the bicycle company make if they sell 10 bikes in a month?

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e Write an equation for the monthly profit

the company makes if they produce and

sell b bikes (You can assume that they

will sell all the bikes they make.)

f What is the profit of producing and

selling 15 bikes in a month?

g How many bikes does the company have

to produce and sell in a month to make

$45,000 profit?

h How many bikes does the company have

to produce and sell in a month to make

$150,000 profit?

SOLuTION

a Define the variables for the problem

b 5The number of bikes produced each month (Remember that a maximum of

20 bikes can be produced each month.)

C 5 The monthly cost, in dollars, to produce b bikes

Each bike cost $5750 for materials and other expenses, so multiply b by 5750, and

then add on the fixed costs, to get the total monthly cost This gives the following equation

A monthly production of 20 bikes will result in a total monthly cost of $169,500

c Define the variables for the problem Recall that b was already defined in part a.

b 5The number of bikes produced each month

R 5 The monthly revenue, in dollars, from selling b bikes

The bicycle company can sell each bike for an average price of $13,995, so the

revenue can be calculated by using the equation

e Profit is calculated by taking the revenue and subtracting any business costs.

b 5The number of bikes produced each month

P 5 The monthly profit, in dollars, from producing and selling b bikes.

Use the equations for revenue and cost written earlier

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Simplify by distributing the sign and combining like terms

g The amount of profit desired is given Substitute 45,000 for P and solve for b

Therefore, the correct answer is that the company cannot make $150,000 profit in a month with its current production capacity.

Distribute the negative sign Combine like terms.

Add 54,500 to both sides.

Divide both sides by 8245.

Add 54500 to both sides.

Divide both sides by 8245.

The answer was rounded, so the check

is not exact, but it is close enough.

Skill Connection

using the Distributive Property

in Subtraction

When subtracting an expression

in parentheses, remember that

subtraction can be represented as

adding the opposite

a 2 b 5 a 1 (21)b

Since subtraction is defined as

adding the opposite, we see that

we use the distributive property

first to distribute the factor of

21 This will ensure that we

subtract all the terms of the second

Often, we do this without writing

the 21 We will say to distribute

the negative sign, implying the

21 that is not seen in the original

expression or equation.

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A local chiropractor has a small office where she cares for patients She has $8000 in

fixed costs each month that cover her rent, basic salaries, equipment, and utilities For

each patient she sees, she has an average additional cost of about $15 The chiropractor

charges her patients or their insurance company $80 for a visit

a Write an equation for the total monthly cost when n patient visits are done in a month

b What is the total monthly cost if the chiropractor has 100 patients visit during a month?

c Write an equation for the monthly revenue when n patients visit a month

d Write an equation for the monthly profit the chiropractor makes if she has n patient

visits in a month

e What is the monthly profit when 150 patients visit in a month?

f How many patient visits does this chiropractor need to have in a month for her

profit to be $5000.00?

Example 2 shows that you should check each answer to determine whether or not

it is a reasonable answer Sometimes this requires some common sense; other times, a

restriction that is stated in the problem should be considered

In both of the previous examples, it is important to pay attention to the definition of each variable The definitions of the variables helps to determine which variable value was given

and which variable is to be solved for Often, you will have to define the variables Use

meaningful variable names to make it easy to remember what they represent For example,

●t 5time in years

●h 5hours after 12 noon

● p 5population of San Diego 1in thousands2

●P 5profit of IBM 1in millions of dollars2

●S 5Salary 1in dollars per hour2Units, or how a quantity is measured, are very important in communicating what

a variable represents The meaning of P 5 100 is very different if profit for IBM is

measured in dollars and not millions of dollars The meaning of S 5 6.5 is different

if S represents your salary for your first job out of college; it would be great if S were

measured in millions of dollars per year and not dollars per hour Units can make a large

difference in the meaning of a quantity When defining variables, always include units

When solving an equation that represents something in an application, always check that the answer found is reasonable for the situation Use the following Concept

Investigation to practice determining which answers might be reasonable and which

would not make sense in the situation given

In each part, choose the value that seems the most reasonable for the given situation

Explain why the other given value(s) do not make sense in that situation

1 If P is the population of the United States in millions of people, which of the

following is a reasonable value for P ?

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2 If H is the height of an airplane’s flight path in feet, which of the following is a

reasonable value for H ?

a H 5 22000

b H 5 3,500,000

c H 5 25,000

3 If P is the annual profit in dollars of a new flower shop the first year it opens,

which of the following is a reasonable value for P ?

along with many others we will work with throughout the book, make up the Equation

Solving Toolboxes found at the back of the book In Example 3, watch for how these

tools are used throughout the solving process

When solving equations that involve fractions, you can either work with the fractions throughout the solving process or eliminate the fractions during the first step

To clear the fractions during the first step, multiply both sides of the equation by the least common denominator and simplify Then finish solving the equation as you would any other linear equation

Solve each equation Check the answer

a To eliminate the fractions, multiply both sides of the equation by the least common

denominator 6 and then continue solving

Steps to Solving Linear Equations

1 Simplify each side of the equation independently by performing any

arithmetic and combining any like terms

2 Move the variable terms to one side of the equation by using the addition or

subtraction property of equality Combine like terms

3 Isolate the variable term by using the addition or subtraction property of

Use to simplify when an equation

contains grouping symbols.

5 5

70 5

x 5 14

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Step 1 Simplify each side of the equation independently by performing any arithmetic

and combining like terms

Step 2 Move the variable terms to one side of the equation by using the addition or

subtraction property of equality Combine like terms

This step is already complete since there are no variable terms on the right side

Divide both sides by 4 to isolate the variable.

6 1

5

607 42

6 07

7 5 7 The answer checks.

Therefore, the answer is x 537

4 59.25 This answer can be written as an improper fraction or a decimal as shown

b For this equation, distribute the 1

4 first and then eliminate the fractions by multiplying both sides of the equation by the least common denominator 4

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4 b54a12 x 2 6b Multiply both sides of the equation by the least common denominator 4

4a14 xb 1 4a54 b524a12 xb 2 4(6) Reduce and multiply.

2 5

17

2 The answer checks.

c To start solving, distribute the negative 2 through the parentheses

Because of rounding, the two sides are not exactly the same, but they are very close

Solve each equation Check the answer

Subtract 5 from both sides of the equation to get the variable term isolated.

Add 2.8t to both sides of the equation

to get the variable terms together Add 17.3 to both sides of the equation

to isolate the variable term.

Divide both sides of the equation by

7.8 to isolate t.

Check the answer.

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Equations that contain more than one variable are called literal equations Formulas

are literal equations used to express relationships among physical quantities For

example, D 5 rt is a formula to calculate the distance traveled when given the rate of

travel and the time traveled at that rate This formula can be solved for one of the other

variables to make it easier to find the rate given the distance and time or to find the time

given the distance and rate Solving literal equations and formulas for other variables

uses the same steps as those used in solving other equations

Solve the following literal equations for the variable indicated

combine them Leave the left side

A term is a constant, or a product

of a constant and variable(s) Like terms are terms that have exactly the same variable parts raised to the same exponents Like terms will be covered more in Section 3.2 of this text.

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Solve the following literal equations for the variable indicated

a Velocity in free fall: v 5 Gt for t

b Velocity: v 5 v01at for a (Note: v0 is the initial velocity.)

11 During the first day of training on the job, a new candy

maker gets faster at making candies The number of

candies a new employee can produce during an hour can

be represented by C 5 10h 1 20 candies, where h is the

number of hours of training

a Find the number of candies a new employee can

produce in an hour after 1 hour of training

b Find the number of candies a new employee can

produce in an hour after 4 hours of training

c How many hours of training must an employee

receive before being able to produce 150 candies

an hour?

1 Profit equals minus .

2 The break-even point is the point at which

equals

3 A(n) is a quantity that can change.

4 tell us how a quantity is measured.

5 The input variable is also called the

12 The number of students who are enrolled in math classes at

a local college can be represented by E 5 217w 1 600, where E represents the math class enrollment at the college

w weeks after the start of the fall semester

a Find the total enrollment in math classes at the

college at the beginning of the fall semester

(Hint: Because the semester is just starting, w 5 0.)

b During which week will the total enrollment be

430 students?

c What will the total enrollment be in math classes

after 8 weeks?

13 The estimated number of Amazon Prime members

in millions, P, in the United States q quarters since

March 2015 can be represented by the equation

P 5 7.89q 1 33.73.

Source: Consumer Intelligence Research Partners.

a Find the estimated number of Amazon Prime

members in the U.S in March of 2017 (March 2017

is 8 quarters, 2 years, since March 2015.)

b Find the estimated number of Amazon Prime

members in the U.S in March of 2020

c In what quarter will the estimated number of

Amazon Prime members in the U.S be 133 million?

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14 The gasoline prices in Southern California can increase

very quickly during the summer months The equation

p 5 2.399 1 0.3w represents the gasoline prices p in dollars per gallon w weeks after the beginning of summer

a What does gasoline cost after 5 weeks of summer?

b During what week of summer will gasoline cost

$2.759 per gallon?

15 P 5 1.5t 2 300 represents the profit in dollars from

selling t printed T-shirts.

a Find the profit if you sell 100 printed T-shirts

b Find the profit if you sell 400 printed T-shirts

c How many printed T-shirts must you sell to make

$1000 in profit?

16 P 5 5.5b 2 500.5 represents the profit in dollars from

selling b books

a Find the profit if you sell 75 books

b Find the profit if you sell 200 books

c How many books must you sell to make $3600 in

profit?

17 The total cost, C, in dollars for a taxi ride in New York

City can be represented by the equation

when the trip is m miles long

a Determine the cost for a 25-mile taxi ride in New

York City

b How many miles can you ride in a New York City

taxi for $100?

18 A team of engineers is trying to pump out the air in a

vacuum chamber to lower the pressure They know that the following equation represents the pressure in the chamber:

P 5 35 2 0.07s

where P is the pressure in pounds per square inch (psi) of

the vacuum chamber and s is the time in seconds

a What will the pressure be after 150 seconds?

b When will the pressure inside the chamber be 1 psi?

For Exercises 19 through 22, determine which given value seems the most reasonable for the given situation Explain why the other given values do not make sense in that situation

19 P is the population of Kentucky in thousands of people.

22 S is a cook’s monthly salary in dollars from working at

White Castle Hamburgers

a S 5 10.50 b S 5 1600 c S 5 28,000

23 Salespeople often work for commissions on the sales

that they make for the company As a new salesperson

at a local technology company, you are told that you will receive an 8% commission on all sales you make after the first $1000 Your pay can be represented by

p 5 0.08(s 2 1000) dollars, where s is the amount of

sales you make in dollars

a How much total pay will you earn from $2000 in sales?

b How much total pay will you earn from $50,000 in

sales?

c If you need at least $500 per week to pay your bills,

what sales do you have to make per week?

24 At a new job selling high-end clothing to women,

you earn 6% commission on all sales you make after the first $500 Your pay can be represented by

p 5 0.06(s 2 500) dollars, where s is the amount of

sales you make in dollars

a How much total pay will you earn from $2000 in sales?

b How much total pay will you earn from $5000 in sales?

c If you need at least $450 per week to pay your bills,

what sales do you have to make per week?

25 Budget charges $29.95 for the day and $0.55 per mile

driven to rent a 10-foot moving truck

Source: Budget.com.

a Let B be the cost of renting a 10-foot moving truck

from Budget for a day and driving the truck m miles

Write an equation for the cost of renting from Budget

b How much would it cost to rent a 10-foot truck from

Budget if you were to drive it 75 miles?

c How many miles could you drive the truck if you

could pay only $100 for the rental?

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26 A local fitness club has a no-contract basic

membership plan with an initiation fee of $59.99

and a per month charge of $29.99

a Write an equation for the total cost, C, in dollars of this

plan if you continue your membership for m months.

b Use your equation to determine the total cost of this

membership for the first 2 years

c How many months of membership will $1000

purchase at this club?

27 A salesperson is guaranteed $250 per week plus a 7%

commission on the total dollar value of all sales made.

a Write an equation for your total pay per week, P, if

you make s dollars of sales.

b What will your total pay be if you have sales of $2000?

c How many dollars of sales do you need to make to

have a total weekly pay of $650?

28 A salesperson is guaranteed $300 per week plus a 5%

commission on the total dollar value of all sales made.

a Write an equation for your total pay per week, P, if

you make s dollars of sales.

b What will your total pay be if you have sales of $4000?

c How many dollars of sales do you need to make to

have a total weekly pay of $750?

29 You are planning a trip to Las Vegas and want to

calculate your expected costs for the trip You found that

you can take a tour bus trip for up to 7 days, and it will

cost you $225 for the round trip You figure that you can

stay at a hotel and eat for about $150 per day

a Write an equation for the total cost of this trip

depending on the number of days you stay (We will

ignore the gambling budget.)

b How much will it cost for a 3-day trip?

c If you have $1200 and want to gamble half of it,

how many days can you stay in Las Vegas, assuming

that you do not win any money?

30 Your family is planning a trip to Orlando, Florida, to

visit the amusement parks You want to budget for your

expected costs You find round trip flights for your

four-person family that total $1600 You expect the hotel, food, and admissions to cost about $900 per day

a Write an equation for the total cost of this trip

depending on the number of days you stay

b How much will it cost for a 5-day trip?

c If your family can afford to spend $7500 on this

trip, how many days can you stay in Orlando?

31 A professional photographer has several costs involved

in taking pictures at an event such as a wedding Editing and printing proofs of the photos cost $5.29 each The photographer also has to pay salaries of $800 for the day

a Write an equation for the total cost to shoot a

wedding depending on the number of proofs the photographer edits and prints

b How much will it cost the photographer if she edits

and prints 100 proofs?

c How many proofs can the photographer edit and print if

the total cost cannot exceed a budget of $1750?

32 The photographer from Exercise 31 charges her clients

a $7.50 fee for each proof she edits and prints plus a flat fee of $600 for the wedding

a Write an equation for the total revenue for shooting

the wedding depending on the number of proofs she edits and prints

b How much will the photographer charge the client for a

wedding that she edits and prints 100 proofs for?

c Write an equation for the profit made by the

photographer depending on the number of proofs she edits and prints

d How much profit will the photographer make on a

wedding if she edits and prints 100 proofs?

e How many photos must the photographer edit and

print to break even? (Breaking even means that profit 5 0.)

33 A snow cone vendor on the Virginia Beach boardwalk

has several costs of doing business She pays salaries of

$2000 per month and other fixed costs, such as utilities and kiosk rental, of $1150 per month Each snow cone sold costs her 36 cents

a Write an equation for the total cost of selling snow

cones for a month depending on the number of snow cones sold

b What will the monthly cost be if she sells

3000 snow cones in a month?

c How many snow cones can she sell if the total

monthly cost cannot exceed a budget of $4400?

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34 The snow cone vendor from Exercise 33 charges $2.50

per snow cone

a Write an equation for the total revenue from selling

snow cones for a month, depending on the number

of snow cones sold

b How much revenue will the vendor make if she sells

3000 snow cones in a month?

c Write an equation for the profit made by the snow

cone vendor depending on the number of snow cones sold

d How much profit will the vendor make from selling

4500 snow cones?

e How many snow cones must the vendor sell in a month

to break even? (Breaking even means that profit 5 0.)

35 The Squeaky Clean Window Cleaning Company

has several costs included in cleaning windows for a business The materials and cleaning solutions cost about $1.50 per window Insurance and salaries for the day will cost about $530

a Write an equation for the total cost to clean

windows for a day depending on the number of windows cleaned

b How much will it cost if the company cleans

60 windows?

c How many windows can the company clean if the

total cost cannot exceed a budget of $800?

36 The Squeaky Clean Window Cleaning Company from

Exercise 35 charges companies $9 per window cleaned plus a travel charge of $50

a Write an equation for the total revenue for cleaning

windows at a business depending on the number of windows cleaned

b How much will the Squeaky Clean Window Cleaning

Company charge a business to clean 50 windows?

c Write an equation for the profit made by the

Squeaky Clean Window Cleaning Company depending on the number of windows cleaned

d How much profit will the company make from

cleaning 80 windows for a business?

e How many windows must the company clean in a day

to break even? (Breaking even means that profit 5 0.)

For Exercises 37 and 38, compare the two students’

work to determine which student did the work correctly

Explain what mistake the other student made

37 A small publicity company will custom-label water

bottles for your company or event It costs the

publicity company $75 to set up the label design and

55 cents for each bottle and custom label Write an equation for the cost of each order depending on the number of bottles ordered

Javier

C 5 Cost of each order in dollars

b 5 number of bottles ordered

C 5 75 1 55b

Maria

C 5 Cost of each order in dollars

b 5 number of bottles ordered

C 5 75 1 0.55b

38 The same publicity company as in Exercise 37 has the

following revenue equation:

R 5 0.95b

where R is the revenue in dollars from an order of b

custom-labeled bottles of water Write an equation for the

profit the publicity company will earn from an order of b

P 5 0.95b 2 75 1 0.55b

P 5 275 1 1.50b

39 Enviro-Safe Pest Management charges new clients

$150 for an in-home inspection and initial treatment for ants Monthly preplanned treatments cost $38

a Write an equation for the total cost for pest

management from Enviro-Safe Pest Management depending on the number of months a house is treated

b If the house has an initial treatment and then is

treated monthly for 1.5 more years, how much will Enviro-Safe charge?

40 The population of the United States since 2010 can be

estimated by the equation P 5 2.76t 1 309.37, where

P is the population in millions t years since 2010 Source: Based on data from the U.S Census Bureau.

a What was the population of the United States in 2013?

b In what year, was the population of the United

States 326,000,000?

c In what year will the population of the United States

reach 375 million?

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41 A small manufacturer of golf clubs is concerned about

monthly costs The workshop costs $23,250 per month

to run in addition to the $145 in materials per set of

irons produced

a Write an equation for the

monthly costs of this club

manufacturer

b What are the monthly costs

for this company if they make

100 sets of irons?

c How many sets of irons does

this manufacturer need to produce for their costs to

be $20,000?

d If this company wants to break even making

100 sets of irons per month, what should they

charge for each set? (To break even, the company

needs the revenue to equal cost Use the cost from

part b and the fact that revenue can be calculated as

the price times quantity.)

42 You are in charge of creating and purchasing T-shirts

for a local summer camp After calling a local

silk-screening company, you find that to purchase 100 or

more T-shirts, there will be a $150 setup fee and a $5

charge per T-shirt

a Write an equation for the total cost, C, in dollars of

making t T-shirts.

b How much would 300 T-shirts cost?

c How many T-shirts can you purchase with a budget

of $1500?

d If this camp wants to break even selling 300

T-shirts, what should they charge for each T-shirt?

(To break even, the camp needs the revenue to

equal cost Use the cost from part b and the fact

that revenue can be calculated as the price times

quantity.)

43 Rockon, a small-town rock band, wants to produce

a EP before their next summer concert series They

have looked into a local recording studio and found

that it will cost them $1500 to produce the master

recording and then an additional $1.50 to make each

EP up to 500

a Write an equation for the total cost, C, in dollars of

producing n EPs.

b How much will it cost Rockon to make 250 EPs?

c If Rockon has $2000 to produce EPs, how many can

they order?

d If Rockon has $3000 to produce EPs, how many can

they order?

44 The percent P of companies that are still in business

t years after the fifth year in operation can be represented by the equation

P 5 2 3t 1 50.

(Hint: Be very careful with how the variables are

defined.)

a What percentage of companies are still in business

after 1 year in operation?

b What percentage of companies are still in business

after 25 years in operation?

c After how many years are there only 35% of

companies still in business?

45 Use the information from Exercise 25 to answer the

following questions

a If Budget doubled the cost per mile, how would that

change the equation found in Exercise 25 part a?

b Would the cost to drive 75 miles also double?

Explain your reasoning

following questions

a If the membership plan doubled the cost per month,

how would that change the equation found in Exercise 26 part a?

b Would the total cost for 2 years also double?

Explain your reasoning

following questions

a If the salesperson is given a raise by increasing the

guaranteed pay per week $100, how would that change the equation found in Exercise 27 part a?

b If instead of raising the guaranteed pay per week,

the salesperson’s commission rate was increased from 7% to 8%, how would that change the equation found in Exercise 27 part a?

c If the salesperson makes an average of $7000 in

sales per week, which raise would be best for the salesperson?

d If the salesperson makes an average of $4000 in sales

per week, which raise would be best for the company?

e What amount of sales per week would make these

raises result in the same weekly pay?

following questions

a If the salesperson is given a raise by increasing the

guaranteed pay per week $150, how would that change the equation found in Exercise 28 part a?

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b If instead of raising the guaranteed pay per week,

the salesperson’s commission rate was increased from 5% to 6%, how would that change the equation found in Exercise 27 part a?

c If the salesperson makes an average of $20,000 in

sales per week, which raise would be best for the salesperson?

d If the salesperson makes an average of $11,000 in

sales per week, which raise would be best for the company?

e What amount of sales per week would make these

raises result in the same weekly pay?

49 Use the information from Exercises 31 and 32 to

answer the following questions

a If the salaries the photographer has to pay increase

by 50%, how does that change the cost equation?

b How does the increase in salary affect the profit for

the photographer?

c If the photographer wants to cover the increase in

salaries, how much should she increase the charge per proof if the client wants 100 proofs?

50 Use the information from Exercises 33 and 34 to

answer the following questions

a If the kiosk rental increases by 20%, how does that

change the cost equation?

b How does the increase in kiosk rent affect the profit

for the snow cone vendor?

c If the snow cone vendor wants to cover the increase

in rent, how much should she increase the charge per snow cone if she can sell 6000 snow cones per month?

For Exercises 51 through 54, the linear equation is being

solved Complete the table by filling in the missing algebraic

steps to solve the equation or supply the missing reasons for

Isolate the variable term

by using the subtraction property of equality

25x

25 5

22525

The solution

52 24x 1 7 5 28x 2 9

Algebraic Step to

Move the variable terms to one side using the addition property of equality

Algebraic Step to Solve the

2(x 2 5) 1 7 5 23x 1 12 2x 2 10 1 7 5 23x 1 12

Combine like terms on each side of the equation

2x 2 3 5 23x 1 12

13x 13x

Isolate the variable term using the addition property of equality.Solve for the variable using the division property of equality

x 53

54 6x 1 1 5 22(2x 1 1) 2 3

Algebraic Step to Solve

Simplify the right side

by using the distributive property

6x 1 1 5 24x 2 2 2 3 6x 1 1 5 24x 2 5

Move the variable terms to one side using the addition property of equality

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For Exercises 55 through 64, solve each equation Provide

reasons for each step Check the answer.

81 Angular acceleration: v 5 v01 at for a (Note: v

is the Greek symbol omega, and a is the Greek symbol alpha.)

95 If a digital thermometer gave the outside temperature as

73.4 degrees Fahrenheit, would you round to the nearest whole degree? Explain your reasoning

96 If a digital thermometer gave your child’s temperature

as 100.3 degrees Fahrenheit, would you round to the nearest whole degree? Explain your reasoning

97 After calculating the discounted price of a TV, you get

the result 236.5725 How would you round the result

to find the discounted price? Explain your reasoning

98 Using a cost equation, you find that 2200.8 pens can

be produced with a budget of $500 How should you round the number of pens? Explain your reasoning

99 Using a profit equation, you find that 312.25 cars

needed to be washed to make a profit of $400 a week How many cars should the company try to wash a week to make $400 profit? Explain your reasoning

Give your own example of a real-life situation in

which the math rounding rule does not apply Explain why it does not apply

100.

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Introduction to Graphing Equations

In this section, we will graph equations in two variables by hand First, we graph

equations by creating a table of values, plotting those points and connecting the points

with a smooth curve Then, we investigate the different characteristics of a linear

equation and its graph

Equations in two variables are traditionally written with variables x and y The variable x is known as the input, or independent, variable The variable y is known as

the output, or dependent, variable In a table, x is put in the left column, and y is placed

in the right column In this text, we often use variables other than x and y, especially in

application problems We write the points generated as (input, output) or (x, y) When

graphing, the input (independent variable) is plotted on the horizontal axis, and the

output (dependent) variable is plotted on the vertical axis

Fundamentals of Graphing and Slope

LEARNING OBJECTIVES

Graph equations by plotting points

Calculate the slope of a line

Interpret the meaning of slope

Graph lines using the slope-intercept form of a line

1.2

Graph the equations by creating a table of values and plotting the points

a y 5 3x 2 8 b x 5 22y 1 10 c y 5 x213

SOLuTION

a Begin by finding ordered pairs that satisfy the equation Because y is already isolated

in this equation, it is easier to choose values for x and find the y-values that go with

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Plot these points.

All of these points lie on a straight line Draw a line through the points To show this equation has an infinite number of points that satisfy it, extend the line beyond the points drawn and put arrows on the ends of the line This indicates that the line continues in both directions infinitely The line itself actually represents all of the

possible combinations of x and y that satisfy this equation.

b Because x is already isolated in the equation x 5 22y 1 10, it is easier to choose

values for y and find values for x.

When entering an equation,

graphing calculators consider x the

independent (input) variable and y

the dependent (output) variable.

When entering an equation

in the Y5 screen, the dependent

variable must be isolated so that

the equation can be entered as

y 5expression.

To enter the equation

y 5 2x 1 5

first press y= and then in the

Y15 line, enter 2x 1 5.

Use the x,t,,n button for the

variable x.

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Now plot these points and draw a straight line through them.

c The equation y 5 x213 is a little more complicated than the others because the

variable x has an exponent of 2 Find more points to plot to see the graph’s shape

Select values of x and calculate the values of y.

using the TABLE Feature

First enter the equation into the

Y 5 screen To set up the table, press tblset f2 window to get the table setup menu.

Be sure the Indpnt and Depend settings are correct

Indpnt should be set to Ask.

Depend should be set to Auto.

These settings allow you to enter a value for the input variable The calculator then automatically calculates the related output Now press graph

table f5

to get the table feature Now you can

enter the values of x that you want.

When the Indpnt setting is set to Auto, the table will automatically fill with input values based on the TblStart setting and the ∆Tbl (change in table) setting This

is used to generate a table automatically.

For more on this feature, see the calculator guide in Appendix C.

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This graph is more complicated than the lines in the first two parts of this example

By graphing more points, a better idea of the shape of the graph develops In general, plotting three points will be enough for graphing a line, but five or more are needed to

graph more complicated equations The type of equation in this example is nonlinear

and will be addressed more in Chapter 4

PRACTICE PROBLEM FOR EXAMPLE 1

Graph the equations by creating a table of values and plotting the points

a y 5 2x 2 6

b y 5 x228

Graphing equations by plotting points and then connecting them with a smooth line or curve can be used with most equations This technique is the most basic method to graph an equation and can be used in applications as well as in basic algebra problems

In Section 1.1, we were given the equation U 5 19.95 1 0.79m, where U is the cost in dollars to rent a 10-foot truck from U-Haul when it is driven m miles.

a Create a table of points that satisfy this equation

b Create a graph for the equation by plotting the points found in part a Remember to

label the axes with units

SOLuTION

a Since we are investigating an equation about renting a truck and driving it m miles,

choose values of m that make sense for miles driven We start with zero miles

because the rental costs money even if we don’t drive anywhere

b Now plot the points Since the cost depends on the miles driven, the cost is the

dependent variable and is plotted on the vertical axis The miles driven is the

independent variable and is plotted on the horizontal axis Select a scale for each

axis that allows all the points to be plotted We choose a scale of 20 for the vertical axis and a scale of 25 for the horizontal axis These scales allow us to see all the points in the table as well as a little beyond them

using Your TI

Graphing Calculator

Evaluating Equations

The table feature of the calculator

can be used to calculate several

values of an equation very quickly

First enter the equation you are

working with into the Y5 screen

Now go to the table using

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All the points lie on the same line Draw that line and extend it farther out to represent more possible combinations of miles and costs that satisfy this equation.

Linear Equations in Two Variables

So far in this section, we have graphed several lines by hand It is helpful to recognize if

an equation in two variables is linear without first graphing it

What’s That Mean?

Scale

The scale of an axis is the

consistent and even spacing between the tick marks on the axis A reasonable scale for an axis will allow the axis to include the lowest and highest values of the points to be plotted.

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Most equations can be graphed by finding points and plotting them Graphing calculators use this technique to graph an equation that you enter When graphing by hand, an understanding of the basic characteristics of the graph of the equation makes graphing quicker and more accurate In the Concept Investigation below, some of the characteristics of the graph of a line and how they are related to the equation of a line are examined.

DEFINITION

Linear Equation An equation that can be expressed in one of the following

forms is called a linear equation in two variables.

y 5 mx 1 b or Ax 1 By 5 C where m, b, A, B, and C are real numbers, and A and B are not both zero.

We will use the graphing calculator to examine the various characteristics of linear equations Start by setting up your calculator by using the following steps

●Clear all equations from the Y5 screen (Press y= , clear )

●Change the window to a standard window (Press zoom , (ZStandard).)Now the calculator is ready to graph equations The Y5 screen is where equations are put into the calculator to graph them or evaluate them at input values Several simple equations will be graphed to investigate how the graph of an equation for a line reacts

to changes in the equation

1 Graph the following equations with positive coefficients on a standard window

Enter each equation in its own row (Y1, Y2, Y3, )

(Note: To enter an x, you use the x,t,,n button next to the alpha button.)

a y 5 x b y 5 2x

c y 5 5x d y 5 8x

In your own words, describe what the coefficient (number in front) of x does to

the graph Remember to read graphs from left to right

2 Now graph the following equations that have negative coefficients

a y 5 2x

b y 5 22x

c y 5 25x

d y 5 28x

In your own words, describe what a negative coefficient of x does to the graph.

3 Graph the following equations with coefficients that are between zero and one.

The number in front of a variable

expression is the coefficient

For example:

27x 5x2

27 is the coefficient for x.

5 is the coefficient of x2

Remember that a variable that

is by itself (x) has a coefficient

of 1, and a variable with only a

negative sign in front of it (2x)

has a coefficient of 21.

Using Your TI

graphing calculator

Entering Fractions

When entering fractions in the

calculator, it is often best to use

parentheses.

y 5(1>5)x

On many graphing calculators,

parentheses are needed in almost

all situations In some calculators,

To be sure the calculator does

what you intend, using parentheses

is a good idea.

The TI-84 Plus does not need

parentheses in some situations,

but in other situations, they are

required To keep confusion down,

one option is to use parentheses

around every fraction Extra

parentheses do not usually create

a problem, but not having them

where they are needed can cause

miscalculations.

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