sm a1 nlb tml so 12 math...754a saxon algebra 1section overview 12 lesson planner pacing guide...

754A Saxon Algebra 1

S E C T I O N O V E R V I E W

12

Lesson Planner

Pacing Guide

45-Minute Class

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

Lesson 111 Lesson 112 Lesson 113Lab 10 Lesson 114 Lesson 115 Cumulative Test 22


Lesson 116 Lesson 117 Lesson 118 Lesson 119 Lesson 120 Cumulative Test 23

Day 13

Investigation 12Lab 11

Block: 90-Minute Class


Lesson 111Lesson 112

Lesson 113Lesson 114Lab 10

Lesson 115Cumulative Test 22



Lesson 120Cumulative Test 23

Day 7

Lab 11Investigation 12

Lesson New Concepts

111 Solving Problems Involving Permutations

112Graphing and Solving Systems of Linear and Quadratic

Equations

113 Interpreting the Disciminant

LAB 10 Graphing Calculator: Graphing Radical Functions

114 Graphing Square-Root Functions

115 Graphing Cubic Functions

Cumulative Test 22, Performance Task 22

116 Solving Simple and Compound Interest Problems

117 Using Trigonometric Ratios

118 Solving Problems Involving Combinations

119Graphing and Comparing Linear, Quadratic, and

Exponential Functions

120Using Geometric Formulas to Find the Probability of an

Event

Cumulative Test 23, Performance Task 23

LAB 11 Graphing Calculator: Matrix Operations

INV 12 Investigation: Investigating Matrices

Resources for Teaching

• Student Edition• Teacher’s Edition• Student Edition eBook• Teacher’s Edition eBook • Resources and Planner CD • Solutions Manual• Instructional Masters• Technology Lab Masters• Warm Up and Teaching Transparencies• Instructional Presentations CD• Online activities, tools, and homework help www.SaxonMathResources.com

Resources for Practice and Assessment

• Student Edition Practice Workbook• Course Assessments• Standardized Test Practice• College Entrance Exam Practice• Test and Practice Generator CD using

ExamViewTM

Resources for Differentiated Instruction

• Reteaching Masters• Challenge and Enrichment Masters• Prerequisite Skills Intervention• Adaptations for Saxon Algebra 1• Multilingual Glossary• English Learners Handbook• TI Resources

* For suggestions on how to implement Saxon Math in a block schedule, see the Pacing section at the beginning of the Teacher’s Edition.

Resources and Planner CD

for lesson planning support

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Section Overview 12 754B

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2Le ssons 111–120, I nve st igat ion 12

Diff erentiated Instruction

Below Level Advanced Learners

Warm Up ............................. SE pp. 754, 761, 769, 776, 782, 788, 796, 804, 809, 817

Skills Bank ........................... SE pp. 846–883Reteaching Masters.............. Lessons 111–120,

Investigation 12Warm Up Transparencies .... Lessons 111–120Prerequisite Skills ................. Skill 50 Intervention

Challenge ............................... TE pp. 760, 768, 774, 780, 787, 794, 802, 808, 816, 823

Extend the Example ............. TE pp. 757, 765, 771, 777, 784, 792, 799, 806, 812, 819

Extend the Exploration ........ TE pp. 756Extend the Problem ............. TE pp. 760, 768, 773, 779, 780, 781, 786,

787, 794, 795, 802, 807, 814, 815, 816, 821, 826

Challenge and Enrichment .. Challenge: 111–120; Enrichment: 115 Master

English Learners Special Needs

EL Tips ................................ TE pp. 757, 766, 771, 777, 786, 789, 797, 805, 810, 822, 827

Multilingual Glossary .......... Booklet and Online English Learners Handbook

Inclusion Tips ...................... TE pp. 756, 762, 772, 790, 778, 790, 799, 806, 811, 818, 820

Adaptations for Saxon Lessons 111–120, Cumulative Tests 22, 23 Algebra 1

For All Learners

Exploration…. ..................... SE pp. 756, 827, 828Caution…. ............................. SE pp. 756, 762, 791, 797,

811, 827Hints….. .............................. SE pp. 776, 777, 783, 789,

790, 796, 800, 805, 806, 810, 817, 818

Alternate Method ................ TE pp. 812Online Tools

Error Alert ........................... TE pp. 755, 757, 760, 763, 766, 768, 770, 771, 774, 777, 779, 780, 781, 785, 786, 787, 791, 792, 794, 795, 799, 800, 802, 803, 805, 806, 807, 811, 813, 815, 816, 819, 820, 823, 827, 828, 829

SE = Student Edition; TE = Teacher’s Edition

Math Vocabulary

Lesson New Vocabulary Maintained EL Tip in TE

111factorial permutation

theoretical probabilitytree diagram

departure

112quadratic equationsystem of linear equations

sprinkler

113discriminantdouble root

quadratic formulavertex

discriminant

114

refl ectionsquare-root function

domainradicaltransformation

alteration

115

cubic function degree of a polynomiallinear functionquadratic function

limb

116compound interestsimple interest

linear functionexponential function

invested

117

cosecantcosinecotangentsecant

sinetangenttrigonometric ratio

hypotenuselegs of a right triangle

adjacentopposite

118combination factorial

permutationchoices

119

exponential functionlinear functionparent functionquadratic function

steep

120 complement of an event unscramble

INV 12element matrix

scale factor transform

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S E C T I O N O V E R V I E W 1 2

754C Saxon Algebra 1

Math Highlights

Enduring Understandings – The “Big Picture”

After completing Section 12, students will understand:

• How to solve problems involving permutations and combinations.

• How to use trigonometric ratios.

• How to fi nd the solution set of systems of linear and quadratic equations.

• How to interpret the discriminant.

• How to graph exponential functions.

• How to solve problems involving simple and compound interest.

Essential Questions

• How can the probability of an event be applied to the area of geometric fi gures?

• How do permutations and combinations differ?

• How can the discriminant be used to fi nd the number of solutions to a quadratic equation, and how does this relate to the x-intercepts of the graph of the related function equation?

• How does the degree of a polynomial function determine the characteristics of its graph?

• How can trigonometric ratios be used to fi nd missing side lengths and measures of angles?

• How do changes to the parent function affect the graph of the function?

Math Content Strands Math Processes

Equations• Lesson 117 Using Trigonometric Ratios

Functions and Relations• Lesson 115 Graphing Cubic Functions• Lesson 116 Solving Simple and Compound Interest

Problems• Lesson 119 Graphing and Comparing Linear, Quadratic,

and Exponential Functions

Probability and Data Analysis• Lesson 111 Solving Problems Involving Permutations• Lesson 118 Solving Problems Involving Combinations• Lesson 120 Using Geometric Formulas to Find the

Probability of an Event

Quadratic Equations and Functions• Lesson 113 Interpreting the Disciminant

Radical Expressions and Functions• Lab 10 Graphing Calculator: Graphing Radical

Functions• Lesson 114 Graphing Square-Root Functions

Systems of Equations and Inequalities• Lesson 112 Graphing and Solving Systems of Linear and

Quadratic Equations• Lab 11 Graphing Calculator: Matrix Operations• Investigation 12 Investigating Matrices

Connections in Practice Problems Lessons

Coordinate

Geometry 118Geometry 111, 112, 113, 114, 115, 116, 117, 118, 119, 120Measurement 111, 112, 114, 115, 116Probability 113

Reasoning and Communication Lessons

• Analyze 111, 112, 113, 114, 115, 116, 118, 119, 120

• Error analysis 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

• Estimate 111, 118• Formulate 112, 115, 116, 119, Inv. 12• Generalize 111, 113, 115, 117, 120, Inv. 12• Justify 111, 116, 117, 119, 120• Math Reasoning 111, 112, 113, 114, 115, 116, 117,

118, 119, 120• Model 111, 113, 118, 120, Inv. 20• Multiple choice 111, 112, 113, 114, 115, 116, 117,

118, 119, 120• Multi-step 111, 112, 113, 114, 115, 116, 117,

118, 119, 120• Predict 111, Inv. 12• Verify 112, 113, 115, 118, 119, 120• Write 112, 114, 115, 116, 117, 118, 119,

120, Inv. 12

• Graphing Calculator 112, 113, 115, 116, 117, Inv. 12

Connections

In Examples: Avalanches, Baseball, Horizon, Indirect measurement, Probability, Retirement investments, Uniform numbers, Volume of a cube, Zoning

In Practice problems: Accessories, Architecture, Astronomy, Aviation, Baking, Biking, Bonds, Business, Capacity, Chemistry, Construction, Credit cards, Dining, Engineering, Finance, Firefighting, Football, Fundraising, Games, Gardening, Interior decorating, Jewelry, Manufacturing, Mutual funds, Nature, Navigation, Nutrition, Oceanography, Oven temperatures, Packaging, Paper folding, Phone chains, Photography, Physics, Population, Projectile motion, Property, Puzzles, Retirement investments, Space Shuttle, Sports, Structural engineering, Temperature, Tennis, Video rental

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Section Overview 12 754D

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Content Trace

LessonWarm Up:

Prerequisite Skills

New Concepts Where PracticedWhere

Assessed

Looking

Forward

111 Lessons 14, 15, 33

Solving Problems Involving Permutations

Lessons 112, 113, 114, 115, 117, 118, 119, 120

Cumulative Test 23

Lessons 118, 120

112 Lessons 9, 29, 55, 84

Graphing and Solving Systems of Linear and Quadratic Equations

Lessons 113, 114, 115, 116, 117, 118, 119, 120

Cumulative Test 23

Lesson 119

113 Lessons 9, 13 Interpreting the Disciminant Lessons 114, 115, 116, 117, 118, 119, 120

Cumulative Test 23

Lesson 119

114 Lessons 69, 76 Graphing Square-Root Functions

Lessons 115, 116, 117, 118, 119, 120

Cumulative Test 23

Lessons 115, 119

115 Lessons 46, 53 Graphing Cubic Functions Lessons 116, 117, 118, 119, 120 Cumulative Test 23

Lessons in other Saxon High School Math programs

116 Lessons 31, 42, Skills Bank 7

Solving Simple and Compound Interest Problems

Lessons 117, 118, 119, 120 Test and Practice Generator CD


117 Lesson 85 Using Trigonometric Ratios Lessons 118, 119, 120 Test and Practice Generator CD


118 Lesson 111 Solving Problems Involving Combinations

Lessons 119, 120 Test and Practice Generator CD


119 Lessons 49, 89 Graphing and Comparing Linear, Quadratic, and Exponential Functions

Lesson 120 Test and Practice Generator CD


120 Lessons 14, 16 Using Geometric Formulas to Find the Probability of an Event

N/A Test and Practice Generator CD


INV 12 N/A Investigation: Investigating Matrices

N/A Test and Practice Generator CD


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S E C T I O N O V E R V I E W 1 2

754E Saxon Algebra 1

Ongoing Assessment

Type Feature Intervention *

BEFORE instruction Assess Prior Knowledge

• Diagnostic Test • Prerequisite Skills Intervention

BEFORE the lesson Formative • Warm Up • Skills Bank• Reteaching Masters

DURING the lesson Formative • Lesson Practice• Math Conversations with the Practice

problems

• Additional Examples in TE• Test and Practice Generator (for additional

practice sheets)

AFTER the lesson Formative • Check for Understanding (closure) • Scaffolding Questions in TE

AFTER 5 lessons Summative After Lesson 115• Cumulative Test 22• Performance Task 22After Lesson 120• Cumulative Test 23• Performance Task 23

• Reteaching Masters• Test and Practice Generator (for additional

tests and practice)

AFTER 20 lessons Summative • Benchmark Tests • End-of-Year Exam

• Reteaching Masters• Test and Practice Generator (for additional

tests and practice)

* for students not showing progress during the formative stages or scoring below 80% on the summative assessments

Evidence of Learning – What Students Should Know

Because the Saxon philosophy is to provide students with sufficient time to learn and practice each concept, a lesson’s topic will not be tested until at least five lessons after the topic is introduced.

On the Cumulative Tests that are given during this section of ten lessons, students should be able to demonstrate the following competencies:

• Simplify rational expressions.• Translate graphs of quadratic functions and absolute–value functions.• Divide polynomials.• Use the discriminant to find the number of solutions without solving.• Solve quadratic equations, radical equations, and absolute-value inequalities.• Use the Fundamental Counting Principle.• Solve systems of linear inequalities.• Extend geometric sequences.

Test and Practice Generator CD using ExamView™

The Test and Practice Generator is an easy-to-use benchmark and assessment tool that creates unlimited practice and tests in multiple formats and allows you to customize questions or create new ones. A variety of reports are available to track student progress toward mastery of the standards throughout the year.

NorthStar Math offers you real-time benchmarking, trackingand student progress monitoring.Visit www.NorthStarMath.com for more information.

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Section Overview 12 754F

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Assessment Resources

Resources for Diagnosing and Assessing

• Student Edition• Warm Up• Lesson Practice

• Teacher’s Edition• Math Conversations with the Practice problems• Check for Understanding (closure)

• Course Assessments• Diagnostic Test• Cumulative Tests• Performance Tasks • Benchmark Tests

Resources for Test Prep

• Student Edition Practice• Multiple-choice problems• Multiple-step and writing problems• Daily cumulative practice

• Standardized Test Practice

• College Entrance Exam Practice

• Test and Practice Generator CD using ExamViewTM

Resources for Intervention

• Student Edition • Skills Bank

• Teacher’s Edition• Additional Examples• Scaffolding questions

• Prerequisite Skills Intervention• Worksheets

• Reteaching Masters• Lesson instruction and practice sheets

• Test and Practice Generator CD using ExamViewTM

• Lesson practice problems• Additional tests

Cumulative TestsThe assessments in Saxon Math are frequent and consistently placed after every fi ve lessons to offer a regular method of ongoing testing. These cumulative assessments check mastery of concepts from previous lessons.

Performance Tasks The Performance Tasks can be used in conjunction with the Cumulative Tests and are scored using a rubric.

After Lesson 115 After Lesson 120 For use with Performance Tasks

Name ____________________________________________ Date ______________ Class_______________

Cumulative Test

© Saxon. All rights reserved. 101 Saxon Algebra 1

21B

1. (95) Kelly rides her bike at 14 mph if there

is no wind. She plans a round trip to a

park that is 35 miles away. If there is a w

mph wind, the time for the outbound trip

is35

14 + whours. The time for the return

trip against a w mph wind is35

14 w

hours. What is the total time for the round

trip?

2. (107) Graph the function f x( ) = x 1 and

give the coordinates of the vertex.

Simplify problems 3–4.

3. (103)108x 4

3 20x3

4. (92)

18x

5x + 206

x + 4

5. (97) Determine whether the ordered pair (–

1, 13) is a solution of the inequalityy < 6x + 7 .

6. (90) Add3x 2

15x+

2x 2

15x. Simplify your

answer.

7. (Inv. 10) Write an equation for the

transformation described below.

Shift f x( ) = 2x 2 1 up 3 units.

Then graph the original function and the

graph of the transformation on the same

set of axes.

8. (93) Divide 12x3 + 24x 2 + 18x( ) ÷ 6x .

9. (110) Use the quadratic formula to solve for

x.

x 2 + 3x 4

10. (109) Graph the system of inequalities

below.

y 2x + 4

y 2x 2

Name ____________________________________________ Date ______________ Class_______________

Cumulative Test


22A

1. (95) Tom’s boat travels at 15 mph if there is

no current. Tom plans a round trip to a

beach that is 40 miles away. If there is a

c mph current, the time for the outbound

trip is40

15 + chours. The time for the

return trip against a c mph current is

40

15 chours. What is the total time for

the round trip?

2. (107) Graph the function f (x) = x + 3 and

give the coordinates of the vertex.

Simplify problems 3–4.

3. (103)32x 4

2 12x3

4. (92)

5x

2x + 610

x + 3

5. (97) Determine whether the ordered pair

(–2, 10) is a solution of the inequalityy < 4x + 2 .

6. (90) Add2x 2

12x+

4x 2

12x. Simplify your answer.

7. (Inv. 10) Write an equation for the

transformation described below.

Shift f x( ) = 2x 2 + 3 down 4 units.

Then graph the original function and the

graph of the transformation on the same

set of axes.

8. (93) Divide 15y 3 + 9y 2 + 9y( ) ÷ 3y .

9. (110) Use the quadratic formula to solve for x.

x 2 5x + 6

10. (109) Graph the system of inequalities

below.

y1

2x + 2

y1

2x 4

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test


23B

1. (111) Celeste has four skirts: green, blue,

purple, and red. Each skirt has 2

matching belts. Draw a tree diagram to

determine the number of possible skirt

and belt outfits that Celeste can wear.

2. (114) Determine the domain of y = x = 9 .

3. (92) Simplify the expression below.

xm

ny

mn

x

4. (85) Use the Pythagorean Theorem to find

side length t to the nearest tenth.


2x + =24 + x 2 = 0

6. (109) Graph the system of linear inequalities

below.

y 3x = 2

y1

2

7. (107) Describe the graph of the function

f x( ) = 4 x .

8. (93) Divide x 2 = 7x + 10( ) ÷ 2 = x( ) .

9. (112) Solve the system of equations below by

substitution.

y = x 2 + 4x = 10

y = 4x + 6

10. (115) Evaluate the function y = 3x3 for

x = –2, –1, 0, 1, and 2. Then graph the

function.

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test


23A

1. (111) Roger has three suits: gray, blue, and

black. Each suit has 2 matching vests.

Draw a tree diagram to determine the

number of possible suit and vest outfits

that Roger can wear.

2. (114) Determine the domain of y = x = 12 .

3. (92) Simplify the expression below.

3b

xab

xy

4. (85) Use the Pythagorean Theorem to find

side length t to the nearest tenth.


=2x + =8 + x 2 = 0

6. (109) Graph the system of linear

inequalities below.

y 2x = 2

y1

2

7. (107) Describe the graph of the function

f x( ) = 5 x .

8. (93) Divide x 2 = 6x + 8( ) ÷ 4 = x( ) .

9. (112) Solve the system of equations below

by substitution.

y = x 2 + 3x = 2

y = 3x + 7

10. (115) Evaluate the function y = =2x3 for

x = –2, –1, 0, 1, and 2. Then graph the

function.

© Saxon. All rights reserved. xi Saxon Algebra 1

Student Rubric

StudentEvalutaion

Knowledge andSkil lsUnderstanding

CommunicationandRepresentation

Process andStrategies

4

I understand and can

justify my reasoning in

more than one way.

I explained my work in

detail and justified my

solution. I described

my work in great

detail and illustrated

my thinking.

I selected the most

appropriate strategy

and used the process

to solve the problem.

3I understand the task

and can show that I

understand.

I can explain my

thinking. I can show

my work. .

I selected a strategy

and followed the

process.

2

I have some

understanding of the

task.

I can explain some of

my thinking. I can

show some of my

work.

I selected a strategy

but became confused

on the process.

1

I need help in

understanding the

task.

I need help in

explaining my thinking.

I need help in showing

my work.

I need help in choosing

a strategy.

© Saxon. All rights reserved. xx SSaxon Algebra 1

TTeacher Rubric

CriteriaPerformance

Knowledge andSkil lsUnderstanding

CommunicationandRepresentation

Process andStrategies

4

The student got it!

The student did it in

new ways and

showed how it

worked. The student

knew and understood

what math concepts

to use.

The student clearly

detailed how he/she

solved the problem.

The student included

all the steps to show

his/her thinking. The

student used math

language, symbols,

numbers, graphs

and/or models to

represent his/her

solution.

The student had an

effective and inventive

solution. The student

used big math ideas

to solve the problem.

The student

addressed the

important details. The

student showed other

ways to solve the

problem. The student

checked his/her

answer to make sure

it was correct

3

The student

understood the

problem and had an

appropriate solution.

All parts of the

problem are

addressed.

The student clearly

explained how he/she

solved the problem.

The student used

math language,

symbols, tables,

graphs, and numbers

to explain how he/she

did the problem.

The student had a

correct solution. The

student used a plan

to solve the problem

and selected an

appropriate strategy.

2

The student

understood parts of

the problem. The

student started, but

he/she couldn’t finish.

The student explained

some of what he/she

did. The student tried

to use words,

symbols, tables,

graphs and numbers

to explain how he/she

did the problem.

The student had part

of the solution, but

did not know how to

finish. The student

was not sure if he/she

had the correct

answer. The student

needed help.

1

The student did not

understand the

problem.

The student did not

explain how he/she

solved the problem.

He/she did not use

words, symbols,

tables or graphs to

show how he/she

The student couldn’t

get started. The

student did not know

how to begin.

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Saxon Algebra 1754

Solving Problems Involving Permutations

Warm Up

111LESSON

1. Vocabulary (Experimental, Theoretical ) probability is found by analyzing a situation and finding the ratio of favorable outcomes to all possible outcomes. Theoretical

2. What is the probability of rolling a number greater than 3 on a number cube labeled 1–6? 1_2

3. What is the probability of rolling a number greater than 7 on a number cube labeled 1–6? 0

Identify each set of events as independent or dependent.

4. Rolling a 5 on one number cube and a 3 on another. independent

5. Drawing a blue marble from a bag, keeping it, and then drawing a red marble. dependent

A tree diagram can be used to determine the number of ways 2 pairs of pants and 4 shirts can be arranged to make different outfits. However, the number of possible outcomes can be determined by multiplying the number of ways the first event can occur by the number of ways the second event can occur.

first event second event total possible outcomes

2 pairs of pants × 4 shirts = 8 outfits

This method is an application of the Fundamental Counting Principle. The Fundamental Counting Principle can be used to determine the number of possible outcomes in situations involving independent events.

Fundamental Counting PrincipleIf an independent event M can occur in m ways and another independent event N can occur in n ways, then the number of ways that both events can occur is

m · n.

Example: A restaurant offers 4 entrées and 5 vegetable dishes. How many meals with one entrée and one vegetable dish are possible?

20 meals may be ordered since 4 · 5 = 20.

Example 1 Using the Fundamental Counting Principle

A 1-topping pizza can be ordered with a choice of 4 different toppings: pepperoni, sausage, mushrooms, or onion. There is also a choice of different types of crust: thin, thick, or traditional. Find the number of ways that a 1-topping pizza can be ordered using the Fundamental Counting Principle.

(14)(14)

(14)(14)

(14)(14)

(33)(33)

(33)(33)

New ConceptsNew Concepts

Math Reasoning

Predict How many different pizzas would be possible if you had 5 choices for toppings and 2 choices for crust?

10 different pizzas

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LESSON RESOURCES

Student Edition Practice Workbook 111

Reteaching Master 111Adaptations Master 111Challenge and Enrichment

Master C111Technology Lab Master 111

It is very important for students to know the difference between permutations and combinations. The permutation and combination formulas:

nPr = n! _______ (n - r)!

nCr = n! _________ r!(n - r)!

give the number of possible permutations/combinations of r objects

from a set of n. Permutations have stricter requirements, such as the order, and will therefore have a larger number of possible outcomes than combinations. Students can remember this by:

nPr = nCr · r!

MATH BACKGROUND

Warm Up1

754 Saxon Algebra 1

111LESSON

Problems 4 and 5

Remind students of the difference between independent and dependent events.

2 New Concepts

In this lesson, students learn to solve problems involving permutations.

Example 1

Point out to students that using a tree diagram for reasonably sized sets can be very helpful to prevent counting errors.

Additional Example 1

A school basketball team uses three colors on their uniforms: red, blue, and white. Each uniform has a top and bottom. Determine the number of possible color combinations the team can have on their uniforms. 9

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Lesson 111 755

SOLUTION

Determine the number of ways each event can occur and then find their product.

4 types of toppings × 3 types of crust = 12 possible pizza combinations

Check Use a tree diagram to verify that there are 12 possible pizza combinations.

PepperoniThinThickTraditional

ThinThickTraditional



Pepperoni ThinPepperoni ThickPepperoni Traditional

Sausage ThinSausage ThickSausage Traditional

Mushroom ThinMushroom ThickMushroom Traditional

Onion ThinOnion ThickOnion Traditional

Sausage

Mushrooms

Onions

OutcomesCrustTopping

The tree diagram verifies that there are 12 possible outcomes.

When a group of people or objects are arranged in a certain order, the arrangement is called a permutation. The unique ways that 5 different colored blocks can be arranged are examples of permutations.

The factorial operation can be used to find different ways to arrange a set of n different items, where the first item may be selected n different ways, the second item may be selected n - 1 ways, and so on.

Factorial

The factorial n! is defined for any natural number n as n! = n(n - 1)…(2)(1).

Zero factorial is defined to be 1. 0! = 1.

Example: 5! = 5 · 4 · 3 · 2 · 1

There are n! ways to position n students in a line. For example, the number of ways 6 students can be positioned in a line can be described by 6!. As each position in the line is filled, the number of students that can be chosen to fill each position decreases by 1.

1st 2nd 3rd ...6 Students 5 Students 4 Students

Notice only 5 students can be chosen for the 2nd position because 1 student has already filled the 1st position. Continuing this pattern shows that 6 students can be arranged in order 6!, or 6 · 5 · 4 · 3 · 2 · 1 = 720, different ways.

Online Connection

www.SaxonMathResources.com

Reading Math

The expression 8! is read “eight factorial.”

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Lesson 111 755

Error Alert When drawing tree diagrams, a common mistake is to forget to apply the same number of choices to every independent event.

Remind students that for independent events, the tree diagram should be symmetrical.


Saxon Algebra 1756

Example 2 Simplifying Expressions with Factorials

a. Find 7!.

SOLUTION

7!

= 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 Write the factors of 7! and multiply.

b. Find 9!

_ 4!

.

SOLUTION

9!

_ 4!

= 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 ___

4 · 3 · 2 · 1 Write the factors of 9! and 4!.

= 9 · 8 · 7 · 6 · 5 = 15,120 Multiply.

Exploration Exploration Finding Possibilities When Order is Important

a. On an index card, list all possible ways that the 4 colored ribbons can be arranged.

b. On a second index card, list all possible ways that any two of the four colored ribbons can be arranged. 12 ways; Sample: RB, RG, RY, BR, BG, BY, YR, YG, YB, GR, GY, GB

When choosing 3 of 8 contestants as finalists in a competition, order doesn’t matter. However, in naming a first, second, and third place from the 8 contestants, the order does matter. Since order is important it is a permutation.

PermutationThe number of permutations of n objects taken r at a time is given by the formula nPr = n!

_ (n-r)!

.

Example 3 Finding the Number of Permutations

a. Your school is running a recycling campaign in which 6 classes are competing to see who can collect the most recyclable materials. In how many ways can the classes finish in first through sixth place?

SOLUTION

This is a permutation of 6 things taken 6 at a time.

nPr = n! _

(n - r)! Write the formula.

6P6 = 6! _

(6 - 6)! =

6! _ 0!

Simplify.

= 6 · 5 · 4 · 3 · 2 · 1 __

1 Write the factors of 6! and 0!.

= 720 Multiply.

a. 24 ways; Sample with red, blue, yellow, and green ribbons: RBYG, RBGY, RYBG, RYGB, RGBY, RGYB, BRYG, BRGY, BYRG, BYGR, BGRY, BGYR, YRBG, YRGB, YBRG, YBGR, YGRB, YGBR, GRBY, GRYB, GBRY, GBYR, GYRB, GYBR

a. 24 ways; Sample with red, blue, yellow, and green ribbons: RBYG, RBGY, RYBG, RYGB, RGBY, RGYB, BRYG, BRGY, BYRG, BYGR, BGRY, BGYR, YRBG, YRGB, YBRG, YBGR, YGRB, YGBR, GRBY, GRYB, GBRY, GBYR, GYRB, GYBR

Materials

• index cards

• 4 different colored ribbons

Caution

Remember that 0! is equal to 1, not 0.


Students may have diffi culty with permutations. For Example 3a have students draw blank lines on their paper to represent the places of the competition. Then ask,

“How many classes can fi ll fi rst place?” 6

Ask, “Once the fi rst place position is fi lled, how many classes can fi ll second place?” 5

Fill in the blank lines as the students answer the questions.

6 5

Then have the students fi ll in the rest of the blanks. Have them solve Example 3b using the same method.

INCLUSION

756 Saxon Algebra 1

Example 2

Point out to students the shorthand, !, for factorials.


a. Find 9!. 362,880

b. Find 3! __ 7! .

1 ____ 840

ExplorationExploration

Emphasize that using props and drawing pictures can be very helpful in solving problems involving probabilities.

Extend the Exploration

a. Given that a yellow ribbon is taken out of the set, list all possible ways that the three remaining colored ribbons can be arranged. Sample: RBG, RGB, BGR, BRG, GBR, GRB

b. List all possible way that any two of the three colored ribbons can be arranged. 6 ways; Sample: RB, BR, RG, GR, GB, BG

Example 3

Remind students that the order matters in a permuation.


a. Eight people are competing in the 100-meter dash. In how many ways can the runners fi nish in fi rst through eighth place? 40,320

b. How many ways can the eight runners win a medal (fi nish in the top three places)? 336


Lesson 111 757

b. A total of 6 classes are competing to see who can collect the most recyclable materials. In how many different ways can the classes finish in first and second place?

SOLUTION

This is a permutation of 6 things taken 2 at a time.

nPr = n! _

(n - r)! Write the formula.

6P2 = 6! _

(6 - 2)! =

6! _ 4!

Simplify.

= 6 · 5 · 4 · 3 · 2 ·1 __

4 · 3 · 2 · 1 = 6 · 5 Write the factors of 6! and 4!. Then simplify.

= 30 Multiply.

Example 4 Application: Uniform Numbers

The 15 members of a softball team have uniform numbers 1 through 15. They are introduced randomly at a pep rally. What is the probability that the first 4 players introduced will have uniform numbers 1, 2, 3, and 4 in that order?

SOLUTION

Of the possible permutations only 1, 2, 3, 4 is favorable.

The probability is represented by:

number of ways to choose 1, 2, 3, 4

____ number of ways to choose 4 numbers

= 1 _

15P4 =

1 __ 15 · 14 · 13 · 12

= 1 _

32,760

Lesson Practice

a. While trying to schedule a flight for vacation, you are given two choices for departure and four choices for the return flight. How many ways can you schedule your flights for the trip? 8

b. A video game character has 6 choices each for hair color, face, attitude, and outfit as well as a choice of male or female. How many different characters are possible? 2592

c. Find 5!. 120

d. Find 6!

_ 3!

. 120

e. You are selecting your class schedule for the school year. If there are 7 periods and each of the seven classes are taught each period, how many possible ways are there for your schedule to be determined? 5040

f. There are 10 people in the Activities Club. In how many different ways can a president, vice-president and treasurer be selected from the club members? 720

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

Math Reasoning

Generalize Another way to think about permutations of n Pr is to multiply the first r numbers of n! . So 5P3 would be 5 · 4 · 3 = 60. Explain how to find 7P2 and then find its value.

Sample: Multiply 7 times 6; 42

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Explain the meaning of the word departure. Say:

“The departure time of a fl ight is the time a plane is leaving the airport.”

“Why should someone arrive at the airport several hours before the departure time?” Sample: They will need to check in with the airline and go through security checkpoints.

ENGLISH LEARNERS

Have students read a bus schedule to familiarize themselves with arrival and departure times.

Lesson 111 757

Example 4

Explain to students that the probability of an event with one possible permutation is the reciprocal of the number of ways that event can happen.

Extend the Example

If the fi rst 4 players that were introduced have uniform numbers 1, 2, 3, and 4, what is the probability that the next 4 being introduced will have numbers 5, 6, 7, and 8 in that order? 1

_____ 7920


There are 10 people running for president of the student body government—each identifi ed by a number. If the candidates are introduced randomly at a debate, what is the probability that the fi rst 4 introduced will be candidates 10, 9, 8, 7 in that order? 1

_____ 5040

Lesson Practice

Problem d

Scaff olding When simplifying a fraction involving factorials, suggest that students expand the factorials and divide out like terms one at a time.

Problem f

Error Alert It is possible that some students will attempt to draw the tree diagram for this problem. Remind students that with large numbers, the Fundamental Counting Principle is much more practical.


Practice Distributed and Integrated

Saxon Algebra 1758

*1. Draw a tree diagram to represent the possible outcomes of flipping a coin three times. See Additional Answers.

*2. Multiple Choice Evaluate 10!. A

A 3,628,800 B 362,880 C 55 D 9

*3. Video Rental For movie night, you want to rent one drama, one comedy, and one science fiction movie. The video store has 5 new releases for drama, 6 new releases for comedy, and 3 new releases for science fiction. How many possible movie combinations are there? 90

4. Simplify the rational expression 3d

_ 2x3 - 5d

_ 2x3 if possible. -d_x3

Find the zeros of each function.

5. y = x2 - 8x + 16 4 6. y = 3x2 + 36x - 39 -13, 1

*7. Model Draw a tree diagram to determine the number of possible outcomes of earning an A, B, or C in history, English, and math classes. See Additional Answers.

*8. Justify Explain how to find the number of outfits possible if you have 5 shirts and 4 pairs of pants to choose from. Sample: Multiply 5 times 4 to get 20 possible outfits.

*9. Use the quadratic formula to solve c2 + 16c - 36 = 0. Check the solutions. 2, -18

*10. Estimate Find the best whole number estimate for the solutions to 70 - 52x = -x2. 1, 51

11. Find and correct the error the student made in graphing y - 2.5 > 0

y - 4 < -2x

.

x

y

O

4

2

4

-2

-2-4

-4

12. Graph the system 4x - 2y < 6

y + 1 ≥ 2x

. See Additional Answers.

13. Solve r2 - 24r = -144 by completing the square. r = 12

(111)(111)

(111)(111)

(111)(111)

(51)(51)

(96)(96) (96)(96)

(111)(111)

(111)(111)

(110)(110)

(110)(110)

(109)(109)

Sample: The student should have made both lines dashed because the points on the boundary lines are not solutions.

Sample: The student should have made both lines dashed because the points on the boundary lines are not solutions.

(109)(109)

(104)(104)

g. A popular TV series ran for 10 seasons. You are buying the seasons from an online DVD service. If each season arrives at random, what is the probability that the first 5 seasons you receive in the mail are the first 5 seasons that were made, in the correct order? 1_

30,240

(Ex 4)(Ex 4)


758 Saxon Algebra 1

Check for Understanding

The questions below help assess the concepts taught in this lesson.

“How many ways can 8 people be positioned in a line?” 8! or 40,320

“How can one fi nd the number of permutations of r objects from a group of n?” Sample: nPr = n! ______

(n - r)! .

Practice3

Math ConversationsDiscussion to strengthen understanding

Problem 6

After factoring the expression to fi nd the zeros, remind students that quadratic equations can have zero, one, or two solutions.

Problem 11

Guide the students by asking them the following questions.

“How is y - 2.5 > 0 graphed?” Sample: Graph the line y = 2.5 with a dashed line and shade the region above it.

“How is y - 4 < -2x graphed?” Sample: Graph the line y = -2x + 4 with a dashed line and shade the region below the line.


Lesson 111 759

*14. Error Analysis Two students used the quadratic formula to solve a quadratic equation. Which student is correct? Explain the error.

Student A Student B

8a = -10a2 + 18a - 10a2 + 1 = 0

x = -(-10) ± √ �� -102 - 4(8)(1)

___ 2(8)

x = 10 ± √ �� 100 + 32

__ 16

x = 10 ± √ �� 132

_ 16

x = 10 ± 2 √ � 33

_ 16

x = 5 ± √ � 33

_ 8

8a = -10a2 + 110a2 + 8a - 1 = 0

x = -8 ±

√ �� 82 - 4(10)(-1) ___

2(10)

x = -8 ± √ �� 64 + 40

__ 20

x = -8 ± √ �� 104

__ 20

x = -8 ± 2 √ � 26

__ 20

x = -4 ± √ � 26

_ 10

*15. Space Shuttle The external tank of the space shuttle separates after 8.5 minutes at a velocity of 28,067 kilometers per hour. Can the formula -4.9t2 + v0t + y0 = 0 be used to find the distance above earth? Explain. No. Sample: The formula involves the initial velocity and height.

16. Measurement The length of a piece of wood must measure between 15 and 17 centimeters and the width must measure between 9 and 11 centimeters. Write a system of linear inequalities to represent the possible dimensions of the wood piece, in inches, given that 1 inch is equal to 2.54 centimeters.

17. Business The total profit on a particular skateboard is represented as p2 - 7p where p is the number of units sold in thousands. How many units need to be sold to have a profit of $23,750? Round to the nearest hundred. 157,700 units

18. Find the next 3 terms of the sequence 1

_ 2187 , 1

_ 729 , 1

_ 243 , 1

_ 81 , ... . 1_27 , 1_

9, 1_

3

19. Chemistry Oxygen evaporation from a body of water increases with the temperature. This process of oxygen depletion can be modeled by the expression

√ � x

_ 6 where x is the temperature in C°. What value of x corresponds to an evaporation of 9 cubic feet of oxygen? x = 2916

20. Graph the function f(x) = 3⎢x�. See Additional Answers.

21. Multiple Choice Which function is not an exponential function? C

A y = 4(3)x B y = -4(3)x C y = 43x D y = 4 ( 1 _ 3 )

x

22. Analyze For an exponential function with a = 5 and b = 3, why is it necessary to put parentheses around the 3 when writing the function rule? Sample: To show that the exponent of x only applies to the value of 3, and not to the product of 5 and 3.

23. Geometry The diagram shows a right triangle with a hypotenuse that is an irrational number. What set of numbers would include the hypotenuse? irrational, real numbers

(110)(110)14. Student B; Sample:Student A did not substitute the correct values for a, b,and c and did not rearrange the equation correctly.

14. Student B; Sample:Student A did not substitute the correct values for a, b,and c and did not rearrange the equation correctly.

(110)(110)

3 43 4

(109)(109)

5.91 ≤ l ≤ 6.69 3.54 ≤ w ≤ 4.335.91 ≤ l ≤ 6.69 3.54 ≤ w ≤ 4.33

(104)(104)

(105)(105)

(106)(106)

(107)(107)

(108)(108)

(108)(108)

12

1

√1

2

1

√(1)(1)


Lesson 111 759

Problem 18

Guide the students by asking the following questions.

“What is happening to the numerators of the fractions as the position of the terms increase?” Sample: the numerators stay the same

“What is happening to the denominators of the fractions as the position of the terms increase?” Sample: the denominators decrease

“By how much are the denominators of the fractions decreasing?” By one third

“What is one third of 81?” 27


Saxon Algebra 1760

24. Evaluate x2 - 8x + 15 and its factors for x = -2. 35 = (-5)(-7)

25. Oven Temperature The actual temperature (t) of Jeannine’s oven varies by no more than 9°F from the set temperature. Jeannine sets her oven to 350°F. Write an absolute-value inequality that models the possible actual temperatures inside the oven. What is the lowest possible temperature? ⎢t - 350� ≤ 9; 341 ≤ t ≤ 359; 341°F

26. Multi-Step The length of a picture is 2 inches greater than its width. A 3-inch-wide border is added to the bottom of the picture for a scrapbook page. a. Write expressions for the width and length of the

picture with the border. x + 3, x + 2

b. The area of the picture with the border is 110 square inches. Find the length and width of the original photo. 8 inches by 10 inches

27. Multi-Step Jasmine wants to plant tulips around the perimeter of her property. The property is the shape of a square. The area of the yard is 21,000 square feet and the area of the house is 1500 square feet. a. Write a formula to find the length of the sides of the property.

b. Solve for x. x = 150 ft

c. Jasmine changes her mind and decides to buy enough bulbs to plant them 6 inches apart along just one edge of the property. How many bulbs will she need if she starts at the first corner and goes to the second corner? 301 bulbs

28. Justify Explain how to simplify 6

_ √ � 5 - 7

.

29. Subtract 2x2

_ x2 - 49

- x - 7 __

x2 - 6x - 7 .

2x3 + x2 + 49__(x - 7)(x + 7)(x + 1)

*30. If f(x) = 1 _ 3 x and g(x) = 3x, which function represents exponential growth and

which function represents exponential decay?

(72)(72)

(101)(101)

(98)(98)x + 2 inches

x inches

3 inches

x + 2 inches

x inches

3 inches

(102)(102)

x2 = 21,000 + 1500x2 = 21,000 + 1500

(103)(103)

28. Sample: First, multiply by a factor of 1 using the conjugate of √ � 5 - 7, which is √ � 5 + 7. Then use the Distributive Property to multiply across the numerators, and the FOIL method to multiply across the denominators. Finally, combine like terms and simplify.

28. Sample: First, multiply by a factor of 1 using the conjugate of √ � 5 - 7, which is √ � 5 + 7. Then use the Distributive Property to multiply across the numerators, and the FOIL method to multiply across the denominators. Finally, combine like terms and simplify.

(95)(95)

(Inv. 11)(Inv. 11)g(x) represents exponential growth and f(x)

represents exponential decay.g(x) represents exponential growth and f(x)

represents exponential decay.

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Ann, Bob, Charlie, and Denise are running for president of the Movie Club. The president will get to pick his/her vice-president. However, Ann and Bob don’t know each other and won’t choose the other. Charlie only wants to work with Ann or Denise. How many ways can the president and vice-president be chosen? Use a tree diagram to solve. 8 ways: (president, vice-president)(Ann, Charlie), (Ann, Denise)(Bob, Denise)(Charlie, Ann), (Charlie, Denise)(Denise, Ann), (Denise, Bob), (Denise, Charlie)

Solving problems involving permutations prepares students for

• Lesson 118 Solving Problems Involving Combinations

• Lesson 120 Using Geometric Formulas to Find the Probability of an Event

CHALLENGE LOOKING FORWARD

760 Saxon Algebra 1

Problem 25

Extend the Problem

“Is the setting on Jeannine’s oven more accurate at 100˚F or 450˚F?”Sample: It is more accurate at 450°F since ±9°F is a smaller fraction of the temperature at 450°F than at 100°F.

Problem 27

Error Alert In part c, it is common for students to calculate the answer by 150

___ 0.5 . That divides the length to 300 equal parts, but if Jasmine wants to plant tulips 6 inches apart, she will need 300 + 1 = 301 tulips.


x

y

8

2 4-2-4

y = 14 - 4x

y = x + 4

x

y

8

16

2 4-2-4

y = x + 6

(3, 9)(-2, 4)

y = x 2

x

y

8

16

2 4-2-4

(1, 1)

y = x 2

y = 2x -1

x

y

8

16

2 4-2-4

y = x 2

y = 3x - 5

Lesson 112 761

Warm Up

112LESSON

1. Vocabulary A set of linear equations with the same variables is called a of linear equations. system

2. Write 2x2 = -x + 8 in standard form. 2x2 + x - 8 = 0

3. Solve 5x - y = 4 + 9x for y. y = -4x - 4

4. Evaluate 50 - 2x2 for x = -5. 0

5. Multiple Choice Which ordered pair is a solution of the system x - y = 7

2x + y = -1

?A

A (2, -5) B (6, -1) C ( 8 _

3 , -

13 _ 3 ) D (3, -4)

The equations y = 14 - 4x and y = x + 4 are a system of linear equations. The solution (2, 6) is a point at which the graphs of the equations intersect.

A system of equations can also consist of a linear equation and a quadratic equation. The graphs of three systems each consisting of a quadratic equation, y = x2, and a linear equation are shown.

System A

y = x2

y = x + 6

System A has two solutions because the graphs of the system intersect at two points.

System B

y = x2

y = 2x - 1

System B has only one solution.

System C

y = x2

y = 3x - 5

System C has no solution because the graphs do not intersect.

(55)(55)

(84)(84)

(29)(29)

(9)(9)

(55)(55)


Online Connection


Graphing and Solving Systems of Linear

and Quadratic Equations

Math Reasoning

Analyze What are the cordinates of the vertex for every equation of the form y = ax2?

(0, 0)

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LESSON RESOURCES




Students have learned how to solve systems of linear equations by graphing. They learned that the solution to a system of linear equations is the point at which the two lines intersect on the coordinate plane, so the ordered pair must be a solution to both equations. Solving with systems of equations consisting of linear and quadratic equations is similar to solving systems of linear equations.

However, when quadratic equations are involved, multiple solutions can be possible (including the special case of two identical equations in a system)—i.e. the graphs can intersect at multiple points.

A special case, where a line touches a parabola at a point without crossing over the parabola, is referred to as a tangent line.

MATH BACKGROUND

Warm Up1

112LESSON

Lesson 112 761

Problem 5

Remind students that the solution(s) for a system of equations must satisfy all of the equations.

2 New Concepts

In this lesson, students learn to graph and solve systems of linear and quadratic equations.

Discuss the defi nition of a system of equations. Explain that the equations must have the same variables and that the solution to the system needs to satisfy all of the equations. The equations can be linear or quadratic equations.


(-2, 4)

(2, 4)

x

y

8

16

2 4-2-4

-8

-16

y = 4

y = x 2

Saxon Algebra 1762

Example 1 Solving by Graphing

Solve each system of equations by graphing. Then check the solution.

a. y = x2

y = 4

SOLUTION Graph the parabola y = x2 and the horizontal line y = 4.

The line intersects the parabola at (2, 4) and(-2, 4). The solution of the system is the ordered pairs (2, 4) and (-2, 4).

Check

y = x2 y = x2

4 � (2)2 4 � (-2)2

4 = 4 ✓ 4 = 4 ✓

b. y = x2

y = -4x - 4

SOLUTION Graph the parabola and the line.

The line intersects the parabola at only one point. The solution is (-2, 4).

Check

y = x2 y = -4x - 4

4 � (-2)2 4 � -4(-2) - 4

4 = 4 ✓ 4 = 4 ✓

c. y = 2x2 - 9

y = 4x - 9

SOLUTION The graphs of y = 2x2 - 9 and y = 4x - 9 show two points of intersection. The coordinates of those two points are (2, -1) and (0, -9).

Check Verify that (2, -1) is a solution.

y = 2x2 - 9 y = 4x - 9

-1 � 2(2)2 - 9 -1 � 4(2) - 9

-1 = -1 ✓ -1 = -1 ✓

Verify that (0, -9) is a solution.

y = 2x2 - 9 y = 4x - 9

-9 � 2(0)2 - 9 -9 � 4(0) - 9

-9 = -9 ✓ -9 = -9 ✓

x

y

8

16

2 4-2-4

-8

-16

y = -4x - 4

(-2, 4)

y = x 2

x

y

8

16

2 4-2-4

-8

-16

y = -4x - 4

(-2, 4)

y = x 2

x

8

16

4-4

yy = 2x2

-9

(0, -9)y = 4x -9

(2, -1)

x

8

16

4-4

yy = 2x2

-9

(0, -9)y = 4x -9

(2, -1)

Caution

Be sure to check all solutions in both of the original equations of the system.


Have students copy and complete the table for the system of equations:y = x2

y = 4

x y = x2 y = 4-3 9 4-2 4 4-1 1 40 0 41 1 42 4 43 9 4

Have students use the table to fi nd the solutions. {(-2, 4), (2, 4)}

INCLUSION

762 Saxon Algebra 1

Example 1

Remind students that the solution(s) to the system of equations is where the graph of the equations intersect.

TEACHER TIPEmphasize that, in general, the graphing technique provides only approximate solutions. The estimated coordinates need to be checked by substituting them into each equation.


Solve each system of equations by graphing. Then check the solution.

a. y = x2

y = 9

x

y

8

4

4 8

-4

-4-8

y = x2y = 9

{(-3, 9), (3, 9)}

b. y = x2

y = 2x

x

y8

4

4 8

-8

-4-8

y = x2

y = 2x

{(0, 0), (2, 4)}

c. y = x2 + 2y = x

x

y

O

4

2 4

-2

-2-4

-4

y = xxy = 2+ 2

no solution


Lesson 112 763

Example 2 Solving with a Graphing Calculator

Solve each system of equations by using a graphing calculator.

a.

y = x2

_ 2 - 3

y = x - 3

SOLUTION

Enter Y1 = x2

_ 2 - 3 and Y2 = x - 3.

The display shows two graphs: a parabola and a line.

Use TRACE to approximate the solutions first. Then confirm the answers using INTERSECT.

The display shows the coordinates of the two points of intersection.

The first point of intersection is (2, -1) and the second point of intersection is (0, -3).

Check

Substitute (2, -1) into both equations.

y = x2

_ 2 - 3

-1 � (2)2

_ 2

- 3

-1 = -1 ✓

y = x - 3

-1 � 2 - 3

-1 = -1 ✓

Substitute (0, 3) into both equations.

y = x2

_ 2 - 3

-3 � (0)2

_ 2

- 3

-3 = -3 ✓

y = x - 3

-3 � 0 - 3

-3 = -3 ✓

b.

y = x2

y = 2x - 2

SOLUTION

Enter Y1 = x2 and Y2 = 2x - 2.

The display shows two graphs: a parabola and a line.

The display shows that the parabola and the line do not intersect, so there are no solutions to this system.

If the calculator is used to find a point of intersection, an error message is displayed.

Graphing

Calculator Tip

For help with graphing systems, refer to the graphing calculator keystrokes in Graphing Calculator Lab 5 on p. 352.

Math Reasoning

Verify Use a graphing calculator to show thatx2

_ 2 - 3 is the same as ( x

2

_ 2 ) - 3. Explain.

Sample: The graphs are the same. The division bar acts as a grouping symbol. Since there are no terms added to the numerator or denominator, the parentheses are not necessary.


Lesson 112 763

Example 2

Point out to students that entries into graphing calculators must be accurate for accurate results.

Error Alert Some students will forget to check the solution with both equations. Remind them that an ordered pair must be a solution to both equations to be a solution to the system.



a. y = x2 - 3y = x - 3{(0, -3), (1, -2)}

b. y = -x2

y = x + 1no solution


x

y

8

16

42-2

-16

-8

(2, 13)

(-2, -7)

y = 5x + 3

y = x2+ 5x 1

xy

2 4

-6

-2-4

-8

y = 2x - 3

(-3, -9)

(-2, -7)

y = x2+ 7x + 3

Saxon Algebra 1764

Example 3 Solving Using Substitution

Solve each system of equations by substitution.

a.

y = x2 + 5x - 1

y = 5x + 3

SOLUTION

x2 + 5x - 1 = 5x + 3 Substitute the quadratic equation into the linear equation.

____-5x - 3 ____-5x - 3 Add the expression -5x - 3 to both sides.

x2 - 4 = 0 Recognize the left side of the equation as adifference of squares.

(x + 2)(x - 2) = 0 Factor.

x + 2 = 0 and x - 2 = 0 Solve both equations.

x = -2 x = 2

Determine the corresponding values of y by substituting the values of x into either equation.

y = 5x + 3 y = 5x + 3

y = 5(-2) + 3 y = 5(2) + 3

y = -10 + 3 y = 10 + 3

y = -7 y = 13

The solutions are the ordered pairs (-2, -7) and (2, 13). The solutions appear at the intersections of the two graphs.

b. y = x2 + 7x + 3 y = 2x - 3

SOLUTION

x2 + 7x + 3 = 2x - 3 Substitute the quadratic equation into the linear equation.

____-2x + 3 ____-2x + 3 Add the expression -2x + 3 to both sides.

x2 + 5x + 6 = 0 Recognize the left side of the equation as atrinomial that can be factored.

(x + 3)(x + 2) = 0 Factor.

x + 3 = 0 and x + 2 = 0 Solve both equations.

x = -3 and x = -2

Determine the values of y.

y = 2x - 3 y = 2x - 3

y = 2(-3) - 3 y = 2(-2) - 3

y = -6 - 3 y = -4 - 3

y = -9 y = -7

The solutions are the ordered pairs (-3, -9) and (-2, -7).

Math Reasoning

Verify Show that the ordered pairs (-2, -7)

and (2, 13) are solutions to Example 3a and that the ordered pairs (-2, -7) and(-3, -9) are the solutions to Example 3b.

Example 3a:y = x2 + 5x - 1

-7 � (-2)2 +

5(-2) - 1-7 � 4 - 10 - 1-7 = -7 �

y = x2 + 5x - 113 � (2)2 + 5(2) - 113 � 4 + 10 - 113 = 13 �

Example 3b: y = x2 + 7x + 3-7 � (-2)2 +

7(-2) + 3-7 � 4 - 14 + 3-7 = -7 �

y = x2 + 7x + 3-9 � (–3)2 + 7(–3)

+ 3-9 � 9 - 21 + 3-9 = -9 �

SM_A1_NL_SBK_L112.indd Page 764 1/18/08 6:47:49 PM epg1 /Volumes/ju114/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L11

764 Saxon Algebra 1

Example 3

Point out to students that since a system of equations will have the same variables, they can substitute one into the other.



a. y = x2 + 3x + 3y = -x - 1 {(-2, 1)}

b. y = x2 + 2y = 4x - 1{(1, 3), (3, 11)}


Lesson 112 765

Example 4 Application: Avalanches

A ski patrol fires an explosive arrow to trigger a controlled avalanche. The path of the arrow is modeled by the equation y = - x2

_ 1600

+ 2x and the shape of the mountainside is modeled by y = 3x

_ 4 where x is the horizontal

distance and y is the vertical distance. At what altitude will the arrow strike the mountain? (Assume all dimensions are in feet.)

SOLUTION

Understand The path of the arrow is modeled by a parabola. The mountainside is modeled by a straight line.

y = - x2

_ 1600

+ 2x

y = 3x

_ 4

Plan The equation of the arrow’s path and the equation of the shape of the mountain form a system of equations.

Solving this system will determine the points at which the two graphs intersect.

Solve One way of solving the system isby graphing the two equations. The cannon is located at the base of the mountain, so both graphs pass through (0, 0). The non-origin solution to the system is (2000, 1500). The altitude at which the arrow will strike the side of the mountain is 1500 feet.

Check

y = 3x _ 4 y = -

x2

_ 1600

+ 2x

1500 � 3(2000)

_ 4 1500 � -

20002

_ 1600

+ 2(2000)

1500 � 6000 _

4 1500 � -

4,000,000 _

1600 + 4000

1500 = 1500 ✓ 1500 � -2500 + 4000

1500 = 1500 ✓

Lesson Practice

Solve each system of equations by graphing.

a.

y = x2

y = 16

b.

y = x2

y = 6x - 9

c.

y = x2

y = -2x + 3

mountain

cannon

path of arrow

mountain

cannon

path of arrow

x

y

1000 2000

500

0

1000

1500

Horizontal distance (ft)

Ve

rtic

al d

ista

nc

e (

ft)

x

y

1000 2000

500

0

1000

1500

Horizontal distance (ft)

Ve

rtic

al d

ista

nc

e (

ft)

(Ex 1)(Ex 1)

a.

x

y

4

8

12

2 4-2-4

(-4, 16) (4, 16)a.

x

y

4

8

12

2 4-2-4

(-4, 16) (4, 16)

b.

x

y

4

8

12

2 4-2-4

(3, 9)

b.

x

y

4

8

12

2 4-2-4

(3, 9)

See Additional Answers.

Math Reasoning

Write Describe the meaning of the x-coordinate in the solution (2000, 1500).

Sample: The arrow will strike the mountain at (2000 ft, 1500 ft).The arrow will be 2000 horizontal feet from the cannon when it strikes the mountain.


Lesson 112 765

Example 4

Point out to students that the intersection point of the equation that describes the path of the arrow and the slope of the mountain, when x > 0, is the solution.

Extend the Example

Have students solve the system algebraically. Remind them to use substitution and have them show their work. Sample:

- x2 _____

1600 + 2x = 3x ___

4

x2 - 3200x = -1200x

x2 - 2000x = 0

x(x - 2000) = 0

x = 0 or 2000

y = 3(2000)

_______ 4 = 1500


If two arrows fi red at the same position can be respectively modeled by y = - x2

_____ 1600

+ 2x and

y = - x2 ____

400 + 6x where x is the

horizontal distance and y is the vertical distance. At what approximate altitude will their two paths cross? 1422 feet



Saxon Algebra 1766


d. y = x2

_ 2 + 1 e. y = -2x2 - 1

y = - 3x _ 2

y = -x - 2

(-1, 1.5) and (-2, 3)

(1, -3) and (-0.5, -1.5)


f.

y = x2 - 3x - 17

y = -3x + 8

g.

y = x2 + 7x + 5

y = 2x - 1

(5, -7) and (-5, 23) (-2, -5) and (-3, -7)

h. Physics A gardener places a sprinkler at the bottom of a gently rising hillside described by the equation y = 2x

_ 5 . The equation y = - x

2

_ 25

+ x represents the path of the water. If the water splashes onto a rock on the hillside, what is the rock’s altitude? (Assume all dimensions are in feet.) 6 feet

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

*1. Solve this system by graphing: y = -x2 + 12

y = -x + 6

.

*2. Multiple Choice Which system of equations has no solution? C

A y = x2 + 2

y = 3

B y = x2 - 2

y = 3

C y = -x2 + 2

y = 3

D y = -x2 - 2

y = -3

3. Simplify the rational expression c -2 f -5 + 6

_ c2f 5 , if possible. 7_c2f 5

4. Write a compound inequality that represents all real numbers that are greater than -4 and less than 8. -4 < x < 8

*5. Architecture In a European castle, a room with an arched ceiling is covered by a slanted roof. The ceiling is modeled by the equation y = -x2 + 4 and the roofline by the equation y = -2x + 5. Assume that the dimensions are in meters. What are the coordinates for the point of intersection of the roof with the ceiling assuming that the vertex of the parabola is (0, 4)? (1, 3)

*6. Analyze A system of three equations consists of the quadratic equation y = x2 and two linear equations that do not describe the same line. What is the maximum number of ordered pairs in the solution set? Explain.

7. A six-sided number cube is rolled three times. How many outcomes are possible?216

*8. Error Analysis Two students are finding the value of 6P6. Which student is correct? Explain the error. Student B; Sample: 0! = 1, not 0.

Student A

6P6 = 6! _

(6 - 6)!

= 6! _ 0!

= undefined

Student B

6P6 = 6! _

(6 - 6)!

= 6! _ 0!

= 720 _

1 = 720

(112)(112)

y

4

8

2 4-2-4

(-2, 8)

(3, 3)x

y

4

8

2 4-2-4

(-2, 8)

(3, 3)x

(112)(112)

(51)(51)

(73)(73)

(112)(112)

(112)(112)

6. one; Sample: Because the two linear equations can only intersect at one point, that point must also be the point where they both intersect the parabola. So, the maximum number of points of intersection for all three equations is one.

6. one; Sample: Because the two linear equations can only intersect at one point, that point must also be the point where they both intersect the parabola. So, the maximum number of points of intersection for all three equations is one.

(111)(111)

(111)(111)

1.


Explain the meaning of the word sprinkler. Say:

“A sprinkler is a lawn tool that waters an area of lawn by spraying water.”

ENGLISH LEARNERS

Draw the parabolic path of a sprinkler on the board and show how it is able to water an area of a lawn. Say:

“What do the paths of the water from a lawn sprinkler look like?” Sample: a parabola

766 Saxon Algebra 1

Lesson Practice

Problem e

Error Alert When entering the coeffi cients for equations on a graphing calculator, students sometimes will make the mistake of forgetting the minus sign of negative coeffi cients. Remind them to be careful and to double-check their inputs.

Problem f

Scaff olding After solving for x, suggest that students substitute the value of x into the linear equations to get y.



“Explain what the possible solutions are for a system of a linear and a quadratic equation.” Sample: two points, one point, or no solution

“If an ordered pair is a solution to a system of two equations, what does that mean when the two equations are graphed?” Sample: The ordered pair will be the point where the two equations intersect.

Practice3


Problem 3

Remind students that x-a is the same as 1

___ xa .


Lesson 112 767

9. Geometry A triangle can be classified according to its sides or according to its angles. There are three side length categories—equilateral, isosceles, and scalene—and three angle categories—acute, obtuse, and right. a. How many possibilities are there for classifying triangles according to both

sides and angles? 9

b. How many of these triangles are not possible? Which ones are they? 2; equilateral obtuse triangle and equilateral right triangle

10. Multi-Step There are 7 runners on the track team. Runners will be selected randomly for the first, second, third, and final positions on the 4-member relay team. a. How many different relay teams can be formed? 840 relay teams

b. What is the probability that a runner at random is chosen to be on the relay team? 4_7

11. Use the quadratic formula to solve x2 - 60 + 17x = 0. Check the solutions. 3, -20

12. Multiple Choice What are the solutions to 2a2 + 20a - 30 = 0 ? C

A 20 ± 4 √ � 10 B -20 ± √ � 10

C -5 ± 2 √ � 10 D -5 ± √ � 10

* 13. Measurement A rectangle has sides of length x feet and 2x + 2 feet with an area of 24 square feet. Cassandra uses the quadratic formula and finds that x equals 3 and -4. She determines that this means the sides of the rectangle are -4 by -6 or 3 by 8. Why is she incorrect? Sample: She’s using measurements, therefore negative values of x are irrelevant.

14. Construction Suzanne would like to place a fence around her rectangular yard, which has a perimeter of 200 feet. The fencing for the front length of the house will cost $5 per foot and the fencing for the side and back of the yard will cost $3 per foot. Her total cost is $720. What are the dimensions of her property? 40 feet by 60 feet

15. Find the next 3 terms of the sequence -0.032, 0.16, -0.8, 4, …. -20, 100, -500

16. Paper Folding Solange folds a piece of paper, making two rectangles. When she folds it again, she makes 4 rectangles. Each fold doubles the number of rectangles. A sequence describing this process is 2, 4, 8, …. If someone folds a piece of paper is 12 times, how many rectangles did the 12 folds form? 4096

17. Solve the equation √ � x

_ 6 = 12. Check your answer. x = 5184; √ �� 5184_

6= 72_

6=12

18. Population The exponential function y = 11.35(1.00183)x can model the approximate population of Ohio from 2000 to 2006, where x is the number of years after 2000 and y represents millions of people. What was the population in 2003? 11,412,000 people

19. Evaluate the function f(x) = -2(4)x for x = -2, 0, and 2. - 1_8, -2, -32

20. Multiple Choice Which system has no solutions? D

A y < 2

y < 1

B y > 2

y > 1

C y < 2

y > 1

D y > 2

y < 1

(111)(111)

(111)(111)

(110)(110)

(110)(110)

3 43 4

(110)(110)

(110)(110)

(105)(105)

(105)(105)

(106)(106)

(108)(108)

(108)(108)

(109)(109)


Lesson 112 767

Problem 10

Guide students by asking them the following questions.

“How can the question be rephrased using the word permutation?” Sample: How many permutations of 4 runners can be made from 7 total?

“What is the permutation formula? What are the values of n and r?” nPr = n! ______

(n - r)! ; n = 7, r = 4

Problem 12


“When should the factoring method not be used to solve a quadratic equation?” Sample: when factors are diffi cult to fi nd

“Name a method, other than factoring, that can be used to solve this equation.” Sample: completing the square, substituting into the quadratic formula


Saxon Algebra 1768

21. Analyze What inequality symbols should go into the boxes so that the solution set lies between the lines and does not include the boundary points?

y 3 _ 5 x + 7

y 3 _ 5 x + 1

22. Find the zeros of the function y = x2 - 6x - 72. 12, -6

23. Graph the inequality 4x - y ≤ -5.

24. Multi-Step A girl takes 4 hours to complete a job. Her mother can complete the same job in 3 hours. Her little sister takes 6 hours to complete it. a. Write an equation representing how long it takes the three of them to complete

the job working together. t_4

+ t_3

+ t_6

= 1

b. How long will it take to complete the job, in hours, if all three famiy members work together? 4_3 hours

c. How many minutes is that? 80 minutes

25. Biking Dustin and Roberto leave their house at the same time. Dustin rides his bike 49 feet east. Roberto rides his bike 81 feet south. Use the formula (49)2 + (81)2 = x2 to find the distance between Dustin and Roberto. ≈ 94.667 ft

*26. Formulate In the system y = x2 - 3

y = a , a is a real number. What is the minimum value

of a so that the system will have two solutions? a > -3

27. Multi-Step A race-car driver is driving at a rate of √ �� 10,800 miles per hour. How long does it take the driver to go 85 miles? Give the answer as a rational expression in simplest form. (Hint: distance = rate times time) a. Write the equation to find the driver’s travel time using the given values.

b. Find the solution. t = 17√ � 3_

36

28. Write Tell how to remove any coefficients of the x2-term in a quadratic equation before completing the square.

*29. Describe the transformation of f (x) = -x2 + 2 from the parent quadratic function.

*30. Charlotte invested $1000 in an account that doubles her balance every 7 years.Does this situation model exponential growth or decay? Express the function that represents this situation. After 42 years, how many times will her balance have doubled? What will that balance be after 42 years? exponential growth; f(x) = 1000 · 2x; 6; $64,000

(109)(109)

(96)(96)

(97)(97)

4

x

8

4 8-4-8

-8

4

x

8

4 8-4-8

-8

(99)(99)

49 ft

81 ftx

49 ft

81 ftx

(102)(102)

(112)(112)

(103)(103)

t = 85_√ �� 10,800

t = 85_√ �� 10,800

(104)(104)

28. Sample: Divide each term of the quadratic equation by the coefficient of the quadratic term. The coefficient of the quadratic term must be 1 in order to complete the square.

28. Sample: Divide each term of the quadratic equation by the coefficient of the quadratic term. The coefficient of the quadratic term must be 1 in order to complete the square.

(Inv 10)(Inv 10)

29. The graph of the function is reflected about the x-axis (opens downward) and is shifted up two units.

29. The graph of the function is reflected about the x-axis (opens downward) and is shifted up two units.

(Inv 11)(Inv 11)

Sample: The first inequality should have < and the second inequality should have >.

23.


A system of equations can consist of two quadratic equations. The solution(s) to these systems will still be at the points where the graphs intersect. Solve the system.

y = x2 + 2y = -x2 + 4{(1, 3), (-1, 3)}

Graphing and solving systems of linear and quadratic equations prepare students for

• Lesson 119 Graphing and Comparing Linear, Quadratic, and Exponential Functions


768 Saxon Algebra 1

Problem 24

Extend the Problem

“If the girl’s father takes 2 hours to complete the job and joins the three of them, how long will it take the four of them to fi nish the job?”t __ 4 + t __

3 + t __

2 + t __

6 = 1;

t = 4 __ 5 hours or 48 minutes.

Problem 28

Error AlertA common mistake that students make when they divide to remove the negative coeffi cients of the x2-term in quadratic equations is forgetting to change the signs of the other coeffi cients.


Lesson 113 769

Warm Up

113LESSON

1. Vocabulary The is the number or expression under a radical symbol. radicand

Evaluate each expression for the given values.

2. -x2 - xy - y for x = -5 and y = -1 -29

3. b2 + 3ab - a for a = -7 and b = -2 53

4. ab - 5b2 for a = 3 and b = 4 -68

5. 7y2z + 9 for y = -3 and z = -1 -54

The quadratic formula is one method used to solve quadratic equations. Recall the quadratic formula for a quadratic equation of the form ax2 + bx + c = 0 is:

x = -b ± √ �� b2 - 4ac

__ 2a

In the formula, the expression under the radical sign, b2 - 4ac, is called the discriminant.

Consider the graphs below and the value of the discriminant for each equation.

0 = x2 - 4x + 3

b2 - 4ac a = 1, b = -4, c = 3

= (-4)2 - 4(1)(3) Substitute.

= 4

There are 2 x-intercepts. The discriminant is positive.

0 = x2 - 4x + 4

b2 - 4ac a = 1, b = -4, c = 4

=(-4)2 - 4(1)(4) Substitute.

= 0

There is one x-intercept. The discriminant is zero.

0 = x2 - 4x + 5

b2 - 4ac a = 1, b = -4, c = 5

(-4)2 - 4(1)(5) Substitute.

= -4

There are no x-intercepts. The discriminant is negative.

(13)(13)

(9)(9)

(9)(9)

(9)(9)

(9)(9)


x

y

O

4

4 8

-4

-4-8

-8

x

y

O

4

4 8

-4

-4-8

-8

x

y

O

4

4 8

-4

-4-8

-8

x

y

O

4

4 8

-4

-4-8

-8

x

y

O

4

4 8

-4

-4-8

-8

x

y

O

4

4 8

-4

-4-8

-8

Interpreting the Discriminant

Online Connection



LESSON RESOURCES


Reteaching Master 113Adaptations Master 113 Challenge and Enrichment

Master C113

MATH BACKGROUND

When the discriminant is a perfect square, there are one or two solutions to the equation a x 2 + bx + c = 0 because if b 2 - 4ac ≥ 0, the square root can be taken, and it may be positive or negative. When factoring trinomials, the discriminant can be used to determine if the trinomial is factorable. If a polynomial with integer coeffi cients has a discriminant that is a

perfect square, that polynomial is factorable over the integers. This knowledge can save time if students are having diffi culty fi nding factors that work.

Warm Up1

113LESSON

Lesson 113 769

Problems 2–5

Remind students that it is a good habit to place negative values in parentheses to avoid confusion when substituting into expressions.

2 New Concepts

In this lesson, students learn to interpret the discriminant in the quadratic formula.

Discuss the defi nition of the discriminant. Explain that the discriminant can be used to fi nd the number of solutions to a quadratic equation, which represents the number of x-intercepts for its graph.


Saxon Algebra 1770

The value of the discriminant indicates the number of solutions.

Using the Discriminant

For the quadratic equation ax2 + bx + c = 0 where a ≠ 0, find the value of the discriminant, b2 - 4ac, to determine the number of real solutions, which represents the number of x-intercepts of the graph of its related function.

If b2 - 4ac < 0, then there are no real solutions and no x-intercepts.

If b2 - 4ac = 0, then there is one real solution and one x-intercept.

If b2 - 4ac > 0, then there are two real solutions and two x-intercepts.

If b2 - 4ac = 0, then there is one real solution, which means there is one x-intercept. The real solution is the x value at the vertex of the parabola, which will be on the x-axis. The solution is called a double root of the equation.

Example 1 Finding the Number of Solutions Without Solving

Use the discriminant to find the number of real solutions to the equation. Then state the number of x-intercepts of the graph of the related function.

a. x2 - 3x + 9 = 0

SOLUTION

b2 - 4ac

= (-3)2 - 4(1)(9) Substitute.

= 9 - 36 Simplify.

= -27

There are no real solutions, so the graph has no x-intercepts.

b. 2x2 - 3x - 4 = 0

SOLUTION

b2 - 4ac

= (-3)2 - 4(2)(-4) Substitute.

= 9 + 32 Simplify.

= 41

There are two real solutions, so the graph has two x-intercepts.

c. x2 + 8x + 16 = 0

SOLUTION

b2 - 4ac

= 82 - 4(1)(16) Substitute.

= 64 - 64 Simplify.

= 0

There is one real solution, so the graph has one x-intercept.

Math Reasoning

Analyze What does the discriminant tell about the real solutions of a quadratic equation? What does the discriminant not tell about the solutions of a quadratic equation?

Sample: The discriminant tells the number and nature of the solutions of a quadratic equation. The discriminant does not tell the actual values of the solutions of a quadratic equation.

SM_A1_NLB_SBK_L113.indd Page 770 5/3/08 1:07:41 AM elhi-2 /Volumes/ju115/HCAC057/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L113

770 Saxon Algebra 1

Example 1

Point out to students that, to avoid confusion, they should always write quadratic equations in standard form before calculating the discriminant.


Use the discriminant to fi nd the number of real solutions to the equation. Then state the number of x-intercepts of the graph of the related function.

a. x2 - 3x + 4 = 0 b2 - 4ac = -7; there are no real solutions; y = x2 - 3x + 4 has no x-intercepts

b. 2x2 + 2x - 5 = 0b2 - 4ac = 44; y = 2x2 + 2x - 5 has two x-intercepts and two real solutions

c. x2 + 2x + 1 = 0b2 - 4ac = 0; y = x2 + 2x + 1 has one x-intercept and one real solution

Error Alert When calculating the discriminant, students can sometimes have problems keeping track of the sign of the -4ac term. Remind them that the term is negative to start with and will change signs every time a negative value in a or c is introduced.

TEACHER TIPHave students confi rm the number of solutions by graphing the equations on their graphing calculators. Remind students that the solutions are the x-intercepts of the graph of the function.



Lesson 113 771

Example 2 Application: Baseball

A baseball is thrown in the air with an initial velocity of 20 feet per second from 5 feet off the ground. Use the equation h = -16t2 + 20t + 5 to model the situation. Will the ball reach a height of 30 feet?

SOLUTION

h = -16t2 + 20t + 5

30 = -16t2 + 20t + 5 Substitute 30 for h.

0 = -16t2 + 20t - 25 Set the equation equal to 0.

Use the discriminant to determine if the ball will reach a height of 30 feet.

b2 - 4ac = 202 - 4(-16)(-25)

= 400 - 1600

= -1200

Since the discriminant of the equation is negative, there are no solutions. The ball will not reach a height of 30 feet.

Lesson Practice

Use the discriminant to find the number of real solutions to the equation. Then state the number of x-intercepts of the graph of the related function.

a. x2 - 2x - 35 = 0 144; 2 real solutions, 2 x-intercepts

b. 4x2 + 20x + 25 = 0 0; 1 real solution, 1 x-intercept

c. 2x2 - 3x + 7 = 0 -47; no real solutions, no x-intercepts

d. A football is punted from 2 feet off the ground with an initial velocity of 60 feet per second. Use the equation y = -16t2 + 60t + 2 to model the situation. Will the ball reach a height of 45 feet? The discriminant is 3728, so there are 2 real solutions. The ball will reach a height of 45 feet because its maximum height is 58.25 feet.

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

*1. Find the value of the discriminant of the equation 3x2 - x + 2 = 0. -23

2. The new rectangular basketball court at the high school has a width of 9x2 + x + 36 and a length of 4x2 + 2x + 2. What is the perimeter of the new court? 26x2 + 6x + 76

3. Solve 6⎢z - 3� = 18. {0, 6}

4. Find 8!. 40,320

(113)(113)

(53)(53)

(74)(74)

(111)(111)

Math Reasoning

Generalize What are the values of a, b, and c in the quadratic equation x2 -4 = 0?

a = 1, b = 0, and c = -4


For this lesson, explain the meaning of the word discriminant. Say:

“A discriminant is a characteristic that enables things, people, or classes to be distinguished from one another.”

Connect the meaning of the vernacular use of discriminant with its mathematical use

by explaining that the value of b2 - 4ac distinguishes the type of solution that a quadratic equation has.

Discuss the defi nition of “discriminate.” Explain that discriminate does not necessarily imply a negativity or injustice.

ENGLISH LEARNERS

Lesson 113 771

Example 2

Point out that y is the height of the baseball and t is the time after the ball is thrown.

Extend the Example

“What is the maximum height the baseball will reach?” 11.25 feet


If a baseball is thrown in the air with an initial velocity of 45 feet per second from 5 feet off the ground, will the ball reach a height of 30 feet? yes

Lesson Practice

Problems a–c

Scaff olding Before calculating the discriminant, suggest that students write down what a, b, and c are equal to.

Problem d

Error Alert For this particular problem, students may forget that y is the height and leave out the 45 in the equation 45 = -16t2 + 60t + 2.



“What is the standard form for a quadratic equation?” ax2 + bx + c = 0

“What is the discriminant?” the radicand of the quadratic formula, b2 - 4ac

“What does the discriminant tell you about the actual values of the solutions of the quadratic equation?” Sample: Nothing, it only tells you the number and nature of the solutions.


Saxon Algebra 1772

*5. Multiple Choice Which is a possible value for the discriminant of the equation graphed? A

A -5 B 0

C 3 D 5

*6. Model Draw the graph of a quadratic equation that has a discriminant that is greater than zero.

*7. Generalize Describe the values of the discriminant that indicate two real solutions. Sample: all positive values for b2 - 4ac

*8. Solve this system y = -

x2

_ 2 + 8

y = -2x + 10

by graphing.

*9. Error Analysis Two students are solving the system of equations y = x2 + 3

y = -3x + 1

by substitution. Which student is correct? Explain the error.

Student A

y = x2 + 3 y = -3x + 1x2 + 3 = -3x + 1

x2 + 3 + 3x - 1 = 0x2 + 3x + 2 = 0

(x + 2)(x + 1) = 0 So, x = -2, x = -1, and the solutions are (-2, 7) and (-1, 4).

Student B

y = x2 + 3 y = -3x + 1x2 + 3 - 3x + 1 = 0

x2 - 3x + 4 = 0no solution

10. Geometry For safety reasons, a guy wire must connect the top of a utility pole to the ground at a particular angle. The utility pole is located at the base of a hill described by the equation y = - x

2

_ 25

+ 2x. The equation for the correct angle of the wire is y = -x + 14. At what altitude on the hill should the ground stake be located? (Assume all dimensions are in feet.) 9 feet

*11. In designing a necklace, a goldsmith places a gold wire on a workbench so that the wire takes on the shape of a parabola described by the equation y = x

2

_ 2 . The

goldsmith then lays a straight wire across the first so that the second follows the equation y = x _

6 + 6. Use a graphing calculator to determine the coordinates for the

points of intersection. Round answers to the nearest whole number.

(113)(113)

(113)(113)

6. Sample:6. Sample:

(113)(113)

(112)(112)

8.

x

y

O

4

6

8

10

12

14

2

2 4

-2

-2-4

(2, 6)

8.

x

y

O

4

6

8

10

12

14

2

2 4

-2

-2-4

(2, 6)(112)(112)

9. Student A; Sample: Student B added the linear equation to the quadratic equation rather than substituting it for y.

9. Student A; Sample: Student B added the linear equation to the quadratic equation rather than substituting it for y.

(112)(112)

(112)(112)

(-3, 5) and (4, 7)(-3, 5) and (4, 7)


Use the following strategy with students who have diffi culty with large group instruction. After students determine how many x-intercepts a quadratic equation has by calculating the discriminant, have them check by graphing it on their graphing calculators.

INCLUSION

772 Saxon Algebra 1

Practice3


Problem 5

Emphasize that the discriminant only sheds light on the number and nature of solutions for the quadratic equation and tells nothing of the values of the solutions.

Problem 10

Point out to students that there are other ways they can solve this problem: by graphing or substitution.


Lesson 113 773

*12. Error Analysis Two students are finding the number of ways to choose a president and a vice president from a list of eight candidates. Which student is correct? Explain the error. Student A; Sample: Student B did not use the correct formula for permutations.

Student A

8P2 = 8! _

(8-2)!

= 8! _ 6!

= 56

Student B

8P2 = 8! _ 2!

= 20,160

13. Dining A restaurant offers a choice of 3 sandwiches, 3 chips, and 5 soft drinks. How many different meal combinations are offered? 45

14. Probability A CD has 9 tracks. The CD player is set to play the songs randomly so that each song plays only once. What is the probability that the first 3 songs are the first 3 tracks in order? 1_

504

15. Solve the equation √ � x + 2 = 8. Check your answer. x = 36; √ �� 36 + 2 = 6 + 2 = 8

16. Architecture An architect is designing a structure that merges two different right triangles along the hypotenuse of each triangle. The hypotenuse of one triangle is √ �� x + 2 units long and the hypotenuse of the second is √ �� 2x - 4 . At what value of x are the two lengths equal? x = 6

17. Graph the function f(x) = ⎢x + 4�.

*18. Multi-Step A plot of land is 143 square feet with dimensions of x and x + 2. What is the perimeter of the plot of land? a. Use the quadratic formula to find the dimensions of the plot of land.

b. What is the perimeter of the plot of land? 48 feet

19. Multi-Step Emmanuel throws a football into the air. Its movement forms a parabola given by the quadratic equation h = -16t2 + 14t + 50, where h is the height in feet and t is the time in seconds. a. Find the time t when the ball is at its maximum height. Round to the nearest

hundredth. t = 0.44 seconds

b. Find the time t when the ball hits the ground. Round to the nearest hundredth. t = 2.26 seconds

c. Find the maximum height of the arc the ball makes in its flight. Round to the nearest tenth. h = 53.1 feet

20. Firefighting A forest ranger is stationed at the Delilah Lookout fire tower in the Sequoia National Forest in California. The distance d (in miles) he can see to

the horizon can be estimated by the formula d = √ � 3h

_ 2 , where h is the height of

the observer’s eyes (in feet) above sea level. If Delilah Lookout is located at an elevation of 5176 feet above sea level, write a radical expression that shows the distance the ranger can see to the horizon. 2√ �� 1941 feet

(111)(111)

(111)(111)

(111)(111)

(106)(106)

(106)(106)

(107)(107)

17.

x

y

O

8

4 8

-4

-4-8

-8

17.

x

y

O

8

4 8

-4

-4-8

-8

(110)(110)

11 feet by 13 feet11 feet by 13 feet

(100)(100)

(103)(103)


Lesson 113 773

Problem 14

Extend the Problem

“Three of the tracks on the CD are instrumentals. What is the probability that the fi rst three songs are all instrumentals?”

3 __ 9 · 2 __

8 · 1 __ 7 = 1 __

3 · 1 __

4 · 1 __ 7 = 1 ___

84

Problem 17

Error Alert Some students may translate the parent function f (x) = ⎪x⎥ four units to the right since there is a plus sign in front of the 4. Show them that since the function to be graphed isf (x) = ⎪x + 4⎥ then -4 is substituted for h in the function f (x) = ⎪x - h⎥ . Therefore, the parent function is translated 4 units to the left.


Saxon Algebra 1774

21. Graph the system: y ≥ -

3 _ 5 x + 3

y ≥

3 _

4 x + 3

.

22. Analyze Compare -4.9t2 + v0t + y0 = 0 and -4.9t2 + v0t = 0.

23. Write the inequality that is graphed on the coordinate plane. y > x - 6

*24. Projectile Motion A projectile is shot up in the air from the ground with an initial velocity of 84 feet per second. Using y = -16t2 + 84t, write an equation to model the situation and use the discriminant to determine if the projectile will reach a height of 200 feet. Use the equation 200 = -16t2 + 84t. Since 0 = -16t2 + 84t - 200 and the discriminant is -5744, the projectile will not reach 200 feet.

25. Find the roots of 36x = 9x2 + 36. 2

26. Finance The amount of money Ricardo has after x years of investing $100 at his local bank is f(x) = 100 (1.065)

x . Which graph could represent this function? A

x

y

O

80

40

4 8-4-8

B

x

y

O

160

240

8 16-8-16

C x

y

O8 16-8

-160

-240

27. Multi-Step The time in minutes t it takes for a projectile to strike the ground is described by the equation -4.9t2 - 29.4t + 34.3 = 0. a. Write the quadratic equation in the form x2 + bx = c. t2 + 6t = 7

b. Find the real-number solutions by completing the square. t = -7 or 1

c. At what time does the object strike the ground? Explain your answer. 1 minute; Time cannot be negative.

28. Verify The fifth term of a geometric sequence is -1. The first is -81. Randy thinks the common ratio is 1 _

3 . Robin says it could be - 1 _

3 . Could both be correct? Explain.

29. If a quadratic function has been vertically compressed, does that mean the parabola is wider or narrower than the parent quadratic function f(x) = x2? wider

30. For all real values of the domain, describe the relationship between the graphs of an exponential growth and an exponential decay function. They are mirror images of each other reflected about the y-axis.

(109)(109)

(110)(110)

22. Sample: The first equation has a variable for the initial height while the second equation assumes that the initial height is 0.

22. Sample: The first equation has a variable for the initial height while the second equation assumes that the initial height is 0.

(97)(97)

4

-8

x

y

O

8

8

-4

-4

4

-8

x

y

O

8

8

-4

-4

(113)(113)

(98)(98)

(108)(108)BB

(104)(104)

(105)(105)

28. yes; Sample: If the common ratio is 1_3, the fifth term is -81( 1_

3 )4 = -1. If the common

ratio is - 1_3, the fifth term is -81(- 1_

3 )4 = -1.

28. yes; Sample: If the common ratio is 1_3, the fifth term is -81( 1_

3 )4 = -1. If the common

ratio is - 1_3, the fifth term is -81(- 1_

3 )4 = -1.

(Inv 10)(Inv 10)

(Inv 11)(Inv 11)

21.

2

x

y

O

4

2 4

-2

-2

-4


Mathematicians defi ne the square root of a negative number to be “imaginary”. The letter i is used to denote the square root

of -1. So, √ �� -1 = 1i. For example, √ �� -4

= 2i, √ �� -9 = 3i, and so forth. Use the discriminant to fi nd the number of real number solutions of the quadratic equation x 2 + 1 = 0 . If there are no real number solutions, then fi nd the imaginary solutions. ±i

Interpreting the discriminant prepares students for



774 Saxon Algebra 1

Problem 23

Error AlertA common mistake that students make in identifying or graphing an inequality on the coordinate plane is the incorrect use of the solid or dotted line. Remind them that the dotted line is not included in the shaded region and is drawn when < or > is used.

Problem 25


“What is the standard form for this equation?” 9x2 - 36x + 36 = 0

“How can 9x2 - 36x + 36 = 0 be simplifi ed?” Sample: divide both sides by 9; x2 - 4x + 4 = 0

“How many real roots does it have?” Sample: The discriminant is zero, therefore it has one double root.

“How would you determine the root?” Sample: use the quadratic formula; factor the equation; graph the equation


Lab 10 775

L AB

10A graphing calculator can be used to graph radical functions and to locate points on the graph.

Graph the function y = 2 √ �� x - 1.

1. To enter the equation into the Y= editor, press

the key. Then press 2

1 .

2. Graph the function by pressing 6:ZStandard.

3. Press and use the key to move along

the x-axis until the cursor locates a point on the graph.

The first point on the graph appears to be (1.064, 0.505).

4. Investigate actual points on the graph of

y = 2 √ �� x - 1 .

Press 0 1. Then press

.

The first point on the graph is (1, 0). Note that this is different from (1.064, 0.505); the graph does not appear to pass through the point (1, 0). Therefore, it is important to use the Table feature to determine values of the function.

Lab Practice

a. Graph the function y = 3 √ �� x + 2 . At what point does the graph start?

b. Graph the function y = 2 √ �� x + 2 . At what point does the graph start?

(-2, 0)(-2, 0)

a.a.

(-2, 0)(-2, 0)

b.b.

Graphing Radical Functions

Graphing Calculator Lab (Use with Lesson 114)

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Lab 10 775

10LAB

Materials

• graphing calculator

Discuss

Graphing radical equations may be the fi rst time students consider the domain of the equation using the trace and table features.

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Saxon Algebra 1776

Warm Up

114LESSON

1. Vocabulary Radicals that have the same radicands and roots such as 2 √ � 7 + 4 √ � 7 are ________, and radials that have different radicands and/or roots such as 4 √ � 7 + 2 √ � 11 are ________. like radicals, unlike radicals

Add or subtract.

2. -6 √ � 2 + 8 √ � 2 2√ � 2 3. 31 √ � 5 - 13 √ � 5 18√ � 5

Find each product.

4. (7 + √ � 6 )(4 - √ � 9 ) 7 + √ � 6 5. ( √ � 3 - 12)2 147 - 24 √ � 3

The square root of a number x is the number whose square is x.

√ � 9 = 3 32 = 9

The square root of x can be a function. For the function y = √ � x when x is 9, y is 3 since the square root of 9 is 3. Use the table to make connections with the graph.

x y

0 0

1 1

4 2

9 3

16 4

x

y

O

4

6

2

8 124

A square-root function is a function that contains a square root of a variable.

Example 1 Graphing a Square-Root Function

Make a table of y = 2 √ � x + 1. Then graph the function.

SOLUTION

Evaluate the function when x is 0, 1, 4, and 9.

y = 2 √ � 0 + 1 = 2(0) + 1 = 1

y = 2 √ � 1 + 1 = 2(1) + 1 = 3

y = 2 √ � 4 + 1 = 2(2) + 1 = 5

y = 2 √ � 9 + 1 = 2(3) + 1 = 7

In order for a square root to be a real number, the radicand cannot be negative.

(69)(69)

(69)(69) (69)(69)

(76)(76) (76)(76)


x y

0 1

1 3

4 5

9 7

x y

0 1

1 3

4 5

9 7x

y

O

4

6

2

4 62 8

x

y

O

4

6

2

4 62 8

Graphing Square-Root Functions

Online Connection


Math Language

A function is a mathematical relationship that pairs each value in the domain with exactly one value in the range.

Hint

Try choosing x values that are perfect squares. This may make it easier to graph.

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LESSON RESOURCES



Master C114

The domain of a square-root function is the set of real numbers for which the radical expression is a real number. The radicand must be greater than or equal to zero to ensure the value will be a real number. If the radicand contains a negative number, then the square root is an imaginary number in the complex number system. The complex number system will be studied in more advanced mathematics courses.

A square root function contains a term that is the square root of a variable. A function that contains a square-root term without a variable is not considered a square-root function.

Equations with radicands containing variables must be evaluated in the same manner as other equations. However, the use of perfect squares in a table of values is recommended, as this will produce results more easily graphed.

MATH BACKGROUND

Warm Up1

776 Saxon Algebra 1

114LESSON

Problems 2–5

Briefl y review evaluating expressions with square roots prior to graphing functions that contain square roots.

2 New Concepts

In this lesson, students will graph square-root functions by using tables, transformations, and a graphing calculator.

Example 1

This example introduces students to graphing square-root functions.


Make a table of y = 3 √ � x - 2. Then graph the function.

x 0 1 4 9 16

y -2 1 4 7 10

x

y

O

8

12

4

4 8 12 16

-4


Lesson 114 777

Example 2 Determining the Domain of a Square-Root Function

a. Determine the domain of y = √ �� x - 4 .

SOLUTION

The domain is the values for x that make the radicand greater than or equal to zero. Solve x - 4 ≥ 0.

x - 4 ≥ 0 Set the radicand greater than or equal to 0.

x ≥ 4 Solve for x by adding 4 to both sides.

The domain is the set of all real numbers greater than or equal to 4.

b. Determine the domain of y = 3 √ �� x

_ 2 + 4 - 7.

SOLUTION

x _

2 + 4 ≥ 0 Set the radicand greater than or equal to 0.

x _

2 ≥ -4 Subtract 4 from both sides.

x ≥ -8 Multiply both sides by 2.

The domain is the set of all real numbers greater than or equal to -8.

All square-root functions look similar to the graph of y = √�x , which is called the parent function. A transformation of a function is an alteration of the parent function that produces a new function.

Compare the parent function y = √�x to the function y = √�x + 3.

x y = √ � x y = √ � x +3

0 0 3

1 1 4

4 2 5

9 3 6

16 4 7

x

y

O

4

6

2

8 12 164

y = √x + 3

y = √x

The function y = √�x + 3 has been shifted 3 units up from the parent function y = √�x . Transformations that involve vertical and horizontal shifting are called translations.

Transformations of the Graph of f(x) = √ � x

Vertical translation: The graph of f(x) = √ � x + c is c units up from the parent graph if c > 0 and the graph is c units down from the parent graph if c < 0.

Horizontal translation: The graph of f(x) = √ �� x - c is c units to the right of the parent graph if c > 0 and the graph is c units to the left of the parent graph if c < 0.

Math Language

Remember that the domain is the set of possible input values for a function.

Hint

When graphing square-root functions, always plot the smallest value for x so that the graph will be complete.


For this lesson, explain the meaning of the word alteration. Say:

“An alteration is a change. It is the result of making something different. For example, an alteration in the schedule allowed time for a school sporting event.”

Ask a volunteer to use the word alteration in a sentence. Sample: After an alteration, my jacket has pockets.

ENGLISH LEARNERS

Lesson 114 777

Example 2

These examples introduce students to determining the domain of a function that contains a square root.

Error Alert Watch for students who are careless in copying the equations and do not include the appropriate variables and numbers under the radicand. If students exhibit diffi culty, have them practice identifying which variables and/or numbers are under the radical in each problem before solving.

Extend the Example

Determine how changing x to -x changes the domain in example 2a. Sample: The order of the inequality and the sign of 4 both change so that the domain becomes

x ≤ -4.


a. Determine the domain of y = √ �� x + 9 . The domain is the set of all real numbers such that x ≥ -9.

b. Determine the domain of

y = 2 √ �� x _ 4 - 9 + 4.

The domain is the set of all real numbers such that x ≥ 36.


Saxon Algebra 1778

Example 3 Translating the Square-Root Functions

a. Describe the transformations applied to the parent function to form y = √ � x - 3.

SOLUTION

This function can be written in the form f(x) = √�x + c by changing -3 to + (-3). The function can be written as y = √�x + (-3) which is a translation of the parent function that shifts the graph 3 units down.

b. Describe the transformations applied to the parent function to form y = √��x + 2 .

x

y

4

2

2 4

-2

-2

y = √x

y = √x + 2

SOLUTION

This function y = √ �� x + 2 is written in the form f(x) = √ �� x - c , where c is -2, which is a translation of the parent function 2 units left.

Reflections of the Graph of f (x) = √ � x

If f(x) = √ � x , then g(x) = - √ � x is a reflection of the graph of f across the x-axis.

If f(x) = √ � x , then g(x) = √ �� -x is a reflection of the graph of f across the y-axis.

Example 4 Reflecting a Square-Root Function

a. Describe the transformations applied to the parent function to form y = - √ � x .

SOLUTION

The graph of y = - √ � x is a reflection of the parent function over the x-axis.

b. Describe the transformations applied to the parent function to form y = √ �� -x + 3.

SOLUTION

The graph of y = √ �� -x + 3 is a reflection of the parent function over the y-axis, and then a vertical shift of 3 units up.

x

y

O

4

2

-2

2

y = √x

y = √x + (–3)

x

y

O

4

2

-2

2

y = √x

y = √x + (–3)

x

y

O

2

4 8

-2

-4-8

y = √xy = √ x + 3

y = √x

x

y

O

2

4 8

-2

-4-8

y = √xy = √ x + 3

y = √x

Math Reasoning

Analyze What is the domain of g(x) = √��-x ?

x ≤ 0


Use the following strategy with students who have memory diffi culties. Have students use a graphing calculator to graph each of these basic square-root functions:

y = √ � x y = √ � x + 2 y = √ � x - 2

y = √ �� -x y = √ �� -x + 2 y = √ �� -x - 2

y = - √ � x y = - √ � x + 2 y = - √ � x - 2

Students can make sketches and label the graphs for future reference. Consulting these examples will help them determine the translations of other graphs in this lesson.

INCLUSION

778 Saxon Algebra 1

Example 3

These examples show how transformations are applied to square-root functions.


a. Describe the transformation applied to the parent function to form y = √ � x + 2.translation of 2 units up from the parent function

b. Describe the transformation applied to the parent function to form y = √ �� x - 4 .translation 4 units to the right of the parent function

Example 4

These examples investigate the effect of the inclusion of a negative sign in the function.


Describe the transformation applied to the parent function to form each of the given functions.

a. f(x) = √ �� -x refl ected over the y-axis

b. f(x) = √ �� -x - 5 refl ected over the y-axis with a horizontal shift to the left 5 units; f(x) = √ �� -x - 5 = √ �� -(x + 5)

Example 5

This example describes distance utilizing a square-root function.


When conditions are slightly foggy, the distance d (in kilometers) that Meliza can see from a height of h meters is approximately d = √ �� 12h . Find the distance she can see from a height of 1800 m. Round to the nearest kilometer. d ≈ 147 km



Lesson 114 779

Example 5 Application: Horizon

The distance d (in kilometers) that Meliza can see on a clear day to the horizon from a height of h meters is approximately d = √ �� 15h . Find the distance she can see from a height of 2160 meters.

SOLUTION

Evaluate d = √ �� 15h for h = 2160 m.

d = √ �� 15(2160)

= √ �� 32400 = 180

180 km is the distance she can see from a height of 2160 m.

Lesson Practice

a. Graph y = 3 √ �� x + 1 using a table.

Determine the domain of each of the following functions.

b. f(x) = √ � x

_ 3

x ≥ 0 c. f(x) = √ �� x - 2 x ≥ 2

Describe the transformations applied to the parent function to form the given function.

d. f(x) = √ � x - 2 e. f(x) = √ �� x - 2

f. f(x) = - √ �� x + 3 g. f(x) = √ �� -x - 4

h. Physics An acorn fell from a tree limb. The function t = 0.45 √ � x represents how many seconds it takes something to fall from a height of x meters to the ground. Estimate how long it would take the acorn to fall if the limb were 8 meters above the ground. Sample: about 1.27 seconds

(Ex 1)(Ex 1)

a.

x y

-1 0

0 3

3 6

8 9

15 12

;

x

y

O

8

4

2 4 6

12y = 3√x + 1

a.

x y

-1 0

0 3

3 6

8 9

15 12

;

x

y

O

8

4

2 4 6

12y = 3√x + 1

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

d. a shift of 2 units downd. a shift of 2 units down

(Ex 3)(Ex 3)e. a shift of 2 units to the righte. a shift of 2 units to the right (Ex 4)(Ex 4)

f. a reflection over the x-axis, then a shift of 3 units to the left

f. a reflection over the x-axis, then a shift of 3 units to the left

(Ex 4)(Ex 4)

g. a reflection over the y-axis, then a shift of 4 units down

g. a reflection over the y-axis, then a shift of 4 units down

(Ex 5)(Ex 5)

Solve.

1. ⎢z + 5� + 11 = 10 {} 2. 10x2 = 70x 0, 7

3. 24x = 32x2 0, 3_4 4.

5 _

x + 1 -

2 _ x

= 5 _

10x x = 1

*5. Multiple Choice Evaluate the equation y = √ �� x + 6 - 1 for x = 2. C

A √ � 2 B √ � 7 C 2 √ � 2 - 1 D no solution

*6. Oceanography A good approximation of the speed of a wave in deep ocean water is given by the equation y = √ �� 10d . In this equation, y is the wave’s speed in meters per second and d is the ocean’s depth in meters. What is the speed of a wave if the depth is 400 meters? Round to the nearest whole number. ≈ 63 meters per second

*7. Analyze Given the function f(x) = √ ��

4x _

3 - 1 , for what values of x will f(x) be

greater than 5? Show your work. See Additional Answers.

*8. Analyze Explain how to graph f(x) = √ �� x - 2 + 3 in terms of its parent function.

(74)(74) (98)(98)

(98)(98) (99)(99)

(114)(114)

(114)(114)

(114)(114)

(114)(114)

8. Sample: Translate the parent function, f(x) = √ � x,2 units to the right and then translate the resulting graph 3 units up.

8. Sample: Translate the parent function, f(x) = √ � x,2 units to the right and then translate the resulting graph 3 units up.


Lesson 114 779

Lesson Practice

Problem a

Scaff olding Have students discuss what values they would choose for the table. If opinions vary, allow groups of students to try each suggestion, then decide as a class which would be easiest to graph.

Problem h

Error Alert Make sure that students use a correct estimated value for √ � 2 . Consider fi nding this value as a class before students work to solve the problem.



What is the square-root parent function? y = √ � x

What is the purpose of a table in graphing a square-root function? A table helps to organize and fi nd values that can easily be graphed.

Practice3


Problem 6

Extend the Problem

How does the speed of a wave at a depth of 175 meters compare with the speed of a wave at a depth of 400 meters? Round to the nearest whole number. It is slower: 42 m/s < 63 m/s


Saxon Algebra 1780

9. Find the value of the discriminant of the equation 2x2 - 5x - 4 = 0. 57

* 10. Error Analysis Two students are using the discriminant to find the number of real solutions to the equation 5x2 - 3x = 2. Which student is correct? Explain the error.

Student A

5x2 -3x = 2 b2 - 4ac = (-3)2 - 4(5)(2)

= 9 - 40 = -31

As the discriminant is negative, there are no x-intercepts.

Student B

5x2 -3x = 2 5x2 - 3x - 2 = 0

b2 - 4ac = (-3)2 -4(5)(-2) = 9 + 40 = 49

As the discriminant is positive, there are two x-intercepts.

*11. Geometry The length of a rectangle is x + 12 inches and the width is x + 8 inches. Is there a value for x that makes the area of the rectangle 50 square inches? Explain your reasoning. See Additional Answers.

*12. Multi-Step The equation 288 = (3 + x)(6 - x) can be used to determine if the base of a rectangular box with a length of (3 + x) inches and a width of (6 - x) inches can have an area of 288 square inches. a. Write the equation setting it equal to zero. 0 = -270 + 3x - x2

b. Use the equation to find the values of a, b, and c. a = -1, b = 3, c = -270

c. Find the value of the discriminant. -1071

d. Can a box with these dimensions be made? Explain. no; There is no base possible because the discriminant of the equation is negative.

13. Solve this system by graphing: y = x2 + 3

y = -2x + 3

. See Additional Answers.

*14. Error Analysis Two students are solving the system of equations y = x2 + 4x

y = -4

by substitution. Which student is correct? Explain the error.

Student A

y = x2 + 4x y = -4x2 + 4x = -4

(x2 + 4x) -4 = 0x2 + 4x - 4 = 0

no solution

Student B

y = x2 + 4x y = -4x2 + 4x = -4

(x2 + 4x) + 4 = 0x2 + 4x + 4 = 0

(x + 2)(x + 2) = 0 So, x = -2, and the solution is (-2, -4).

*15. Physics Miguel is standing at the base of a ramp. He tosses a ball into the air. The path of the ball is described by the equation y = -x2 + 7x. The equation y = x represents the ramp. At what altitude does the ball strike the ramp? Assume that dimensions are in feet. 6 feet

16. Measurement On what scale would the distance between the x-coordinates in the

solution set of the system y = x

2

_ 2

y = 4x - 6 be 8 centimeters? 1 unit:2 cm

(113)(113)

(113)(113)

10. Student B; Sample: The values of a, b, and c are found when the equation is set equal to 0.

10. Student B; Sample: The values of a, b, and c are found when the equation is set equal to 0.

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(113)(113)

(112)(112)

(112)(112)

14. Student B; Sample: Student A did not add 4 to both sides when setting the equation equal to zero.

14. Student B; Sample: Student A did not add 4 to both sides when setting the equation equal to zero.

(112)(112)

3 43 4

(112)(112)


Use a graphing calculator to graph these functions:

f (x) = 3 √ � x

f (x) = 4 √ � x

f (x) = 5 √ � x

f (x) = 6 √ � x

Is there a pattern to the graphs? Explain your answer. yes; Sample: Graphs of odd radicals fall on both sides of the y-axis. Graphs of even radicals fall only to the right of the y-axis.

CHALLENGE

780 Saxon Algebra 1

Problem 10

Extend the Problem

Ask students to graph the equation to check that there are two solutions. Yes, there are two x-intercepts;

x

y4

2 4

-2

-2-4

-4

Problem 11

Error AlertStudents may use the quadratic formula and solve for x. Remind students that it is suffi cient to only use the discriminant.


Lesson 114 781

17. Graph the function f(x) = ⎢x� - 2. See Additional Answers.

18. Temperature The temperature outside yesterday was 65°. Today the temperature changed by ⎢5°�. Give the possible temperatures outside today. 60° or 70°

19. Determine if the set of ordered pairs {(6, 3), (4, 2), (2, 1), (8, 4)} satisfies an exponential function. no

20. Engineering A small bridge has a weight limit of 8000 pounds. A photographer wants to photograph at least 5 vehicles on the bridge. The cars weigh about 1800 pounds each and the motorcycles weigh about 600 pounds each. There must be at least one car and four motorcycles in the photo. Graph a system of linear equations to describe the situation. Give two combinations of cars and motorcycles that are solutions. See Additional Answers.

21. Use the quadratic formula to solve x2 = 19x - 60. Check the solutions. 4, 15

22. Multiple Choice There are three numbers in a locker combination: 19, 22, and 28. How many different ways can the numbers be arranged? B

A 3 B 6 C 12 D 24

*23. Write Explain what types of situations apply to permutations.

24. Multi-Step In a bowling lane, the distance (d) from the foul line to the center of the Number 1 pin should be 60 feet and should vary from this length by no more than 1 _

2 inch.

a. Convert 60 feet to inches. 720 inches

b. Write and solve an absolute-value inequality that models the acceptable distances from the foul line to the center of the Number 1 pin.

c. If the diameter of the base of the Number 1 pin is 4 1 _ 8 inches, what is the shortest

possible distance between the foul line and the front of the Number 1 pin?

25. Evaluate y = √ � 2x + 3 for x = 8. 7

26. Property Mr. Kinsey’s property is in the shape of a right triangle. The legal description states that the property has an area of 900 yd2 and that the base of the property is 30 yards longer than the height. What are the actual dimensions of the property? base: 60 yards, height: 30 yards

27. Multi-Step A company gives its employees a 4% raise at the beginning of every year. This year, Jordan earns $32,000. a. Write a rule that can be used to find Jordan’s salary after n years. 32,000(1.04)

n

b. How many years will it take for Jordan to earn $40,000? 6

c. What will Jordan’s salary be in 12 years? Round to the nearest cent. $51,233.03

28. Analyze Write the radical equation √ �� x + 3 = 2x so that the equation has no radical and is equal to zero. 4x2 - x - 3 = 0

29. Has the graph of the parent quadratic function been stretched or compressed to produce the graph f(x) = 4x2 + 2?

30. Describe the similarity and difference between the graphs of f(x) = 3 x and g(x) = 4 · 3 x .

(107)(107)

(107)(107)

(108)(108)

(109)(109)

(110)(110)

(111)(111)

(111)(111)

23. Sample: When you are trying to fi nd the number of ways to pick items and the order of the items matters.

23. Sample: When you are trying to fi nd the number of ways to pick items and the order of the items matters.

(101)(101)

24b. |d - 720| ≤ 1_2;

719 1_2

≤ d ≤ 720 1_2

24b. |d - 720| ≤ 1_2;

719 1_2

≤ d ≤ 720 1_2

717 7_16

inches717 7_16

inches

(114)(114)

(104)(104)

(105)(105)

(106)(106)

(Inv 10)(Inv 10)

29. Because the coeffi cient of x2, (i.e., 4) is greater than 1, the graph has been vertically stretched (which means the graph is narrower than the parent quadratic function).

29. Because the coeffi cient of x2, (i.e., 4) is greater than 1, the graph has been vertically stretched (which means the graph is narrower than the parent quadratic function).

(Inv 11)(Inv 11)

30. Both are exponential functions with the same shape, but g(x) has been vertically stretched by a factor of four.

30. Both are exponential functions with the same shape, but g(x) has been vertically stretched by a factor of four.


LOOKING FORWARD

Graphing square-root functions will prepare students for

• Lesson 115 Graphing Cubic Functions


Lesson 114 781

Problem 21

Error AlertMake sure that students have correctly written the equation in standard form before applying the quadratic formula. Check that they have identifi ed a, b, and c correctly and copied them correctly into the formula.

Problem 27

Extend the Problem

Jordan’s boss earned $82,431.12 this year and gets a 7% raise every year. What will she earn in 9 years? $151,546.25

Problem 29

Guide students by asking them the following questions.

“What is the parent function?” y = x2

“Will the graph of this function be wider or narrower than that of the parent function? How do you know?” It will be narrower because the coeffi cient of the x2 term is greater than 1.


Saxon Algebra 1782

Warm Up

115LESSON

1. Vocabulary The polynomial x - 5x2 + 3x3 - 1 written in form is 3x3 - 5x2 + x - 1. standard

Find the degree of each polynomial expression and write the polynomial in descending order.

2. 8 + x2 + 2x 2; x2 + 2x + 8 3. 2x3 - 6x + x4 4; x4 + 2x3 - 6x

4. Simplify (125) 1

_ 3 . 5

5. Multiple Choice Which value is equivalent to 3 √ �� -343 ? B

A 7 B -7 C 114.3 D -114.3

A cubic function is a polynomial function in which the greatest power of any variable is 3. In other words, a cubic function is a polynomial function of degree 3.

The degree of a polynomial function determines many characteristics of its graph.

Function Type Graph Degree x-Intercepts

(Maximum) End Behavior

Linearx

y4

2

2 4

-2

-2-4

-4

1 1Ends go in opposite directions.

Quadraticx

y4

2

4

-2

-4O 2 2

Ends go in the same direction.

Cubic

x

y

2

2-2

-4

O

3 3Ends go in opposite directions.

(53)(53)

(53)(53) (53)(53)

(46)(46)

(46)(46)


Graphing Cubic Functions

Online Connection


Reading Math

The equation y = x3 is read, “y is equal to xcubed” or “y is equal to xto the third power.”


When the leading coeffi cient of a polynomial function with an odd degree is positive, the end behavior of the function isy +∞ as x +∞, and y -∞ as x -∞. When the leading coeffi cient is negative, the end behavior of the function is y -∞ as x +∞, and y +∞ as x -∞.

Once the end behavior of a function is determined, it can be used to help graph the function.

The degree of a polynomial function determines the general shape of its graph, the maximum number of possible solutions of the function, and along with the coeffi cient of the highest degreed term, the end behavior of its graph.

When the leading coeffi cient of a polynomial function with an even degree is positive, the end behavior of the function is y +∞ as x ±∞. If it is negative, the end behavior of the function is y -∞ as x ±∞.

LESSON RESOURCES



Master C115, E115Technology Lab Master 115

MATH BACKGROUND

Warm Up1

782 Saxon Algebra 1

115LESSON

Problem 5

Review cube roots and their meaning. Challenge students to use reasoning rather than calculations to determine the correct choice.

2 New Concepts

In this lesson, students will investigate the graphs of cubic functions.

Discuss the shape of the graph of a quadratic function, such as y = x2 + 4. Have students note that the shape can be generally described as a parabola. Now graph a simple cubic function, such as y = x3. Ask students how they would describe the shape of the graph of a cubic function. Sample: an “S”, a backward “S”, a roller coaster

Error Alert Make sure that students take care in drawing their graphs. Emphasize that they should locate the points and draw a curved line to connect these points. The solution at times will depend on their precision in drawing these curves.


Lesson 115 783

The parent function for cubic polynomials is y = x3. The graph of y = -x3 is related to the graph of the parent function y = x3.

Example 1 Graphing Cubic Functions

Evaluate the cubic parent function y = x3 and the function y = -x3 for x = -2, -1, 0, 1, and 2. Then graph the functions.

SOLUTION Make tables of values. Then plot points to graph the functions.

x -2 -1 0 1 2

y -8 -1 0 1 8

x -2 -1 0 1 2

y 8 1 0 -1 -8

Example 2 Solving Cubic Equations by Graphing

a. Solve 0 = x3 - 1 by graphing.

SOLUTION

To solve 0 = x3 - 1, begin by graphing the related function y = x3 - 1.

Then find the x-intercepts of y = x3 - 1 since these are the x-values where y = 0.

The only x-intercept is near 1, so the approximate solution to 0 = x3 - 1 is x ≈ 1.

b. Solve 2 = -2x3 - 7 by graphing.

SOLUTION

Write 2 = -2x3 - 7 so that one side is equal to zero. Then graph the related function and find its x-intercepts.

Subtracting 2 from both sides of 2 = -2x3 - 7 gives the equation 0 = -2x3 - 9. To solve 0 = -2x3 - 9, graph the related function y = -2x3 - 9.

The only x-intercept is between -1 and -2, at about -1.7.

The approximate solution to 2 = -2x3 - 7 is x ≈ -1.7.

x

y8

4

4 8-4-8

y = x3

O x

y8

4

4 8-4-8

y = x3

O

x

y8

4

4 8-4-8

y = x3

-4

-8

x

y8

4

4 8-4-8

y = x3

-4

-8

x

y4

2

2 4-2-4

-4

O x

y4

2

2 4-2-4

-4

O

x

y

O

4

2 4

-4

-2-4

x

y

O

4

2 4

-4

-2-4

Math Reasoning

Generalize Which transformation changes the graph of y = x3 into the graph of y = -x3?

Sample: a refl ection over the y-axis

Math Reasoning

Verify In Example 2a, how can you check whether x = 1 is the exact solution or not?

Sample: Substitute x = 1 into the original equation and see whether the resulting equation is true.

Hint

Another way to solve 2 = -2x3 - 7 is to graph y = -2x3 - 7 and y = 2, and then to find the x value(s) at their point(s) of intersection.


Lesson 115 783

Example 1

Whenever students are not familiar with a given function, a table of values can be used to fi nd points to plot to determine the shape of the graph.


Evaluate the function y = 2x3 for x = -2, -1, 0, 1, and 2. Then graph the function.

x -2 -1 0 1 2

y -16 -2 0 2 16

x

y

O

20

10

2 4

-10

-2-4

-20

Example 2

These examples show students how a cubic equation can be solved by graphing the related function.


a. Solve 0 = x 3 + 2 by graphing.x ≈ -1.25

x

y8

4

4 8-4-8

b. Solve 4 = -3 x 3 + 6 by graphing. x ≈ 0.9

x

y

O4 8

-4

-4-8

-8


Saxon Algebra 1784

Example 3 Solving Cubic Equations Using a Graphing

Calculator

Solve -2x2 = 1 _ 2 x3 - 1 by graphing on a graphing calculator.

SOLUTION

Write -2x2 = 1 _ 2 x3 - 1 so that one side is equal

to zero. Then graph the related function and find its x-intercepts.

Adding 2x2 to both sides of the original equation gives the equation 0 = 1_

2x3 + 2x2 - 1.

Use the graphing calculator to graph the related function y = 1_

2x3 + 2x2 - 1.

The graph shows that there are three x-intercepts. Trace to estimate their values.

The approximate solutions are x ≈ -3.9, x ≈ -0.8, and x ≈ 0.7.

For better estimates, use the Zero function. To the nearest hundredth, the solutions are x ≈ -3.87, x ≈ -0.79, and x ≈ 0.66.

Example 4 Application: Volume of a Cube

A cube of pure gold weighing 100 pounds would have a volume of about 143 cubic inches. Use a graphing calculator to estimate the side length of a 100-pound cube of gold.

SOLUTION

The formula for the volume of a cube is V = s3. To graph this equation on a graphing calculator, let y represent V and x represent s. Then graph y = x3.

Adjust the window to make sure that 143 is included in the y-values.

Window Settings

Xmin = 0

Xmax = 12

Ymin = 0

Ymax = 200

Trace to estimate the x-value where y = 143.

When y = 143, x ≈ 5.2.

The side length of a cube of gold weighing 100 pounds would be about 5.2 inches—about the width of a DVD case.

Graphing

Calculator Tip

For help with graphing functions, refer to the graphing calculator keystrokes in Lab 3 on p. 305.

Math Language

Remember that the zeros of a function are its x-intercepts or solutions.


784 Saxon Algebra 1

Example 3

This example focuses on the use of a graphing calculator to solve cubic equations.


Solve 3 = - 2 __ 3 x3 + 6x2 by

graphing on a graphing calculator. The x-intercepts are approximately -0.68176, 0.73801, and 8.94374.

Note: The answer graph in Additional Example 3 uses the following window settings: Xmin = -2, Xmax = 10, Xscl = 1, Ymin = -10, Ymax = 69, Yscl = 10, and Xres = 1.

Example 4

This example shows how a cubic equation can describe the volumes of different sized cubes.


A cube of pure silver weighing 25 pounds would have a volume of about 66 cubic inches. Use a graphing calculator to estimate the side length of a 25-pound cube of silver. side length ≈ 4.09 inches



Lesson 115 785

Solve and check.

1. x - 2 _ x + 7

= x - 6 _

3x + 21 x = 0 2.

x - 4 _ x + 1

= x + 5 _ 2x + 2

x = 13

*3. Graph the cubic function y = 1 _ 3 x3. Use it to solve the equation 0 = 1

_ 3 x 3 .

*4. Multiple Choice Which equation represents a cubic function? C

A y = 3x - 4y B y = 6x2 + 2

C y = x3 - 4x + 1 D y = 10x4 + 3x2 - 5

*5. Capacity The volume of a box is represented by the equation V = x3 - 4. Use a table or graph to find the value of x that corresponds to a volume of 23 cubic units. x = 3

*6. Games The volume of a whiffle ball is represented by the equation V = 4 _ 3 πr3. Use

a graphing calculator to graph the equation and then use the graph to estimate the volume of air in a ball with a radius of 2 inches.

*7. Write Describe the characteristics of the graph of a cubic function.

*8. Formulate Write an example of a cubic function. Sample: y = 10 x3

9. Evaluate y = √ � 4x - 5 for x = 3. Round to the nearest tenth. y ≈ -1.5

*10. Error Analysis Two students are evaluating the equation y = √ �� 2x - 5 + 2 for x = 6. Which student is correct? Explain the error. Student B; Sample: Student A incorrectly subtracted the 5 from 6.

Student A Student B

y = √ �� 2x - 5 + 2y = √ �� 2 · 6 - 5 + 2y = √ � 2 + 2

y = √ �� 2x - 5 + 2y = √ �� 2 · 6 - 5 + 2y = √ �� 12 - 5 + 2y = √ � 7 + 2

(99)(99) (99)(99)

(115)(115)

(115)(115)

(115)(115)

(115)(115)

(115)(115)

7. Sample: The ends of the graph go in opposite directions, it is a smooth curve, and the graph crosses the x-axis at least once and at most three times.

7. Sample: The ends of the graph go in opposite directions, it is a smooth curve, and the graph crosses the x-axis at least once and at most three times.

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(114)(114)

(114)(114)

Lesson Practice

a. Graph y = x3 + 1.

b. Solve 0 = -4x3 by graphing. See Additional Answers.

c. Solve 3 = -x3 + 8 by graphing. See Additional Answers.

d. Solve x2 - 1 _ 4 = 1 _

4 x3 by graphing on a graphing calculator.

e. The volume of a rectangular prism is represented by the equation V = x3 + 4. Use a graphing calculator to find the volume when x = 25.5 units. See Additional Answers.

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3) See Additional Answers.See Additional Answers.

(Ex 4)(Ex 4)

3.

x

y8

4

2 4

-4

-2-4

-8

O

; x = 0

6. ;

33.51 cubic inches

a.

x

y

O

8

4

4 8-4-8

SM_A1_NL_SBK_L115.indd Page 785 1/18/08 8:10:37 PM admini1 /Users/admini1/Desktop

Lesson 115 785

Lesson Practice

Problem a

Scaff olding Have students make a table of values before drawing their graphs.

Problem d

Error Alert Remind students that they will have to insert parentheses around the fractions. Omitting the parentheses will cause an error in the calculation.



“On the graph of a cubic equation, what is represented by the x-intercept(s)?” The x-intercept(s) represent the solution(s) or zero(s) of the equation.

“In general, describe the graph of a cubic function.” The ends always go in opposite directions (up and down); there can be 1, 2, or 3 x-intercepts or solutions.

Practice3


Problem 4


“What characteristic identifi es a cubic function?”A cubic function has a degree of 3.

“Does choice A include a cubic term?” No

“Does choice D include a cubic term?” No

“Which choice is correct?” Choice C

Problem 9

Error AlertMake sure that students do not include -5 as part of the radicand.


Saxon Algebra 1786

11. Multi-Step An apple fell from a tree limb. The function t = 0.45 √ � x represents how long it takes an object to fall from a height of x meters. a. Graph the function. (Hint: Increment the x-axis by 1 and the y-axis by 0.1, and

if a graphing calculator is not used, then use the following values for x: 0, 4, 9, and 16.)

b. Use the graph to estimate how long it took the apple to fall if the limb was 12 meters above the ground. Sample: ≈1.6 seconds

12. Use the discriminant to find the number of real solutions of the equation 6x2 + 2x - 1 = 0. d = 28; two real solutions

* 13. Error Analysis Two students are using the discriminant to find the number of real solutions to the equation 2x2 + 3x - 4 = 0. Which student is correct? Explain the error. Student B; Sample: The value of c is -4, not 4.

Student A Student B

2x2 + 3x - 4 = 0 b2 - 4ac = 32 - 4(2)(4) = 9 - 32 = -23As the discriminant is negative, there are no x-intercepts.

2x2 + 3x - 4 = 0 b2 - 4ac = 32 - 4(2)(-4) = 9 + 32 = 41As the discriminant is positive, there are two x-intercepts.

14. Gardening The length of a garden is 6 + x meters and the width is 10 - x meters. Write an equation to model the area of the garden, and use the discriminant to determine if there is a value for x that will allow the area of the garden to be 50 square meters. A = (6 + x)(10 - x); 50 = 60 + 4x - x2 and the discriminant is 56; Yes, the garden can have an area of 50 square meters.

15. Measurement The length of a fence is 15 - x feet and the width is 12 + x feet. Can the fence enclose an area of 200 square feet? Explain.

16. Determine if the set of ordered pairs ⎧ ⎨ ⎩ (-3, 1 _

8 ) , (-1, 1 _

2 ) , (-2, 1 _

4 ) , (-4, 1 _

16 )

⎫

⎬ ⎭

satisfies an exponential function. yes

17. Graph the system 21x + 7y ≥ -14

1 _ 2 y ≤ -x + 2

.

18. Geometry If the area of the triangle is 48 square units, what are the lengths of the base and the height to the nearest whole number? h = 8 units, b = 12 units

x + 4

x

*19. Graph the cubic function y = 3x3. Use it to solve the equation 0 = 3x3.

20. A new 3-digit area code is being created for new telephone numbers. If the first digit must be even but not 0, the second digit is 0 or 1, and the third digit can be any number except 0, how many new area codes are possible? 72 area codes

(114)(114)

See Additional Answers.See Additional Answers.

(113)(113)

(113)(113)

(113)(113)

3 43 4

(113)(113)

15. no; Sample: The equation 200 =(15 - x)(12 + x)

represents the area of the rectangle; 200 = 180 + 3x - x2

and 0 = -x2 + 3x - 20. The discriminant of this equation is 32 - 4(-1)(-20) =

9 - 80 = -71. Since the discriminant is negative, there is no value for xthat makes the equation true.

15. no; Sample: The equation 200 =(15 - x)(12 + x)

represents the area of the rectangle; 200 = 180 + 3x - x2

and 0 = -x2 + 3x - 20. The discriminant of this equation is 32 - 4(-1)(-20) =

9 - 80 = -71. Since the discriminant is negative, there is no value for xthat makes the equation true.

(108)(108)

(109)(109)See Additional Answers.See Additional Answers.

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(115)(115) See Additional Answers.See Additional Answers.

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“The arms and legs on a person are also called limbs because they extend from the body.”

Have students use the word limb in a sentence. Sample: Lightning struck the tree limb.

For problem 11 explain the meaning of the word limb. Say:

“A tree limb is a part of a tree that extends out from the trunk of the tree. It can have leaves and fruit on it.”

Draw a tree on the board and label the larger branches limbs. Then draw a stick person on the board. Say:

ENGLISH LEARNERS

786 Saxon Algebra 1

Problem 14

Error AlertStudents may not multiply the binomials. Remind students to write the product in standard form before fi nding the discriminant.

Problem 19


“Use a graphing calculator to show these graphs: y = 2 x 3 , y = 8 x 3 , and y = 5.5 x 3 . What do these graphs have in common?” The x-intercept for each is 0.

“Predict the x-intercept of the function y = 6 1 __ 3 x 3 .” The

x-intercept would be 0.

“Write a cubic function with an x-intercept of 0.” y = 4x 3

Problem 20

Extend the Problem

A zip code consists of fi ve digits 0 – 9. If the fi rst digit must be a 6 or 7, the second digit must be less than 8, and the fourth digit cannot be a 3, 5, or 6, how many zip codes are possible? 11,200


Lesson 115 787

21. Multiple Choice Which system of equations has the solution (-1, 1)? C

A y = x2 y = x + 6

B y = x2 y = 6

C y = x2 y = -2x - 1

D y = x2 y = -x + 6

*22. Analyze A system of three equations consists of a quadratic, given by y = x2 - 3, and two linear equations. One linear equation intersects the parabola at two points. If the second linear equation is parallel to the first, how many solutions does the system have? Explain.

23. Accessories Candida has plans to shop for hair bows and does not plan on spending more than $20. Each big bow costs $5 and each small bow costs $2. Write an inequality and graph it to describe the situation. See Additional Answers.

24. Solve -x2 + 2 = -7x by using a graphing calculator. Round to the nearest tenth. x = -0.3 and 7.3

25. Multiple Choice Solve x2 + 7 = -42. C

A 7 B ±7 C no solution D ±7 √ � 1

26. Phone Chains In order to relay information quickly, staff at a school use a phone chain. The superintendent first notifies 3 people of a snow day. In the second set of calls, these 3 people each call 2 people. Each person called then calls 2 other people. How many sets of calls need to be made to notify 96 people? 6 sets of calls

27. Multi-Step A square frame is to be made so that its side length is √ �� x + 1 . a. What is the perimeter of the square? 4√ �� x + 1 units

b. For what value of x will the perimeter of the frame be equal to 8 units? x = 3

28. Generalize Look at the function f(x) = - 0.5⎢x� . How can you find the direction of the “V” without graphing it? Sample: The negative sign indicates that the “V” will open downward.

29. Football The distance d from the goal post in feet of a football during a field goal kick is represented by the function d = ⎪60t - 90⎥ where t is the time in seconds. If the ball were kicked at 80 feet per second how would the graph change? Thegraph would be compressed.

30. Write, in order, the function that grows the slowest to the one that grows the fastest: exponential, linear, quadratic. linear, quadratic, exponential

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22. none; Sample: The second parallel line could intersect the parabola, at least once. However, since it never intersects the other linear equation, there can be no solution to the system.

22. none; Sample: The second parallel line could intersect the parabola, at least once. However, since it never intersects the other linear equation, there can be no solution to the system.

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Graphing cubic functions will be further developed in other Saxon Secondary Mathematics courses.

LOOKING FORWARD

Describe the end behavior of each graph: linear equation ends go in opposite directionsquadratic equation ends go in the same directioncubic equation ends go in opposite directions

Use this knowledge to predict the end behavior of the graph of y = x 4 . Then graph y = x 4 to check your prediction. ends go in the same direction

CHALLENGE

Lesson 115 787

Problem 23

Error AlertStudents may not use two variables to represent the two kinds of bows. Remind them that the bows have different prices and cannot be represented by the same variable.

Problem 25


“What is the degree of the equation?” The variable is squared, so the degree is 2.

“How can x2 be isolated?” Subtract 7 from both sides of the equation to get x 2 = -49.

“What answer choice or choices can be eliminated at this point? Why?” All answer choices except C can be eliminated. The term x2 is equal to –49. There is no real number that when squared will equal a negative number.

Problem 27

Extend the Problem

For what value of x will each side of the frame be equal to 8 units? x = 63


Saxon Algebra 1788

Warm Up

116LESSON

1. Vocabulary Two equivalent ratios form a . proportion

2. Change 24% to a fraction and a decimal. 6_25 , 0.24

3. Change 1 _ 40

to a decimal and a percent. 0.025, 2.5%

4. Find 25% of 250. 62.5

5. 36 is what percent of 1125? 3.2%

Money that is borrowed or invested is called principal. Interest is money paid for the use of that money. If money is borrowed, interest is paid. If money is invested, interest is earned.

Simple interest is interest paid on the principal only. To find simple interest, use the formula I = Prt.

Simple Interest FormulaI = Prt

I the amount of interestP the principalr the annual rate, a percent expressed as a decimalt the time in years

Example 1 Finding Simple Interest

a. An account is opened with $4000. The bank pays 5% simple interest annually. How much interest will be earned in 3 years?

SOLUTION

Use the simple interest formula.

The principal P is 4000. The rate r is 5%, or 0.05. The time t is 3.

I = Prt Write the formula, then evaluate.

= 4000(0.05)(3) Substitute the values of the variables.

= 600 Simplify.

The account will earn $600 interest in 3 years.

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(SB 7)(SB 7)

(SB 7)(SB 7)

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Solving Simple and Compound Interest

Problems

Online Connection


Math Language

Even though the account value grows as interest is earned, simple interest is only paid on the original amount deposited into the account.


Simple interest is money paid as a percent of the original amount, or principal. Simple interest is paid only on that original principal, and not on any accrued interest.

Compound interest is money paid on the total amount in an account, which means that as the account accrues interest, more interest is paid.

The compound interest formula is an exponential growth function where P corresponds to the original amount a,

(1 + r __ n ) corresponds to the base b, and nt corresponds to the exponent x in the function f (x) = abx.

The compound interest formula also relates to the formula for the nth term of a geometric sequence A(n) = ar n-1. The principal P corresponds to a, the fi rst term of the sequence, (1 + r __ n ) corresponds to the common ratio r, and nt corresponds to the exponent n - 1.

MATH BACKGROUNDLESSON RESOURCES




Warm Up1

788 Saxon Algebra 1

116LESSON

Problems 2 and 4

These problems review changing a percent to a decimal, a skill students will need in this lesson.

2 New Concepts

In this lesson, students learn to calculate simple and compound interest using formulas.

Discuss the terms principal and interest, and the vocabulary words simple interest and compound interest. Explain that simple interest is calculated on the original principal only, while compound interest is calculated on the principal plus any previous interest earned.


Lesson 116 789

b. $12,500 is invested for 15 years at 4% simple interest. How much money will be in the account after 15 years?

SOLUTION

Use the simple interest formula.

The principal P is 12,500. The rate r is 4%, or 0.04. The time t is 15.

I = Prt Write the formula, then evaluate.

= 12,500(0.04)(15) Substitute the values of the variables.

= 7500 Simplify.

The account will earn $7500 interest in 15 years.

Add this interest to the original amount invested to find the total amount in the account.

12,500 + 7500 = 20,000

There will be $20,000 in the account after 15 years.

c. $6000 is borrowed at 8.5% simple interest. The total amount of interest paid is $2040. For how many years was the money borrowed?

SOLUTION

Use the simple interest formula and solve for t.

The principal P is 6000. The interest I is 2040. The rate r is 8.5% or 0.085.

I = Prt Write the formula.

2040 = 6000(0.085)t Substitute the values of the variables.

2040 = 510t Simplify.

4 = t Divide both sides by 510.

The money was borrowed for 4 years.

d. After 18 months, $738 had been earned on an $8200 investment. What was the interest rate?

SOLUTION

Use the simple interest formula and solve for r.

The principal P is 8200. The interest I is 738. The time t is 18 _

12 = 1.5 years.

I = Prt Write the formula.

738 = 8200 · r · 1.5 Substitute the values of the variables.

738 = 12,300r Simplify.

0.06 = r Divide both sides by 12,300.

Convert 0.06 to a percent. The interest rate was 6%.

Hint

The time in the simple interest formula must be in years. There are 12 months in 1 year. To change the units from months to years, divide by 12.


For Example 1b, explain the meaning of the word invested. Say:

“Invested means that money has been put into a business, real estate, stocks, or bonds in order to make more money.”

Discuss ways that money is invested by the purchase of land or homes. Explain that stocks are investments in privately owned companies, and that bonds are investments in a government agency.

ENGLISH LEARNERS

Lesson 116 789

Example 1

This example demonstrates the application of the formula for fi nding simple interest.


a. An account is opened with $3500. The bank pays 4.5% simple interest annually. How much interest will be earned in 8 years? $1260.00

b. $15,000 is invested for 30 years at 6.25% simple interest. How much will be in the account after 30 years? $43,125

c. $7500 is borrowed at 7.75% simple interest. The total amount of interest paid is $3487.50. For how many years was the money borrowed? 6 years

d. After 66 months, $1384.62 had been earned on an investment of $5750. What was the interest rate? 4.4%


Saxon Algebra 1790

The amount in an account grows faster with compound interest. Compound interest is interest that is paid on both principal and on previously-earned interest. The compound interest formula gives the total amount accumulated after a given number of years.

Compound Interest Formula

A = P (1 + r _ n )

nt

A the total amount after t yearsP the principalr the annual rate, a percent expressed as a decimalt the time in yearsn the number of times interest is compounded each year

Example 2 Finding Compound Interest

a. $5000 is invested at 6% compounded annually. Find the value of the investment after 10 years.

SOLUTION

The principal P is 5000. The rate r is 6% or 0.06. The time t is 10 years.

A = P (1 + r) t Write the formula, then evaluate.

= 5000 · (1 + 0.06) 10 Substitute the values of the variables.

= 5000 · (1.06) 10 Simplify inside the parentheses.

= 5000 · 1.790847697 Simplify the power, and do not round.

= 8954.24 Multiply, and round to the nearest penny.

The value of the investment will be $8954.24.

b. $5000 is invested at 6% compounded quarterly. Find the value of the investment after 10 years.

SOLUTION

The principal P is 5000. The rate r is 6% or 0.06. The time t is 10 years and n = 4 because quarterly means four times per year.

A = P (1 + r _ n )

nt Write the formula, then evaluate.

= 5000 (1 + 0.06 _

4 )

4(10)

Substitute the values of the variables.

= 5000 · (1.015) 40 Use the order of operations to simplify.

= 5000 · 1.814018409 Simplify the power and do not round.

= 9070.09 Multiply and round to the nearest penny.

The value of the investment will be $9070.09.

Hint

Use a calculator to evaluate the power and to multiply the result by the principal.

Math Reasoning

Justify Explain why the formula for interest compounded annually is A = P (1 + r) t .

Sample: When interest is compounded annually, it is compounded once a year; n = 1. So,

A = P(1 + r_n )nt

= P(1 + r_1 )1t

= P(1 + r)t


For students who may need further help, write out the formula to fi nd factors other than the interest paid:

Interest ______________ Principal × rate

= time

Interest _______________ Principal × time

= rate

Allow students to refer to these diagrams as needed during the lesson.

Use the following strategy with students who have diffi culty with written symbols. Help students remember the formulas by writing out the formulas using complete words.

The formula for simple interest would look like this:

Interest = Principal × rate × time

INCLUSION

790 Saxon Algebra 1

TEACHER TIP Students should realize that while the formula for simple interest gives a result that shows the interest earned, the formula for compound interest yields a cumulative account balance (principal plus interest). Help students recognize this difference so they can more accurately answer the questions.

Example 2

Discuss with students the difference in the value of the investment when compounded annually and quarterly.


a. $13,000 is invested at 4.5% compounded annually. Find the value of the investment after 8 years. $18,487.31

b. $7500 is invested at 5.8% compounded semi-annually. Find the value of the investment after 20 years.$23,532.94


Lesson 116 791

Example 3 Comparing Simple and Compound Interest

a. An account has $1000 and earns 20% simple interest. Make a table to find the total amount in the account after 1, 2, 5, and 10 years.

SOLUTION

Years Prt = I Total in Account1 (1000)(0.20)(1) = 200 $1000 + $200 = $12002 (1000)(0.20)(2) = 400 $1000 + $400 = $14005 (1000)(0.20)(5) = 1000 $1000 + $1000 = $2000

10 (1000)(0.20)(10) = 2000 $1000 + $2000 = $3000

b. An account has $1000 and earns 20% interest compounded annually. Make a table to find the total amount in the account after 1, 2, 5, and 10 years.

SOLUTION

Years A = P (1 + r) t Total in Account

1 A = 1000(1 + 0.20)1 $1200

2 A = 1000(1 + 0.20)2 $1440

5 A = 1000(1 + 0.20)5 $2488.32

10 A = 1000(1 + 0.20)10 $6191.74

c. Use the table in a to graph the account earning simple interest and the table in b to graph the account earning compound interest on the same coordinate plane. Compare the growth of the two accounts over time.

SOLUTION

$2,000

$0

$4,000

$6,000

$8,000

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Years

2 4 6 8 10 12

Compound Interest

Simple Interest

Simple interest grows linearly because it adds the same amount each year. Compound interest grows exponentially because it pays interest on the previously-earned interest as well as the principal. The account earning compound interest grows more rapidly than the account earning simple interest.

Caution

Be sure to add the interest paid to the original principal to find the total amount in the account.


Lesson 116 791

Example 3

This example presents a method of comparing simple interest and compound interest earned over varying time periods.

Error Alert Be sure students recognize that the formula for compound interest can be modifi ed to A = P(1 + r ) t only when the interest is paid once per year, or annually. When paid over any other time period, the formula that divides the rate by the time period must be used.


a. $2000 earns 12.6% simple interest. Make a table to fi nd the total amount in the account after 1, 2, 5, and 10 years.

Years Prt = I Total

1 252 $2252

2 504 $2504

5 1260 $3260

10 2520 $4520

b. $3000 earns 8.6% interest compounded quarterly. Make a table to fi nd the total amount in the account after 1, 2, 5, and 10 years.

Years A = P (1 + r __ n ) nt

1 $3266.44

2 $3556.54

5 $4590.80

10 $7025.16


Saxon Algebra 1792

Example 4 Application: Retirement Investments

Two people plan to retire at age 65. A 25-year-old woman invests $2000 in a bond that pays 7% per year, compounded annually. A 45-year-old man invests $5000 in a bond that pays 7% per year, also compounded annually. Whose investment will be worth more when they reach retirement age and by how much?

SOLUTION

Use A = P (1 + r)t to calculate the value of the investment for each person.

For the 25-year-old woman, P = 2000, r = 0.07, and t = 40.

A = 2000 (1 + 0.07)40 Substitute.

A = 2000(1.07)40 Add inside the parentheses.

A = 29,948.92 Simplify using the order of operations.

The total value of her account will be $29,948.92.

For the 45-year-old man, P = 5000, r = 0.07, and t = 20.

A = 5000(1 + 0.07)20 Substitute.

A = 5000(1.07)20 Add inside the parentheses.

A = 19,348.42 Simplify using the order of operations.

The total value of his account will be $19,348.42. The woman’s investment will be worth $10,600.50 more.

Lesson Practice

a. An account is opened with $5600. The bank pays 4% simple interest annually. How much interest will be earned in 10 years? $2240

b. $25,000 is invested for 12 years at 6% simple interest. How much will be in the account after 12 years? $43,000

c. $4500 is borrowed at 2.5% simple interest. The total amount of interest paid is $562.50. For how many years was the money borrowed? 5 years

d. After 15 months, $130 had been earned on a $2600 investment. What was the interest rate? 4%

e. $12,000 is invested at 4% compounded annually. Find the value of the investment after 30 years. $38,920.77

f. $12,000 is invested at 4% compounded quarterly. Find the value of the investment after 30 years. $39,604.64

g. An account has $2500 and earns 12% simple interest. Complete the table to find the total amount in the account after 1, 2, 5, and 10 years.

Years Prt = I Total Amount in Account

1 $300 $2800

2 $600 $3100

5 $1500 $4000

10 $3000 $5500

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

Math Reasoning

Analyze Why was the man’s account value less than the woman’s?

Sample: The man had 20 fewer years of compounded interest.


792 Saxon Algebra 1

Example 4

Have students fi nd the amount of each investment when compounded quarterly in order to compare compounding for different intervals of time. woman’s: $32,102.35; man’s: $20,031.96

Extend the Example

What amount would the man have to invest for the value of his investment at his retirement to equal the value of the woman’s investment at her retirement? $7739.37


A 30-year-old man invests $4000 in a bond that pays 5.5% interest per year, compounded semi-annually. How much will the investment be worth at his retirement age of 65? $26,717.03

Lesson Practice

Problem b

Error Alert Make sure that students remember to add the amount of interest to the principal to correctly answer the question. Suggest that students write the formula to emphasize that it computes interest only.

Problem d

Scaff olding Have students fi rst write the formula for simple interest, transforming it to show how they would fi nd the interest rate:

I ___ Pt = r. They can then substitute

the given information and fi nd the solution.



Lesson 116 793

h. A second account has $2500 and earns 12% compounded annually. Complete the table to find the total amount in each account after 1, 2, 5, and 10 years.

Principal Rate YearsTotal

Amount in Account

$2500 12% 1 $2800

$2500 12% 2 $3136

$2500 12% 5 $4405.85

$2500 12% 10 $7764.62

i. Use the table in problem g to graph the account earning simple interest and the table in problem h to graph the account earning compound interest on the same coordinate plane. Compare the growth of the two accounts over time.

j. Retirement Investments Two people plan to retire at age 60. A 30-year-old man invests $4000 in a bond that pays 5% per year, compounded annually. A 40-year-old man invests $6000 in a bond that pays 5% per year, also compounded annually. Whose investment will be worth more when they reach retirement age and by how much? The 30-year-old man’s investment will be worth more by $1367.98.

(Ex 3)(Ex 3)

i.

Years

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t

2 4 6 8 10

2000

4000

6000

8000

0

i.

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2 4 6 8 10

2000

4000

6000

8000

0

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

*1. $900 is invested at 3% simple interest for 5 years. How much interest is earned? $135

*2. Write Explain the difference between simple and compound interest.

*3. Formulate The graph shows the value of a money market account that pays compound interest. How much principal was originally invested? $200

x

y

200

400

600

800

2 4 6 8

Years

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t

4. Population The exponential function y = 3.45 (1.00617) x can model the approximate population of Oklahoma from 2000 to 2006, where x is the number of years after 2000 and y represents millions of people. Assuming the model does not change, predict when the population will reach 4 million? 2025

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2. Sample: Simple interest is just paid on the principal. Compound interest is paid on the principal and interest earned.

2. Sample: Simple interest is just paid on the principal. Compound interest is paid on the principal and interest earned.

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The account earning compound interest increases more rapidly.


Lesson 116 793



What is the difference between simple interest and compound interest? Simple interest is paid on the principal only; compound interest is paid on the principal and any interest earned.

How can the formula for compound interest be used to fi nd the principal? Write the formula. Divide the amount in the account (A) by the rate and time portion of the

formula: A _______ (1 + r __ n ) nt

.

Practice3


Problem 1


“What is the principal in this problem?” $900

“What is the interest rate? 3%

What is the period of time?” 5 years

“How is this information substituted into the formula?” The P stands for the principal, or $900. The r stands for the interest rate of 3%, or 0.03. The t stands for the length of time of the investment, or 5.


Saxon Algebra 1794

*5. Multiple Choice $600 is invested at 11% simple interest. What is the value of the investment after 14 years? B

A $924 B $1524 C $2586.26 D $92,400

*6. Mutual Funds Over the past 20 years, a mutual fund averages paying 10% interest compounded annually. If a woman had invested $3000 originally, how much would her account be worth now? $20,182.50

*7. Graph the cubic function y = -3x3. Use the graph to find the roots of the equation.

8. Error Analysis Two students write the equation “y equals x cubed plus five.” Which student is correct? Explain the error. Student A; Sample: The word “cubed” means to the third power, not to the second power.

Student A Student B

y = x3 + 5 y = x2 + 5

9. Geometry The formula for the volume of a cube is V = s3. Graph the equation and find the volume of the cube if the side length is 2 units.

*10. Multi-Step The volume of a packing container is given by the function y = x3 + 5. a. Make a table of values for the equation.

b. Graph the equation.

c. Find the volume when x is 3 feet. 32 cubic feet

11. Evaluate y = 3 √ �� 7x + 2 - 7 for x = 2. 5

12. Physics The speed at which an object in free fall drops is modeled by the equation y = 8 √ � x . In this equation, y is the speed in feet per second and x is the distance fallen in feet. What is the speed of an apple after it falls a distance of 8 feet? Round to the nearest tenth. ≈ 22.6 feet per second

*13. Error Analysis Two students are determining the domain and range of the function f (x) = √ �� x - 5 + 1. Which student is correct? Explain the error.

Student A Student B

f(x) = √ �� x - 5 + 1 x - 4 ≥ 0x ≥ 4; y ≥ 0

f(x) = √ �� x - 5 + 1 x - 5 ≥ 0x ≥ 5; y ≥ 1

14. Measurement The function s = √ � A gives the side length of a square with area A. What is the side length of a square that has an area of 625 square feet? 25 feet

15. Graph the system y ≥ 2 _

5 x - 4

y ≤ 0 .

16. Use the quadratic formula to solve 46 + 16x = -x2. Find approximate answers to four decimal places. -3.7574, -12.2426

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7.

x

y

16

32

2 4

-16

-2-4

-32

O

; x = 07.

x

y

16

32

2 4

-16

-2-4

-32

O

; x = 0

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x

y

O

8

4

4 8-4-8

; 8 cubic units

9.

x

y

O

8

4

4 8-4-8

; 8 cubic units

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x y

-2 -3

-1 4

0 5

1 6

2 13

10a.

x y

-2 -3

-1 4

0 5

1 6

2 13

10b.

x

y

O

16

8

-8

-16

2 4-4

10b.

x

y

O

16

8

-8

-16

2 4-4

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(114)(114) 13. Student B; Sample: Student A just removed the radical sign and then set the entire right side greater than or equal to zero.

13. Student B; Sample: Student A just removed the radical sign and then set the entire right side greater than or equal to zero.

3 43 4

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15.

x

y4

2

2 4

-2

-2-4O

15.

x

y4

2

2 4

-2

-2-4O

(110)(110)


Have students solve this problem:

Antwaan deposited $5000 in a savings account that paid 5.75% interest compounded semi-annually. After 6 years, he withdrew $2500. What is the total amount in the account 13 years after the account was opened? $6730.10

CHALLENGE

794 Saxon Algebra 1

Problem 6

Error AlertMake sure that students remember to change the percent to a decimal before completing the calculations.

Problem 12

Extend the Problem

What is the speed in feet per second of a coin that is dropped from the roof of a building 450 feet tall? 169.7 feet per second

Problem 16

Have students give the answer in simplest radical form for practice simplifying radical expressions. -8 + 3 √ � 2 , -8 - 3 √ � 2


Lesson 116 795

17. Sports The American League Central Division in Major League Baseball has 5 teams. How many different ways are there for the teams to finish first through fifth? 120

18. Solve this system by graphing: y = 2x2 - 6x + 1

y = -x - 4

.

19. Multiple Choice How many x-intercepts does the equation y = 4x2 + 8x - 2 have? A 0 B 1 C 2 D 3 C

20. Write Explain what the discriminant tells about the graph of a quadratic equation. Sample: The discriminant tells how many times the graph of aquadratic equation crosses or touches the x-axis.

21. Solve 4x2 + 8 = -6x by using a graphing calculator. Round to the nearest tenth. no solution

22. Solve and graph the inequality ⎢4x - 3� + 1 > 10.

23. Structural Engineering The water pressure p on a dam is a function of the depth of the water x behind the dam: p = 4905 √ � x . For what value of x is the pressure equal to 44,145? x = 81

24. Multi-Step Graph the function f(x) = ⎢x� - 4, and then translate the function to the left by 2. What is the vertex of this new function? (-2, -4)

*25. $4500 is borrowed at 3.5% simple interest. The total amount of interest paid is $1260. For how many years was the money borrowed? 8 years

*26. Credit Cards A man uses a credit card to make a $1200 purchase. The credit card charges 22% annual interest compounded monthly and requires no payments for the first year. At the end of one year, how much will he owe? $1492.32

27. Justify Why is f(x) = 4(-2)x not an exponential function?

28. Multi-Step Study the numbers in the sequence.

3, √ � 3 , 1, √ � 3

_ 3 , 1 _

3 , . . .

a. Find the pattern. Divide each term by √ � 3.

b. What is the next term in the sequence? √ � 3_9

29. If f(x) = 3x2 - 12x + 2, where is the axis of symmetry located? Give the x- and y-coordinates of the vertex. x = 2, (2, -10)

30. Identify which function is linear, quadratic, exponential growth, and exponential decay: f (x) = ( 1 _

5 )

x , g(x) = x2, h(x) = 5 x , and j(x) = 5x. f(x) is exponential decay,

g(x) is quadratic, h(x) is exponential growth, and j(x) is linear.

(111)(111)

(112)(112)

18. no solution;

x

y

2

2 4

-2

-2-4O

18. no solution;

x

y

2

2 4

-2

-2-4O

(113)(113)

(113)(113)

(100)(100)

(101)(101)x < -1.5 OR x > 3;

0 2 4 6-2-4x < -1.5 OR x > 3;

0 2 4 6-2-4

(106)(106)

(107)(107)

(116)(116)

(116)(116)

(108)(108)27. because b is negative; Sample: The range values are not all positive or all negative. For example, f(2) = 16 and f(3) = -32.

27. because b is negative; Sample: The range values are not all positive or all negative. For example, f(2) = 16 and f(3) = -32.

(103)(103)

(Inv 10)(Inv 10)

(Inv 11)(Inv 11)

SM_A1_NLB_SBK_L116.indd Page 795 5/14/08 3:26:24 PM pawan /Volumes/ju115/HCAC057/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L116

Solving simple and compound interest problems will be further developed in other Saxon Secondary Mathematics courses.

LOOKING FORWARD

Lesson 116 795

Problem 19


“What is an x-interceept?” A point at which the line of a graph crosses the x-axis.

“What is the shape of this graph?” A parabola that opens upward.

“Where is the vertex located?” The vertex is below the x-axis.

“How many x-intercepts does the equation produce?” 2

Problem 21

Error AlertSome students may not enter the function correctly into their calculators. Remind them that the function must be in standard form and that, depending on the side of the equal sign to which the terms are gathered, sign changes will occur.

Problem 23

Extend the Problem

What is the water pressure on the dam at a depth of 225 feet? 73,575 psi


Saxon Algebra 1796

Warm Up

117LESSON

1. Vocabulary A ratio is the comparison of two quantities using . division

2. If the two legs of a right triangle measure 9 inches and 12 inches, find the length of the hypotenuse. 15 in.

3. In a right triangle, one leg measures 10 inches and the hypotenuse measures 17 inches. Find the length of the other leg. 3√ �� 21 in. or ≈ 13.75 in.

Decide if the following are Pythagorean triples or not.

4. 6, 10, 8 yes 5. 8, 12, 20 no

Recall that a right triangle has one right angle and two acute angles. In the triangle, ∠C is the right angle and ∠A and ∠B are the acute angles.

Using ∠A in the triangle, the leg across from the angle is called the opposite leg and the leg next to ∠A is called the adjacent leg. The hypotenuse is always opposite the right angle and is always the longest side of the triangle.

A

C a

c hypotenuseleg adjacentto ∠A

leg opposite ∠A

b

B

In any right triangle, there are six trigonometric ratios that can be written using two side lengths of the triangle in relation to the angles of the triangle. The three most common trigonometric ratios are sine, cosine, and tangent, abbreviated sin, cos, and tan, respectively.

Sine, Cosine, and Tangent

sine of ∠A = length of leg opposite ∠A

___ length of hypotenuse

= a _ c

cosine of ∠A = length of leg adjacent to ∠A

___ length of hypotenuse

= b _ c

tangent of ∠A = length of leg opposite ∠A

___ length of leg adjacent to ∠A

= a _ b

(85)(85)

(85)(85)

(85)(85)

(85)(85) (85)(85)


Using Trigonometric Ratios

Online Connection


Hint

The three trigonometric ratios of sine, cosine, and tangent can be remembered using the mnemonic device:

SOH-CAH-TOA

(pronounced “sew-ka-toe-a”). Sine equals Opposite leg over Hypotenuse, Cosine equals Adjacent leg over Hypotenuse, and Tangent equals Opposite leg over Adjacent leg. This can also be written as S o _ h C a _ h T o _ a .






The word trigonometry derives from Greek words meaning “triangle measurement.”

Trigonometric ratios are used to fi nd angle and side measurements in right triangles. In a right triangle, the hypotenuse is the longest side and is opposite the right angle. The legs are the two sides adjacent to the right angle. Each angle other than the right angle has a leg that is opposite it and a leg that is adjacent to it. These relationships are

used in the defi nition of the trigonometric ratios: sine, cosine, and tangent. The reciprocals of each of the trigonometric ratios are cosecant, secant, and cotangent, respectively.

When an angle measurement and side length are known, these ratios can be used to fi nd the lengths of the other sides. If the side lengths are known, the inverse functions can be used to fi nd the angle measures.

Warm Up1

796 Saxon Algebra 1

117LESSON

Problem 4

Remind students that the hypotenuse is always the longest side, no matter in what order the sides are listed.

2 New Concepts

In this lesson, students fi nd trigonometric ratios and use them to fi nd missing side lengths. They use trigonometry to fi nd missing angle measures.

TEACHER TIPEmphasize which sides of a right triangle are legs and which side is the hypotenuse. In addition, for each acute angle in the triangle, have students identify the adjacent and opposite sides.


Lesson 117 797

In addition to the three trigonometric ratios previously discussed, there are three other trigonometric ratios called cosecant, secant, and cotangent, abbreviated csc, sec, and cot, respectively.

Cosecant, Secant, and Cotangent

cosecant of ∠A = length of hypotenuse

___ length of leg opposite ∠A

= c _ a

secant of ∠A = length of hypotenuse

___ length of leg adjacent to ∠A

= c _ b

cotangent of ∠A = length of leg adjacent to ∠A

___ length of leg opposite ∠A

= b _ a

Example 1 Finding Trigonometric Ratios

a. Using the right triangle, find sin B, cos B, and tan B.

SOLUTION

sin B = opposite leg

__ hypotenuse

= 5 _ 13

cos B = adjacent leg

__ hypotenuse

= 12 _ 13

tan B = opposite leg

__ adjacent leg

= 5 _ 12

b. Using the right triangle, find all six trigonometric ratios for ∠A.

SOLUTION

First find the length of side a using the Pythagorean Theorem.

a 2 + b 2 = c 2

a 2 + 4 2 = 5 2

a 2 + 16 = 25

a 2 = 9

a = 3

sin A = a _ c =

3 _ 5 cos A =

b _ c = 4 _ 5

tan A = a _ b

= 3 _ 4 csc A =

c _ a = 5 _ 3

sec A = c _ b

= 5 _ 4 cot A =

b _ a = 4 _ 3

A

C 12

135

B

A

C 12

135

B

A C

B

4

a5

A C

B

4

a5

Math Reasoning

Generalize Explain the relationship between sine and cosecant; cosine and secant; and tangent and cotangent ratios.

They are reciprocals of each other.

Caution

Writing “sin = 5 _ 13 ” is not

a valid trigonometric ratio because there is no angle measure included with sine.

Caution

Although a = ±3, only the positive value is used because a represents length.


ENGLISH LEARNERS

For this lesson, explain and contrast the meaning of the words adjacent and opposite. Using four volunteers, stand two students next to each other and the other two students opposite each other.

Discuss that adjacent means “next to” and opposite means “across from.”

Have students identify objects in the classroom that are opposite each other and adjacent to each other.

Lesson 117 797

Example 1

Students fi nd the trigonometric ratios for a right triangle.


a. Using the right triangle, fi nd sin B, cos B, and tan B.

A

B

7

2524

sin B = 7 _ 25

; cos B = 24 _ 25

; tan B = 7 _ 24

b. Using the right triangle, fi nd all six trigonometric ratios for ∠A.

A

B15

2520

sin A = 3 _ 5 , cos A = 4 _

5 ,

tan A = 3 _ 4 , csc A = 5 _

3 ,

sec A = 5 _ 4 , cot A = 4 _

3


Saxon Algebra 1798

Example 2 Using a Calculator with Trigonometric Ratios

a. If ∠A = 42°, find sin A, cos A, and tan A to the nearest ten-thousandth.

SOLUTION

Use a calculator to find the value of the trigonometric ratios.

42 sin A ≈ 0.6691

42 cos A ≈ 0.7431

42 tan A ≈ 0.9004

b. If ∠A = 33°, find csc A, sec A, and cot A to the nearest ten-thousandth.

SOLUTION

Use a calculator.

33 OR 1 33

csc A ≈ 1.8361

33 OR 1 33

sec A ≈ 1.1924

33 OR 1 33

cot A ≈ 1.5399

Example 3 Using Trigonometry to Find Missing Side Lengths

Use a calculator to find trigonometric ratio values.

a. Find the value of x. Round to the nearest hundredth.

SOLUTION

Since the missing side is opposite the angle and the adjacent side length is given, use the tangent ratio.

tan 28° = x _ 9

9 · tan 28° = x

4.79 ≈ x

b. Find the value of x and y. Round to the nearest hundredth.

SOLUTION

sin A = opposite leg

__ hypotenuse

cos A = adjacent leg

__ hypotenuse

sin 52° = x _ 12

cos 52° = y _

12

12 · sin 52° = x 12 · cos 52° = y

9.46 ≈ x 7.39 ≈ y

9

x

28°

9

x

28°

y

x

52°

B

C

12

Ay

x

52°

B

C

12

A


798 Saxon Algebra 1

Example 2

Students fi nd the value of trigonometric ratios using a calculator.


a. If ∠A = 38°, fi nd sin A, cos A, and tan A to the nearest ten-thousandth. 0.6157, 0.7880, 0.7813

b. If ∠A = 51°, fi nd csc A, sec A, and cot A to the nearest ten-thousandth. 1.2868, 1.5890, 0.8098

Example 3

Students use trigonometric ratios to fi nd missing side lengths.


Use a calculator to fi nd trigonometric ratio values.

a. Find the value of x. Round to the nearest hundredth.

x

17

32°

x ≈ 10.62

b. Find the value of x and y. Round to the nearest hundredth.

x

y

24

61°

x ≈ 20.99, y ≈ 11.64


Lesson 117 799

Inverse trigonometric functions can be used to find missing angle measures. Ona calculator, these are sin-1, cos-1, and tan-1. Because they are inverse functions, sin-1(sin A) = A, and the same principle follows for cosine and tangent.

Example 4 Using Trigonometry to Find Missing Angle Measures

a. Find the measure of ∠A. Round to the nearest hundredth of a degree.

SOLUTION

Use the cosine ratio since you know the adjacent leg and the hypotenuse.

cos A = 6 _ 11

cos -1 (cos A) = cos -1 ( 6 _

11 )

∠A ≈ 56.94°

b. Find the measures of ∠A and ∠B. Round to the nearest hundredth of a degree.

SOLUTION

Use the tangent ratio since you know the lengths of the legs.

tan A = 2 _ 5 tan B =

5 _ 2

tan -1 (tan A) = tan -1 ( 2 _ 5 ) tan -1 (tan B) = tan -1 ( 5 _

2 )

∠A ≈ 21.80° ∠B ≈ 68.20°

Example 5 Application: Indirect Measurement

If an airplane takes off at a 35° angle with the ground, how far has the plane traveled horizontally when it reaches an altitude of 10,000 feet?

SOLUTION

Use the tangent ratio since the problem involves both legs.

tan 35° = 10,000

_ x

x · tan 35° = 10,000

_ x · x

x · tan 35° = 10,000

x = 10,000

_ tan 35°

≈ 14,281

The plane has traveled about 14,281 feet horizontally.

B

AC 6

11

B

AC 6

11

A C

B

5

2

A C

B

5

2

x

10,000feet

35°x

10,000feet

35°

Hint

You can also find the measure of the second acute angle of a right triangle by subtracting the first angle from 90°. For example, if m ∠A = 21.80°, then m ∠B = 90° - 21.80° = 68.20°.

Math Reasoning

Generalize When do you use the sine function and when do you use the inverse sine ( sin -1 ) function?

Sample: Use the sine function when you know the angle and want to find the side length. Use the inverse sine function when you want to find the angle and you know the side length.


INCLUSION

Have students create a poster of a 3-4-5 right triangle. They can highlight each side of the triangle in a different color and label the acute angles A and B. Then have students list the 6 trigonometric ratios for each angle using the corresponding color when writing each side length.

Lesson 117 799

Example 4

Students fi nd missing angle measures using trigonometric ratios and inverse trigonometric functions.

Error Alert Students may confuse the word inverse with reciprocal. Emphasize that inverse is used to fi nd the measure of an angle.


a. Find the measure of ∠A. Round to the nearest hundredth of a degree.

19

A

12

∠ ≈ 39.17°

b. Find the measures of ∠A and ∠B. Round to the nearest hundredth of a degree.

4 A

B

3

∠A ≈ 36.87°, ∠B ≈ 53.13°

Example 5

Students apply trigonometric ratios to indirect measurement.

Extend the Example

When the plane is at 10,000 feet, how far is the plane from where it left the ground? Round to the nearest foot. approximately 17,434 feet


A tree falls over and rests against a wall 10 feet off the ground. The tree now forms a 28° angle with the ground. How tall was the tree? Round to the nearest tenth of a foot. about 21.3 feet



Saxon Algebra 1800

Lesson Practice

a. Using the right triangle, find sin A, cos A, and tan A.

b. Using the right triangle, find all six trigonometric ratios for ∠B.

c. If ∠A = 49°, find sin A, cos A, and tan A. Round to the nearest ten-thousandth. sin 49° ≈ 0.7547, cos 49° ≈ 0.6561, tan 49° ≈ 1.1504

d. If ∠A = 67°, find csc A, sec A, and cot A. Round to the nearest ten-thousandth. csc 67° ≈ 1.0864, sec 67° ≈ 2.5593, cot 67° ≈ 0.4245

e. Find the value of x. Round to the nearest hundredth. x ≈ 29.06

f. Find the value of x and y. Round to the nearest hundredth. x ≈ 12.63; y ≈ 11.38

17

48°

y

x

AC

B

g. Find ∠A and ∠B. ∠A ≈ 32.28°; ∠B ≈ 57.72°

19

12

AC

B

h. A 10-foot ladder is placed on the side of a building 4 feet away from the base of the building along the ground. Find the measure of the angle the ladder makes with the ground. 66.42°

(Ex 1)(Ex 1)A

26

24

10

C B

A26

24

10

C B

a. sin A = 12_13

, cos A = 5_13

,

tan A = 12_5

a. sin A = 12_13

, cos A = 5_13

,

tan A = 12_5

B

35

bA C

B

35

bA C

(Ex 1)(Ex 1)b. sin B = 4_5 , cos B = 3_

5,

tan B = 4_3, csc B = 5_

4,

sec B = 5_3, cot B = 3_

4

b. sin B = 4_5 , cos B = 3_5,

tan B = 4_3, csc B = 5_

4,

sec B = 5_3, cot B = 3_

4

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

(Ex 5)(Ex 5)

*1. Find sin A, cos A, and tan A. *2. Find sin A, cos A, and tan A.

20 29

21 A

B

C

17

15

8

A

B C

(117)(117)

1. sin A = 20_29

, cos A = 21_29

,

tan A = 20_21

1. sin A = 20_29

, cos A = 21_29

,

tan A = 20_21

(117)(117)

2. sin A = 15_17

, cos A = 8_17

,

tan A = 15_8

2. sin A = 15_17

, cos A = 8_17

,

tan A = 15_8

36°

54°

40

x


800 Saxon Algebra 1

Lesson Practice

Problem d

Error Alert Students may take the reciprocal of 67 and then fi nd the sine. Emphasize that sin 67° must be found. Then, the reciprocal of that value is the cosecant.

Problem h

Scaff olding Draw a picture of the situation. Label the angle A. The hypotenuse should be labeled 10, and the adjacent side, or ground distance, should be labeled 4. Because the adjacent side and the hypotenuse are known, use the inverse cosine to fi nd the value of the angle.



“Explain the relationship among the six trigonometric ratios.” Sample: Sine and cosecant, cosine and secant, tangent and cotangent are reciprocals of each other.

“How can the trigonometric ratios be useful if the angle measure is not known?” Sample: Use the ratios with the inverse trigonometric functions to fi nd the angle measure.


Lesson 117 801

3. Write an equation for a direct variation that includes the point, (24, 3). y = 1_8

x

*4. If ∠A = 77°, find sin A, cos A, and tan A to the nearest ten-thousandth.

*5. Error Analysis Two students are finding the measure of ∠A. Which student is correct? Explain the error.

Student A

tan A = 9 _ 13

tan-1(tan A) = tan-1 ( 9 _ 13

)

A ≈ 34.7°

Student B

tan A = 13 _ 9

tan-1(tan A) = tan-1 ( 13 _ 9 )

A ≈ 55.3°

*6. Geometry In a right isosceles triangle, the acute angles are congruent. Find the measures of the acute angles. Then use the sine or cosine ratio to find the length of a leg of a right isosceles triangle to the nearest hundredth if the hypotenuse is 5 centimeters. 45°; 3.54 cm

*7. Multi-Step You are standing on the roof of a 70-foot-tall building looking across at another building. Use the picture to answer the questions. a. Find the distance from the bottom of the

building where you are standing to the top of the other building. 250 feet

b. Find the measure of ∠A. 16.26°

*8. Nature A tree casts a shadow of 25 feet along the ground. The angle from the ground to the top of the tree is 45°. How tall is the tree? 25 feet tall

25 feet45°

9. If $1100 is borrowed for 2 years at 9% simple interest, how much interest is paid? $198

*10. Navigation A submarine begins diving from the water’s surface at an angle of 7°. How far below the water’s surface is the submarine after it has traveled 3.4 miles?

3.4 milesx7°

*11. Generalize Explain the meaning of opposite leg and adjacent leg to an acute angle in a right triangle.

(56)(56)

(117)(117)

4. sin 77° ≈ 0.9744, cos 77° ≈ 0.2250, tan 77° ≈ 4.3315

4. sin 77° ≈ 0.9744, cos 77° ≈ 0.2250, tan 77° ≈ 4.3315

(117)(117)

13

9

A

B

C13

9

A

B

C

5. Student A; Sample: The tangent ratio is the opposite leg over the adjacent leg and Student B used adjacent over opposite.

5. Student A; Sample: The tangent ratio is the opposite leg over the adjacent leg and Student B used adjacent over opposite.(117)(117)

(117)(117)

70

240

c

B

C A

70

240

c

B

C A

(117)(117)

(116)(116)

(117)(117)

10. about 0.41 miles below the water’s surface10. about 0.41 miles below the water’s surface

(117)(117)Sample: The opposite leg is the leg of a right triangle that is opposite the acute angle and the adjacent leg is the leg that is next to the acute angle, but not the hypotenuse.

Sample: The opposite leg is the leg of a right triangle that is opposite the acute angle and the adjacent leg is the leg that is next to the acute angle, but not the hypotenuse.


Lesson 117 801

Practice3


Problem 3

Have students substitute values of x and y into the equation y = kx to fi nd k.

Problem 6

Suggest that students draw a right isosceles triangle and label the hypotenuse with the given length, 5 centimeters. Ask them if trigonometry is needed to fi nd the measure of the acute angles in the triangle. no; The acute angles measures can be determined without trigonometry because the sum of the measures of the angles in any triangle is 180°.

Problem 7


“What formula is used to fi nd the distance from the bottom of the building to the top of the other building?” the Pythagorean Theorem

“How is the trigonometric ratio determined for fi nding the measure of ∠A?” Since the length of the sides opposite and adjacent to ∠A are given, the tangent ratio is used.

Problem 9

Remind students that the formula for simple interest is I = Prt.


Saxon Algebra 1802

12. Error Analysis A $1500 investment earns 8% simple interest. Two students find the value of the account after 25 years. Which student is correct? Explain the error.

Student A

I = PrtI = 1500 · 0.08 · 25I = 3000$3000

Student B

I = PrtI = 1500(0.08)(25)I = 30003000 + 1500 = 4500$4500

13. Multi-Step A boy plans to invest $100 in an account that pays 10% interest compounded annually for 10 years. Another option is an account that earns 20% interest compounded annually for 5 years. Which will earn him more money, and how much more? 10% for 10 years earns $10.54 more.

14. Graph the cubic function y = x3 + 3. Use the graph to evaluate the equation for x = 0. See Additional Answers.

15. Error Analysis Two students draw a graph of the equation y = 2x3. Which student is correct? Explain the error. Student B; Sample: Student A graphed the parent function.

Student A Student B

x

y

O

8

4

2 4

-4

-2-4

-8

x

y

O

20

10

2 4

-10

-2-4

-20

16. Packaging A rectangular box has a volume of V = x3 + 3 cubic units. Use a table or graph to find the value of x that corresponds to a volume of 30 cubic units. x = 3

17. Use the quadratic formula to solve 2x2 + 9 = 9x. Check the solutions. x = 3 or x = 1.5

18. Find 12P3. 1320 19. Solve x2 + 5 = 9. x = ±2

20. Astronomy Near a planet, a satellite follows a trajectory described by the equation y = x

2

_ 8 + 7 _

4 . The trajectory is intercepted by a radio signal represented by the line

y = - 9x _

8 . At what coordinates will the radio signal intersect the trajectory?

21. Use the discriminant to find the number of real solutions of the equation x2 + 2x - 2 = 0. d = 12; two real solutions

22. Multiple Choice What is the domain of the function f(x) = 2 √ �� x + 6 - 1? B

A x ≥ -5 B x ≥ -6 C x ≥ 6 D x ≥ 0

23. Write Describe the graph of f(x) = √ �� x + 4 in terms of its parent function.

24. Solve and graph the inequality 3⎢8x + 2� < 12. -0.75 < x < 0.25 See Additional Answers.

(116)(116)

12. Student B; Sample: Student A found the interest earned, not the account’s value.

12. Student B; Sample: Student A found the interest earned, not the account’s value.

(116)(116)

(115)(115)

(115)(115)

(115)(115)

(110)(110)

(111)(111) (102)(102)

(112)(112)

20. (-2, 9_4 ) and (-7, 63_

8 )20. (-2, 9_4 ) and (-7, 63_

8 )

(113)(113)

(114)(114)

(114)(114)

23. Sample: The graph of f(x) = √ �� x + 4 can be rewritten in the form f(x) = √ �� x - c by changing + 4 to -(-4).The function is now y = √ �� x - (-4) , which is a translation 4 units left of the parent function.

23. Sample: The graph of f(x) = √ �� x + 4 can be rewritten in the form f(x) = √ �� x - c by changing + 4 to -(-4).The function is now y = √ �� x - (-4) , which is a translation 4 units left of the parent function.

(101)(101)


CHALLENGE

Identify trigonometric ratios that have the

value 5 _ 12 .

12

A

B

13

5tan B, cot A

802 Saxon Algebra 1

Problem 12

Point out to students that the investment total is the principal plus the interest.

Problem 14

Extend the Problem

Use the graph to evaluate the equation for x = -3. -24

Problem 17

Check that students write the equation in standard form and set it equal to 0 before applying the formula.

Problem 18

Remind students that for permutations, order does matter, so there are a greater number of possibilities than with combinations.

Problem 19

When students take the square root of a number, there are two answers: a positive and a negative one.

Problem 22


“Defi ne domain.” Sample: The values of x for which the function is defi ned.

“What restrictions would be on the domain of a square root?” The value in the square root must be positive or 0.

“What values make x + 6 positive or 0?” x ≥ -6

Problem 23

It may be helpful for students to write x + 4 as x - (-4).

Problem 24

Error Alert Students may forget to reverse the inequality sign when dividing by a negative number. Remind them that any time they multiply or divide by a negative number when solving an inequality, the symbol must be reversed.


Lesson 117 803

25. Multi-Step The sum of the squares of two consecutive odd numbers is 74. a. Write expressions for two consecutive odd numbers. x and x + 2

b. Write an equation to represent the problem. x2 + (x + 2)2 = 74

c. What is the possible solution(s)? 5 and 7 or -5 and -7

*26. If ∠A = 81°, find csc A, sec A, and cot A to the nearest ten-thousandth. csc 81° ≈ 1.0125, sec 81° ≈ 6.3925, cot 81° ≈ 0.1584

27. Multi-Step Anita figures that the value of her car, in thousands of dollars,

can be approximated by f(x) = 15 ( 4 _ 5 )

x

, where x is the number of years after the car’s manufacture. Evaluate the function for x = 0, 1, and 2 and then sketch the function. See Additional Answers.

28. Justify Explain why y > 4

y < 4

has no solutions but y ≥ 4

y ≤ 4

does. See Additional Answers.

29. What does a half-life mean? If a substance’s half-life is 25 hours, how many half-lives are there in 150 hours? A half-life is the amount of time it takes for half of a substance to remain; 6 half-lives

30. Tennis A tennis instructor has a budget of $2000 to buy new rackets. He will receive 2 free rackets when he places his order. The number of rackets, y, that he can get is given by y = 2000

_ x + 2, where x is the price per racket. a. What is the horizontal asymptote of this rational function? y = 2

b. What is the vertical asymptote? x = 0

c. If the price per racket is $200, how many rackets will he receive? 12 rackets

(104)(104)

(117)(117)

(108)(108)

(109)(109)

(Inv 11)(Inv 11)

(78)(78)


LOOKING FORWARD

Using trigonometric ratios will be further developed in other Saxon Secondary Mathematics courses.

Lesson 117 803

Problem 25

Error Alert When writing an expression for two consecutive odd numbers, students may write x and x + 1, rather than x + 2. Remind them that consecutive odd numbers differ by 2.

Problem 30

The horizontal asymptote of an equation in the form

y = a _____ x - b

+ c occurs where

y = c. The vertical asymptote occurs at values for which the domain is undefi ned, when x = b.


Saxon Algebra 1804

Warm Up

118LESSON

1. Vocabulary A ( permutation, factorial ) is an arrangement of outcomes in which the order does matter. permutation

Simplify.

2. 7! 5040 3. 6! _

4! 30

Simplify.

4. 7P3 210 5. 9P4 3024

In Lesson 111, you learned about permutations, a selection of items where order does matter. In some cases, however, the final group of items is all that matters, not the order in which the items were selected. A combination is a grouping of items where order does not matter.

Example 1 Comparing Combinations to Permutations

A teacher puts 4 essay questions on a test. They are labeled A, B, C, and D. Students are required to choose 3 questions to answer.

a. How many permutations of the 3 questions are possible?

SOLUTION

First, find the number of permutations.

4P3 = 4! _

(4 - 3)! =

4! _ 1!

= 4 · 3 · 2 · 1 _

1 = 24.

There are 24 permutations of the 3 test questions.

b. How many combinations of the 3 questions are possible?

SOLUTION

As the order of the questions chosen does not matter, choosing ABC is the same as ACB, CAB, CBA, BCA, and BAC. So, to find the number of combinations, list the 24 permutations and then cross out the duplicate sets.

ABC ABD ACB ACD ADB ADC

BAC BAD BCA BCD BDA BDC

CAB CAD CBA CBD CDA CDB

DAB DAC DBA DBC DCA DCB

That leaves 4 combinations of the 3 test questions.

(111)(111)

(111)(111) (111)(111)

(111)(111) (111)(111)


Solving Problems Involving Combinations

Math Reasoning

Analyze Why are there more permutations than combinations?

Sample: When order matters, there are more results.


Combinations are the number of ways a group of objects can be selected when order does not matter. For example, if you are chosen to be on a committee, whether you are chosen fi rst or last does not affect the makeup of the committee. Similarly, if grouping 3 letters, ABC is the same group as BCA. When compared to permutations, with which order does matter, there will be fewer possible groups. With a permutation, ABC is different from BCA.

It may be helpful to make a list of key words that are associated with a permutation and with a combination. However, be careful when instructing students to look for key words. For example, choosing 3 out of 5 people to be in a group is a combination, while choosing 3 out of 5 people to be in 1st, 2nd, and 3rd place is a permutation. It is most important to determine whether or not order matters in the situation.





Warm Up1

804 Saxon Algebra 1

118LESSON

Problem 3

Cancel common factors in the numerator and denominator to simplify the expression.

2 New Concepts

In this lesson, students solve problems involving combinations.

Example 1

Students fi nd the number of permutations and combinations.

Additional Examples

Five students volunteer to work a problem on the board. The teacher selects a student for each of 4 problems.

a. How many permutations of the students are possible? 120

b. How many combinations of the 4 students are possible? 5

TEACHER TIPAnalyze with the students a variety of situations to determine when order matters. That is, determine whether the situation requires permutations or combinations.


Lesson 118 805

In Example 1, there are 6 times as many permutations as combinations. For each set of 3 letters, there are 3 · 2 · 1 = 6 different ways to order the letters. To find the number of combinations, nCr , when selecting r out of n items, divide the number of permutations, nPr , by the number of ways to order r items, r!.

That is, nCr = nPr ___ number of ways to order r items

=

n! _ (n - r)!

_ r!

= n! _ r!(n - r)!

.

Combination FormulaThe number of combinations of n items taken r at a time is

nCr = n! _ r!(n - r)!

.

Example 2 Finding the Number of Combinations

a. At a restaurant 2 side dishes may be chosen. There are a total of 6 side dish choices. How many combinations are there?

SOLUTION

nCr = n! _

r!(n - r)! Use the combination formula.

6C2 = 6! _

2!(6 - 2)! Substitute n = 6 and r = 2.

= 6! _

2!4! Simplify inside parentheses.

= 720 _

2 · 24 = 15 Simplify.

There are 15 ways to choose 2 side dishes.

b. A company delivers fruit to its customers every month. There are 16 different types of fruit. Each customer can choose 12 types of fruit each year. How many combinations can each customer make?

SOLUTION

nCr = n! _

r!(n - r)! Use the combination formula.

16C12 = 16! __

12!(16 - 12)! Substitute 16 for n and 12 for r.

= 16! _

12!4! Simplify inside parentheses.

= 16 · 15 · 14 · 13 · 12! __

12!4! Rewrite 16! as 16 · 15 · 14 · 13 · 12!.

= 16 · 15 · 14 · 13 __

4! Cancel 12!.

= 1820 Simplify.

There are 1820 ways to choose 12 fruits. Online Connection


Math Language

nCr is read “n choose r.” So, 4C3 is read, “Four choose three.”


ENGLISH LEARNERS

For Example 2, explain the meaning of the word choices. Say:

“Choices means the best or most wanted items. For example, favorite color choices for baby girl’s clothing are pink and yellow.”

Have students think about their choices of lunch foods, entertainers, or books. Have them share their thoughts with the class using the word choices in a complete sentence.

Lesson 118 805

Example 2

Students fi nd the number of combinations.

Error Alert Students may confuse permutations and combinations. Emphasize that a general group means a combination. A group with specifi c positions or titles would mean permutations.

Additional Examples

a. In class, there are 8 report topics. A student must research 3 of them during the year. How many combinations are there for the student? 56

b. At a salad bar, there are 20 types of food. Each customer is allowed to choose 15 of them. How many combinations of plates can be made? 15,504



Saxon Algebra 1806

Example 3 Application: Probability

An animal shelter has 20 dogs. Each month 3 of the dogs are randomly chosen to be pets-of-the-month on the website.

What is the probability that Jumbo, Fluffy, and Max will be selected?

SOLUTION

Note that the order in which the dogs are chosen does not matter, so it is a combination.

20C3 = 20! __

3!(20 - 3)! Use the formula. Substitute 20 for n and 3 for r.

= 20!_

3!17!= 1140 Simplify.

There are 1140 ways 3 dogs can be chosen, but only one way that a particular set of 3 dogs, {Jumbo, Fluffy, and Max}, can be chosen. So, the probability of choosing that set is 1

_ 1140 .

Lesson Practice

A teacher selects 2 students from a group of 5 students.

a. How many permutations of the 2 students are possible?

b. How many combinations of the 2 students are possible?

c. To color a map, each student chooses 4 markers from a box of 8. How many combinations are there? 70

d. To pick a parent committee for his class, the teacher chooses 9 parents out of the 22 volunteers. How many combinations can the teacher make? 497,420

e. A cook has 18 possible ingredients for soup. He only uses 4 ingredients. What is the probability he will pick beans, corn, rice, and carrots? 1_

3060

(Ex 1)(Ex 1)20 permutations20 permutations

(Ex 1)(Ex 1)10 combinations10 combinations

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

*1. Photography A photographer wants to take a picture of a group of 4 students from a class of 15. How many different pictures could she take? 1365

Calculate each combination.

*2. 11C4 330 3. 9C7 36 4. 12C5 792

*5. Write Explain the difference between permutations and combinations.

*6. Verify Show that 8C3 = 8P3 ___ number of ways to order 3 items.

.

Sample: 8C3 =8!_

3!(8 - 3)!=

8!_(8 - 3)!

·1_3!

=

8!_(8 - 3)!_

3!=

8P3___number of ways to order 3 items

(118)(118)

(118)(118) (118)(118) (118)(118)

(118)(118)5. Sample: With permutations order matters, but with combinations, order does not matter.

5. Sample: With permutations order matters, but with combinations, order does not matter.

(118)(118)

Hint

Remember, the theoretical probability of an event is

# of favorable outcomes __ total number of outcomes


Materials: pattern blocks

Use pattern blocks to make groups of particular shapes or colors. Then, use them to make particular patterns. Have students use the blocks to show the number of different ways the groups or combinations can be made. Discuss each time whether the order mattered or not, and thus, whether it would be a permutation or combination. Show, each time, that the number of permutations is greater than the number of combinations.

INCLUSION

806 Saxon Algebra 1

Example 3

Students use combinations to fi nd the probability of an event.

Extend the Example

What is the probability that those three will not be chosen together? 1139 _____

1140


A student has 20 stickers and wants to trade 10 of them. A friend randomly chooses 10 of them. What is the probability that those were her favorite 10 stickers? 1 _______

184,756

Lesson Practice

Problems a and b

Scaff olding First, fi nd 5P2. Then, fi nd 5C2. Compare to make sure that the number of combinations is less than the number of permutations.

Problem e

Error Alert Students may fi nd the number of permutations because they think about selecting the ingredients in a particular order. The order in which they were chosen does not matter.



“Explain the difference between a permutation and combination.” Sample: There are more possibilities with a permutation because order does matter. With combinations, order does not matter.

“Explain how to fi nd the probability of a particular group being chosen from a larger population.” Sample: Find the number of ways the number in the group can be chosen from the larger population using combinations. Because there is only one way to form that group, the probability is 1 out of the number of possibilities.


Lesson 118 807

*7. Multiple Choice Find 5C3. A

A 10 B 12 C 20 D 60

*8. Nutrition Teenage girls need 3 servings of dairy products per day. How many combinations can a girl make from 10 different dairy products? 120

9. Find the LCM of (15x - 10) and (3x - 2). 5(3x - 2)

*10. Geometry How many straight line segments can be formed by connecting any 2 of 8 points? 28

*11. Multi-Step A student chooses 9 stuffed animals to give to a charity. He had a total of 25 stuffed animals. a. How many combinations of 9 animals could be chosen? 2,042,975

b. If one stuffed animal has already been chosen, then what is the probability that he chooses his 8 favorite animals.

*12. Find the six trigonometric ratios for ∠A. sin A = 24_

25, cos A = 7_

25, tan A = 24_

7, csc A = 25_

24, sec A = 25_

7, cot A = 7_

24

13. Multi-Step A kite is caught in the top of a 12-foot tree and a string 20 feet long is stretched out to the ground. a. Find the distance along the ground from the base of the tree to the end

of the string. 16 feet

b. Find ∠A, the angle the string makes with the ground. 36.87°

14. Coordinate Geometry A right triangle has coordinate A(2, 1), B(8, 9), and C(8, 1). Plot the points and find the measure of acute angle A. 53.13°

15. If $9200 is borrowed for 3 years at 5% simple interest, how much money will be owed after 3 years? $10,580

16. Bonds A woman invests $20,000 in a bond that pays 6% interest compounded annually. How much interest will she earn in 5 years? $6764.51

*17. Error Analysis A CD earns 4% interest compounded quarterly. A woman deposits $10,000 for 20 years. Two students find the value of her account. Which student is correct? Explain the error.

Student A

A = 10,000(1 + 0.01)80

A = 10,000(1.0180)A = 22,167.15$22,167.15

Student B

A = 10,000(1 + 0.04)20

A = 10,000(1.0420)A = 21,911.23$21,911.23

18. Find 10!

_ 5!

. 30,240 19. Simplify 4 _ √ � 3 - 2

. -4√ � 3 - 8

20. Solve this system by substitution: y = x2 - 5

y = 4x

. (-1, -4) and (5, 20)

(118)(118)

(118)(118)

(57)(57)

(118)(118)

(118)(118)

1_735,471

1_735,471

(117)(117)

2012

bAC

B

2012

bAC

B(117)(117)

(117)(117)y

2 4 6 8

2

4

6

8

0A C

B

x

14. y

2 4 6 8

2

4

6

8

0A C

B

x

14.

(116)(116)

(116)(116)

(116)(116)

17. Student A; Sample: Student B used interest compounded annually, not quarterly.

17. Student A; Sample: Student B used interest compounded annually, not quarterly.

(111)(111) (103)(103)

(112)(112)

725

24

A

BC


Lesson 118 807

Practice3


Problem 9

Remind students to factor fi rst.

Problem 13


“What lengths do you know?” the height of the tree and the length of the string

“What theorem will help you fi nd the distance on the ground?” Pythagorean

“What will you use to fi nd the measure of the angle?” inverse sine

Problem 14

Error AlertWhen fi nding the length of the sides, students may count the number of lines. Explain that a distance is measured by the number of spaces.

Problem 15

Remind students to change the percent to a decimal before multiplying.

Problem 16

Extend the Problem

How much more interest would be earned if it were calculated semi-annually? $113.82

Problem 19

Check that students multiply

by √ � 3 + 2

______ √ � 3 + 2

to remove the radical

from the denominator.


Saxon Algebra 1808

21. Football A punter kicks a football straight up in the air from 2 feet off the ground with an initial velocity of 75 feet per second. Using y = -16t2 + 75t + 2, write an equation to model the situation and use the discriminant to determine if the ball will reach a height of 45 feet.

22. Evaluate y = √ �� 3 _ x + 2 for x = 6. Round to the nearest tenth. y ≈ 1.6

23. Multiple Choice What is the solution to the equation x3 - 27 = 0 ? A

A 3 B 9 C 27 D 0

24. Model What is the equation for the parent function of a cubic equation? y = x3

25. Find the solution of x2 = 45. Round the answer to three decimal places. x ≈ ±6.708

26. Multi-Step Colin is experimenting with a type of cell that multiplies 5 times each day. a. On Monday, there are 500 cells. If all cells survive, how many are there on Friday?

b. Approximately one fourth of the cells die off each day. If the cells still multiply at the same rate, what rule represents a geometric sequence to represent the number of cells remaining each day? a(4 3_

4 )n-1

c. Use the rule to find the number of cells on Friday if there are 500 on Monday. about 254,533

27. Multi-Step Tickets to a school play are sold to teachers for $3 each and to students for $0.50 each. The drama class hopes to earn at least $200 from the ticket sales. The theater seats 250 people. a. Write and graph a system of linear inequalities to describe the situation.

b. If 15 teachers buy tickets, is it possible for the drama class to meet their goal? Explain. No; Sample: There is no ordered pair with 15 teachers that is in the solution set.

28. Estimate Estimate to the nearest whole number the value of the zeros for the equation 2v2 + 20v = 21. 1 and -11

29. Factor completely -88z3 - 2r2z3 - 30rz3. -2z3(r + 11)(r + 4)

30. Radioactive glucose is used in cancer detection. It has a half-life of 100 minutes. How much of a 320 mg sample remains in the body after 10 hours? First determine how many half-lives there are in a 10-hour period if the half-life is 100 minutes. 6 half-lives; 5 mg

(113)(113)

21. Use the equation 45 = -16t2 + 75t + 2. Then 0 = -16t2 + 75t - 43 and the discriminant is 2873, so the ball will reach a height of 45 feet.

21. Use the equation 45 = -16t2 + 75t + 2. Then 0 = -16t2 + 75t - 43 and the discriminant is 2873, so the ball will reach a height of 45 feet.

(114)(114)

(115)(115)

(115)(115)

(102)(102)

(105)(105)312,500312,500

(109)(109)

27a.⎧ ⎨

⎩

3x + 1_2y ≥ 200

x + y ≤ 250;

x

y

Teachers

Stu

den

ts

60 120 180 240

44

88

132

176

220

0

27a.⎧ ⎨

⎩

3x + 1_2y ≥ 200

x + y ≤ 250;

x

y

Teachers

Stu

den

ts

60 120 180 240

44

88

132

176

220

0

(110)(110)

(79)(79)

(Inv 11)(Inv 11)



There are 13 different toppings available at a pizza place. A family chooses 3 different toppings. What is the probability that the toppings are three of the following: pineapple, mushrooms, olives, tomatoes, or peppers? 10

____ 286 = 5 ____ 143

Solving problems involving combinations will be further developed in other Saxon Secondary Mathematics courses.

808 Saxon Algebra 1

Problem 22

Remind students that the radical sign acts as a grouping symbol. Simplify inside the radical before taking the square root.

Problem 27

Guide students to understand that one equation will represent the number of tickets sold. The other equation will represent the number of dollars earned.


Lesson 119 809

Warm Up

119LESSON

1. Vocabulary The highest or lowest point (turning point) on a parabola is called the of the parabola. vertex

2. Find the axis of symmetry for the graph of y = 2x2 + 8x + 2. x = -2

Determine the slope and the y-intercept of each equation.

3. y = 0.5x - 3.5 m = 0.5; b = -3.5 4. -8x + 2y = 10 m = 4; b = 5

5. Multiple Choice Which is the zero of the function y = x2 + x - 2 ? B

A x = 2 B x = -2 C x = -1 D x = 3

A function family is a set of functions whose graphs have similar characteristics. A function family can be formed through transformations of one function, called the parent function. Transformations of the parent function can cause the graph of the function to move vertically or horizontally, be stretched or compressed, or be reflected across an axis.

The following chart gives three examples of parent functions, along with the characteristics of each parent function. Those characteristics can be helpful in identifying functions within that family.

(89)(89)

(89)(89)

(49)(49) (49)(49)

(89)(89)


Linear Function Quadratic Function Exponential Function

Parent Function

f (x) = x f (x) = x2 f (x) = b x

Graphx

y

x

y

x

y

Domain real numbers real numbers real numbers

Range real numbers real numbers ≥ 0 real numbers > 0

Maximum/Minimum none vertex none

Rate of Change constant not constant not constant

Linear Function Quadratic Function Exponential Function

Parent Function

f (x) = x f (x) = x2 f (x) = b x

Graphx

y

x

y

x

y

Domain real numbers real numbers real numbers

Range real numbers real numbers ≥ 0 real numbers > 0

Maximum/Minimum none vertex none

Rate of Change constant not constant not constant

Graphing and Comparing Linear, Quadratic,

and Exponential Functions

Online Connection


Math Reasoning

Write In your own words, describe a parent function.

Sample: A parent function is a basic graph that can be changed in many ways to form a function family.

SM_A1_NL_SBK_L119.indd Page 809 1/18/08 10:45:27 PM admini1 /Volumes/ju114/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L11

LESSON RESOURCES

Student Editition Practice Workbook 119


Master C119

MATH BACKGROUND

The study of functions begins with linear functions because they are the simplest. They are defi ned over all real numbers with a constant rate of change. They are straight lines. Quadratics are a bit more complicated. Their rate of change is not constant and they are not straight lines. In fact, their range is restricted and each one has a maximum or minimum. They are symmetric. Exponential functions have even more limitations. They are not straight lines or symmetric in their shape.

Each type of function has a parent function that shows the characteristics of its family. Understanding the basic shape is critical to recognizing to which family a function belongs. Learning how changes in the equation affect the appearance of the graph is explored further in other Saxon High School Math courses.

Warm Up1

119LESSON

Lesson 119 809

Problem 4

Remind students to transform the equation into slope-intercept form fi rst.

2 New Concepts

In this lesson, students graph and compare linear, quadratic, and exponential functions.

TEACHER TIPHave students describe the shape of each function in their own words before discussing the characteristics of each.


Saxon Algebra 1810

Example 1 Matching Function Families and Graphs

a. Which of the following graphs represents an exponential function?

x

y

O

4

6

2 4-2-4

Graph A

x

y

O

4

6

2 4-2-4

Graph B

x

y

O

4

2

4

-2

-2-4

-4

Graph C

SOLUTION

Graph B displays the shape of an exponential function.

b. Use the graph to identify the function family.

SOLUTION

This graph has the shape of the quadratic function. It is the graph of f (x) = -x2 + 3.

c. Use the graph to identify the function family.

SOLUTION

This graph has the shape of a linear function. It has a slope of 2 and a y-intercept of 1. It is the graph of f (x) = 2x + 1.

Example 2 Matching Function Families and Tables

Use each table of values to identify the function family.

a. x -3 -2 -1 0 1 2 3

f (x) 1

_ 27 1 _ 9 1

_ 3 1 3 9 27

SOLUTION

Plot the points on a graph and connect them using a smooth curve. From the graph you can tell that the function belongs to the exponential function family. It is a graph of f (x) = 3 x . The values for f(x) increase more steeply as x increases, so it shows exponential growth.

x

y

O

4

2

4

-2

-4

-4

x

y

O

4

2

4

-2

-4

-4

x

y

O

4

2

42

-2

-4 -2

-4

x

y

O

4

2

42

-2

-4 -2

-4

x

y

O

16

24

8

8 16-8-16

x

y

O

16

24

8

8 16-8-16

Hint

Recall that y = 6x2 and f(x) = 6x2 describe the same function. The form that uses f(x) is called function notation.

Math Reasoning

Analyze Is there a maximum number of functions that can be in a function family? Explain.

no; Sample: A parent function can be transformed in an infi nite number of ways by adding different numbers to it or multiplying it by different numbers.


ENGLISH LEARNERS

For Example 2, explain the meaning of the word steep. Draw a picture of two mountains on the chalkboard. Be sure one mountain is steep and the other is not steep. Say:

“A steep mountain is diffi cult to walk up because of its sharp incline.”

Have the students decide which mountain in the drawing is steeper.

Next have students use grid paper to draw two lines with positive slopes. Ask:

“Which line is steeper? Which line has the greater slope?”

Explain to students how the slope of a line and the steepness of a line are directly related.

810 Saxon Algebra 1

Example 1

Students match function families with their graphs.


a. Which of the graphs from Example 1 represents a quadratic function? A

b. Use the graph to identify the function family. exponential

x

y

O42

20

40

-4 -2

60

100

80

c.

x

y

O

4

84

-4

-8 -4

-8

Use the graph to identify the function family. linear


Lesson 119 811

b. x -3 -2 -1 0 1 2 3

f (x) 8 3 0 -1 0 3 8

SOLUTION

Plot the points on a graph and connect them using a smooth curve.

x

y

O

4

2

2 4

-2

-2-4

You can see that this function has a graph similar to the graph of f (x) = x2, but it is translated down one unit. This function belongs to the quadratic function family. It is the graph of f (x) = x2 - 1.

c. Identify the table of values that shows a linear function family.

Table 1 x -3 -2 -1 0 1 2 3

f (x) -9 -4 -1 0 -1 -4 -9

Table 2 x -3 -2 -1 0 1 2 3

f (x) 1

_ 125 1

_ 25 1 _ 5 1 5 25 125

Table 3 x -3 -2 -1 0 1 2 3

f (x) 0 2 4 6 8 10 12

SOLUTION

For a function to be linear, it must have a constant rate of change. Determine which of these tables shows a constant rate of change of f (x) as x increases by 1.

Table 1: In the first row, f (-3) = -9 and f (-2) = -4, which is a difference of 5. But the difference between f (-2) and f (-1) is only 3. This is not a constant rate of change, so the function is not linear.

Table 2: The difference between f (1) and f (2) is 20, but the difference between f (2) and f (3) is 100, so this function is also not linear.

Table 3: f (-3) = 0 and f (-2) = 2, which is a difference of 2. Each time x increases by 1, the value of f(x) increases by 2. This is a constant rate of change.

Table 3 shows a linear function.

Caution

A constant rate of change does not have to be a positive number. In the function f(x) = -3x, the constant rate of change is -3. Each time x increases by 1, f(x) decreases by 3.


INCLUSION

Give each student a card with each of the three parent functions on it. Have students trace each new function that they must categorize on a sheet of thin paper. Then, have them translate, rotate, or fl ip it until it lines up with the general shape of a parent function. Emphasize that it will not match identically, but when their shapes look similar, then the functions belong to the same family.

Lesson 119 811

Example 2

Students use a table of values to identify the function family.


Use each table of values to identify the function family.

a. x -2 -1 0 1 2 3f (x) 0.01 0.1 1 10 100 1000

exponential

b. x -2 -1 0 1 2 3f (x) 40 10 0 10 40 90

quadratic

c. Identify the table of values that shows a linear function.

Table 1x -2 0 1 5 7 8

f (x) 7 9 11 13 15 17

Table 2

x -2 -1 0 1 2 3

f (x) -4 -6 -8 -10 -12 -14

Table 3

x -2 0 1 5 7 8

f (x) -3 -2 0 3 7 11Table 2

Error Alert Students may decide that the ordered pairs in a table of values represent a line based only on the f(x) values. Point out to students that linear functions can be identifi ed by their constant rate of change. Remind students that the rate of change is found by comparing the change in f(x) values with the change in corresponding x values.


Saxon Algebra 1812

Example 3 Identifying the Function Family from a Description

For each description, state whether the description best fits a linear, quadratic, or exponential function.

a. The rate of change is always the same. The graph is always decreasing.

SOLUTION

Because the rate of change is always the same, the function is linear.

b. The rate of change is not always the same. The graph is always increasing. The graph is a curve that gets steeper as the x-values increase.

SOLUTION

The rate of change is not always the same, so the function family is not linear. The graph is always increasing, so it is not quadratic.

The function is exponential.

c. The rate of change is not always the same. The graph changes direction at a minimum point at which y = -1.

SOLUTION

The rate of change is not constant, so the function family is not linear. It has a minimum point at which y = -1 where the graph changes direction, so it is quadratic.

d. The graph is always decreasing. The graph is a curve that gets less steep as the x-values increase.

SOLUTION

The graph is always decreasing, so it is not quadratic.

The graph is a curve that gets less steep as x increases, so it is exponential.

Linear, quadratic, and exponential functions can be used to model real-world situations.

Linear models apply to situations with a constant rate of change. An example of a function with a positive rate of change is the distance traveled by a train that travels at a constant speed. An example of a function with a negative rate of change is the amount of water left in a bucket with a constant leak.

Quadratic models may apply in situations with a maximum or minimum value because the graph of a quadratic function changes direction at the vertex. For example, if a ball is thrown up in the air, its height at time t is modeled by a quadratic function.

Exponential models can be used for situations where values are always increasing or always decreasing, but not at a constant rate. Examples of exponential situations are population growth and radioactive decay.

Math Reasoning

Justify The graph of a function crosses the x-axis twice. Could the function be linear? Explain.

no; Sample: A linear function always increases or always decreases, so once it crosses the x-axis, it cannot cross it again.


ALTERNATE METHOD FOR EXAMPLE 3

Have student sketch a graph that fi ts all of the characteristics given. Then, based on the sketch, determine to which family function it belongs.

812 Saxon Algebra 1

Example 3

Students identify the function family from a description.


For each description tell whether the description best fi ts a linear, quadratic, or exponential function.

a. The function is always increasing. Its rate of change is 2. linear

b. The range is restricted to real numbers greater than or equal to 0. quadratic

c. The rate of change is not constant. In fact, sometimes the graph is increasing and sometimes it is decreasing. quadratic

d. The function is always increasing. Its rate of change is not constant. exponential

Example 4

Students identify an appropriate model for a real-world situation.

Extend the Example

Write an equation for the number of bacteria cells in the laboratory dish. y = 1(2)x-1


a. The number of people who hear the news doubles every hour. exponential

b. the wage of a dog walker who earns $2 per hour linear

c. the height of a disc that is thrown upward from a 2-meter tall wall quadratic


Lesson 119 813

Example 4 Identifying an Appropriate Model

Identify the appropriate model for each of the following situations.

a. the height of a ball thrown upward from an initial height of 5 feet

SOLUTION

When a ball is thrown upward, it goes up for a short time and then changes direction and comes down again. Its height has a maximum value. Therefore, a quadratic function models this situation.

b. the cost of a tank of gas when gas costs $3.25 per gallon

SOLUTION

The cost of gas is a constant rate. For every additional gallon, the price of the tank of gas increases by $3.25. Therefore, a linear function models this situation.

c. the number of bacteria cells in a laboratory dish when each cell divides into two cells every day

SOLUTION

Imagine the first day there was only one cell in the dish. The next day, there would be 2 cells, and the following day, 4 cells. The number of cells doubles every day, so it is not a constant rate of change but an increasing rate of change. Therefore, an exponential function models this situation.

Lesson Practice

Identify the function family represented by each graph.

a.

x

y

O

4

2 4

-2

-4

-4

b. x

y

O2 4-2-4

-4

-6

c.

x

y

O

4

6

2 4-4 -2

Use the table of values to identify the function family.

d. x -2 -1 0 1 2f (x) 4 7 10 13 16

linear

e. x -2 -1 0 1 2f (x) 4 2 0 2 4

quadratic

Tell whether the function family is linear, quadratic, or exponential.

f. The graph is always increasing with a constant rate. linear

g. The graph changes direction at a maximum of 3. quadratic

h. The graph is always increasing and it gets steeper as x increases.

(Ex 1)(Ex 1)

a. linear

b. quadratic

c. exponential

a. linear

b. quadratic

c. exponential

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

exponentialexponential

Math Reasoning

Formulate Write a function that could be used to find the cost of gas in Example 4b.

f(x) = 3.25x


Lesson 119 813

Lesson Practice

Problem a

Scaff olding First determine whether the rate of change is constant. Then, if not, see if the graph is always increasing or decreasing.

Problem h

Error Alert Students may decide that the graph represents a linear function from the fi rst part of the statement. Point out that the function is not constant as indicated in the second part of the statement.



“What is a family of functions?” Sample: Functions that have the same general shape.

“Describe the characteristics of a linear function.” Sample: It is always increasing or decreasing at a constant rate of change. It has no maximum or minimum.

“How do quadratic and exponential functions differ?” Sample: The range of the exponential parent function does not include 0. Exponential functions are always increasing or decreasing. Quadratic functions have a maximum or minimum.



Saxon Algebra 1814

*1. Error Analysis Students were asked to write an equation that has a quadratic parent function. Which student is correct? Explain the error.

Student A

f (x) = 2(x - 1)2 - 4

Student B

f (x) = 2x + 5

*2. Identify the function family. exponential

x

y

O

4

6

2 4-2-4

*3. Identify the function family. quadratic

x

y

2

2 4

-2

-2

-4

*4. Write How can a parent function be used to graph a family of functions? Sample:Graph the parent function and then graph a series of transformations of it.

*5. Interior Decorating A type of carpet sells for $12 per square foot. Installation is an additional $500. A function can be written to determine the price for installing carpet in a square room with floor x feet in length. Identify the function family and the parent function. quadratic; f(x) = x2

(119)(119)1. Student A; Sample: Student B wrote an equation that does not have an x2 term in it.

1. Student A; Sample: Student B wrote an equation that does not have an x2 term in it.

(119)(119)

(119)(119)

(119)(119)

(119)(119)

Identify the appropriate model for each of the following situations.

i. the height of an arrow that is shot upwards from the edge of a cliff quadratic

j. the number of radioactive particles remaining in a sample when, at the end of each hour, the number of radioactive particles is half of what it was at the beginning of the hour exponential

k. the amount of money you can earn babysitting when you charge $5 per hour of babysitting linear

(Ex 4)(Ex 4)


814 Saxon Algebra 1

Practice3


Problem 1

Point out to students that the function written by Student A has a binomial factor that is raised to the second power. Have them use the FOIL method to see that the function contains a quadratic term.

Problem 5

Extend the Problem

Find the cost of carpeting a room with a side length of 13 feet. $2528


Lesson 119 815

6. Error Analysis Two students find 18C5. Which student is correct? Explain the error.

Student A

18C5 = 18! _

5!(18 - 5)!

= 8568

Student B

18C5 = 18! _

(18 - 5)!

= 1,028,160

*7. Multi-Step A restaurant charges $16 for a large pizza plus $1.50 per topping. a. Which type of function best describes the cost of a pizza? linear

b. Write a function that describes the cost of a pizza with x toppings.

8. Jewelry A jewelry store sells necklaces for the cost of the chain plus the cost of the beads. Chains cost $20 each, and beads cost $7.50 each. To which function family does the equation that describes the cost of a necklace with x beads belong? linear

*9. Multiple Choice Which of the following describes the graph of y = 5 x ? C

A quadratic B linear C exponential D none of these

*10. Geometry A triangle has base b and height b - 4. If its area is written as a function, to which function family does the function belong? quadratic

11. Identify the function family to which y = 2 - 1100x belongs. linear

*12. Write A fair coin is flipped many times in a row. The probability of all flips resulting in heads is given by P(all heads) = ( 1 _

2 )

x , where x is the number of flips.

What type of function is this? exponential

13. Multi-Step A teacher randomly selects 4 helpers from her class of 22. a. How many ways can 4 helpers be selected? 7315

b. What is the probability that Shawn, Tonia, Torie, and Reid are all chosen? 1_7315

*14. If ∠A = 14°, find sin A, cos A, and tan A to the nearest ten-thousandth. sin 14° ≈

0.2419, cos 14° ≈ 0.9703, tan 14° ≈ 0.2493 15. Error Analysis Two students are finding the value of x in the figure. Which student

is correct? Explain the error.

Student A

sin 63° = x _ 21

21 · sin 63° = x18.71 ≈ x

Student B

cos 63° = x _ 21

21 · cos 63° = x9.53 ≈ x

*16. Aviation An airplane begins making its descent at an angle of 11° with the horizontal. If the plane is at an altitude of 8000 feet and will remain at this angle throughout its descent, how far away is the plane from its landing point? about 41,927 feet

(118)(118)6. Student A; Sample: Student B used the formula for permutations.

6. Student A; Sample: Student B used the formula for permutations.

(119)(119)

f(x) = 1.5x + 16f(x) = 1.5x + 16

(119)(119)

(119)(119)

(119)(119)

(119)(119)

(119)(119)

(118)(118)

(117)(117)

A

x

CB

63°21

A

x

CB

63°21(117)(117)

15. Student B; Sample: The cosine ratio is the adjacent leg over the hypotenuse and x represents the adjacent leg.

15. Student B; Sample: The cosine ratio is the adjacent leg over the hypotenuse and x represents the adjacent leg.

8000x

11°8000

x

11°11°

(117)(117)


Lesson 119 815

Problem 6

Remember that the number of combinations is less than the number of permutations.

Problem 8

Extend the Problem

What is the cost of a necklace with 5 beads? $57.50

Problem 11

Rearrange the equation to be in the form y = mx + b.

Problem 13

Error Alert Students may try to use the formula for permutations. Remind them that with a group of four helpers, the order does not matter.

Problem 15

Review the acronym SOH–CAH–TOA to help students remember the trignometric ratios.


Saxon Algebra 1816

17. Solve this system by graphing: y = x2 - 2

y = 2x - 5

no solution.

18. Use the discriminant to find the number of real solutions of the equation 9x2 - 24x + 16 = 0. d = 0; one real solution

19. Business A baseball-card seller models the costs for attending a sale as the function y = $250 + $0.25 √ �� n + 30 . In this function, n represents the number of cards sold and 30 is the number of cards given out free as prizes. What is the domain of the function? Explain. n ≥ 0; Sample: The domain is n ≥ -30, but in the context of the problem the number of cards cannot be negative.

20. Graph the cubic function y = x3 - 2. Use the graph to evaluate the equation for x = 0.

21. Verify Show that a $500 investment earning 6% interest compounded annually for 3 years will earn more than earning 6% simple interest for 3 years.

22. Multiple Choice $1000 is invested in an account paying 5% interest compounded quarterly. What is the value of the account after 20 years? D

A $1282.04 B $2000 C $2653.30 D $2701.48

23. Find 8C3. 56 24. Simplify 6 _

√ � 7 - 3 √ � 5 . -

3√ � 7_19

- 9√ � 5_19

25. Solve -2m2 - 12m = 10 by completing the square. m = -1 or m = -5

26. Multi-Step The expression - √ �� x - 4 , where x is the time in seconds, represents the change in temperature as water cools when ice cubes are placed in a glass of water. a. For what value of x is the temperature change equal to -4 ? x = 20

b. What must be done to isolate the radical when solving for x?

c. Graph the radical equation y = - √ �� x - 4 .

27. Identify the function family to which y = 10 0 (-3x) belongs. exponential

28. Write Explain why you would not use the quadratic formula for the equation x2 - x - 2 = 0 ? Sample: It can be easily factored so the quadratic formula would be unnecessary work.

29. Verify Use the formula for permutations to verify that 6P2 = 30.

30. Given the function f (x) = ax2 + bx + c, describe the general shapes of the graphs of f (x) when: parabola; line; parabola; horizontal line a. a ≠ 0 and b ≠ 0;

b. a = 0 and b ≠ 0;

c. a ≠ 0 and b = 0;

d. a = 0 and b = 0.

(112)(112)

x

y

O

4

2

4-2-4

-4

17.

x

y

O

4

2

4-2-4

-4

17.

(113)(113)

(114)(114)

(115)(115)

20.

x

y

O

8

4

42-2-4

-8

;

y = -2

20.

x

y

O

8

4

42-2-4

-8

;

y = -2

(116)(116)

21. Sample: The amount for simple interest results in I = 500(0.06)(3) = 90 for an account balance of $590, but compound interest is A = 500(1.06) 3 = 595.51 for an account balance of $595.51.

21. Sample: The amount for simple interest results in I = 500(0.06)(3) = 90 for an account balance of $590, but compound interest is A = 500(1.06) 3 = 595.51 for an account balance of $595.51.

(116)(116)

(118)(118) (103)(103)

(104)(104)

(106)(106)

26 b. Sample: To isolate the radical, both sides of the equation must be multiplied by -1.

26 b. Sample: To isolate the radical, both sides of the equation must be multiplied by -1.

(119)(119)

(110)(110)

(111)(111)

(Inv 10)(Inv 10)

x

y

O

4

2

8 12

-2

-4

26c.

29. 6P2 = 6!_(6 - 2)!

= 6!_4!

=

6 · 5 · 4 · 3 · 2 · 1__4 · 3 · 2 · 1 = 6 · 5 = 30


LOOKING FORWARD

Graphing and comparing linear, quadratic, and exponential functions will be further developed in other Saxon Secondary Mathematics courses.

CHALLENGE

Describe a situation that can be modeled by an exponential function. Sample: The number of pairs of rabbits multiplies every month.

816 Saxon Algebra 1

Problem 17

Remember the solution to a system of equations is where the graphs intersect.

Problem 19

Remind students that the radicand must be positive.

Problem 22

Extend the Problem

How much interest has been earned? $1701.48

Problem 24

Use the conjugate to take the radical out of the denominator.

Problem 27

Error Alert Students may want to describe this as a cubic function because there is a 3 in the exponent. Because there is a variable, x, in the exponent, it must be an exponential function.

Problem 29

Extend the Problem

What is 6C2? 15

Problem 30

Encourage students to substitute values for a and b that fi t the criteria given. Then, analyze the general shape of the graph.


Lesson 120 817

Warm Up

120LESSON

1. Vocabulary In probability, the set of all outcomes that are not in an event is the (complement, converse) of that event. complement

2. If the probability of an event happening is 3 _ 4 what is the probability of the

complement of the event happening? 1_43. What is the probability of tossing a number cube and getting a number

greater than 4? 1_34. Find the area of a rectangle with length of 25 inches and width of

5 inches. 125 sq in.

5. Find the area of a circle with a radius of 2 inches. Use 3.14 for π. 12.56 sq in.

In Lesson 14, you learned that the theorectical probability of an event is the ratio of the number of favorable outcomes to the total number of outcomes. The same definition applies to geometric probability when working with the area of geometric shapes.

Example 1 Finding Geometric Probability with Rectangles

A rectangular vegetable garden is 10 feet by 15 feet. Tomatoes are planted in a rectangular area that is 3 feet by 12 feet. A bird lands in the garden. What is the probability that it lands in the tomato area?

15 feet

10 feetTomatoes

3 feet

12 feet

SOLUTION

favorable outcomes

__ total outcomes

= area with tomatoes __

area of entire garden

= 3 · 12 _

10 · 15 Use A = l · w to find the

area of each figure.

= 36 _

150 Multiply.

= 6 _

25 Simplify.

The probability of a bird landing in the area with tomatoes is 6 _

25 .

(14)(14)

(14)(14)

(14)(14)

(16)(16)

(16)(16)


Using Geometric Formulas to Find the

Probability of an Event

Online Connection


Hint

The favorable outcome is the area with tomatoes. The area of the entire garden represents all possible outcomes.


MATH BACKGROUND

Geometric probability combines two different branches of mathematics, geometry and probability. Geometric fi gures are used to describe a situation. Then, based on those fi gures, a probability can be calculated. As usual, the probability of an event is the ratio of favorable outcomes to total outcomes. Drawing a picture and shading the favorable area is one strategy for working these problems.

Geometric probability can incorporate any geometric fi gure. In addition, any skill in probability can be required. So, it is a review of previously-learned concepts applied in a new way.

LESSON RESOURCES



Master C120

Warm Up1

120LESSON

Lesson 120 817

Problem 2

Remind students that the sum of the probability of an event and its complement is 1.

2 New Concepts

In this lesson, students use geometric formulas to fi nd the probability of an event.

Example 1

Students fi nd geometric probability.


A sheet of paper, 12 inches by 9 inches, is lying on a table. The table is 15 inches by 18 inches. What is the probability that a fl y will land on the paper?

18 in.

12 in.

15 in.9 in.

2 __ 5


Saxon Algebra 1818

Example 2 Finding Geometric Probability with Circles

A target is made of two concentric circles. The outer circle has a radius of 12 inches. The inner circle has a radius of 4 inches. What is the probability that a dart will land in the inner circle?

SOLUTION

favorable outcomes __

total outcomes =

shaded area __ entire area

= π(4)2

_ π(12)2

Use A = π r 2 to find the areas.

= 16π

_ 144π

= 1 _ 9 Simplify.

The probability that a dart will land in the inner circle is 1 _ 9 .

Recall that the formula for the complement of an event is:

1 - P(A) = P(not A)

Example 3 Finding the Probability of the Complement

a. A town is represented by a circle with a diameter of 50 miles. There is a square park with side length 5 miles located within the town. What is the probability that a raindrop would land in the town, but not the park?

SOLUTION

Find the probability of the complement of the raindrop landing in the park.

P(not landing in the park) = 1 - P(landing in the park)

= 1 - area of park

__ area of town

= 1 - 52

_ π(25)2

Use the area formulas.

= 1 - 25 _

625π

Simplify the powers.

≈ 0.99 Subtract.

The probability of a raindrop landing in the town but not the park is 99%.

b. A carnival game has the player release an air-filled balloon towards a square wall. In the middle of the wall is a triangular target. What is the probability that the balloon will not hit the target?

12 in

4 in

12 in

4 in

Park

5 miles

50 milesPark

5 miles

50 miles

3.5 ft.

4 ft.

7 ft.

3.5 ft.

4 ft.

7 ft.

Hint

When finding the ratio of the areas of two circles, leave the areas in terms of π to make simplification easier.

Math Language

The complement of an event is all the outcomes in the sample space that are not included in the event.


To fi nd the complement of an event, the probability of the event can be subtracted from 1. When working with geometric probability, some students may prefer to subtract the areas to fi nd the favorable area, and then use it to fi nd the probability directly.

For example, in Example 3a, students could fi nd the area of the town that excludes the park by subtracting the area of the park from the area of the town. Then, the result is the favorable area and the area of the town is the total area.

INCLUSION

818 Saxon Algebra 1

Example 2


A parachutist jumps from a plane. The target area for his landing is a circle with diameter 40 meters. There is one large pond that covers an area with a diameter of 10 meters. What is the probability that he lands in the water?

40 m

10 m

1 ___ 16

Example 3

Students learn to use geometric formulas to fi nd the probability of the complement of an event.


a. A rectangular train table is 48 inches by 24 inches. The train is 2 inches by 6 inches. If an object is dropped on the table, what is the probability that it will not hit the train?

48 in.

24 in. 6 in.

2 in.

95 ___ 96

b. On the front of a tent is a two-foot square door. The base of the tent is 5 feet and the height is 4 feet. A bird fl ies at the tent. What is the probability that it does not hit the door?

4 feet

5 feet

2 feet

3 __ 5


Lesson 120 819

SOLUTION

Find the probability of the complement of the balloon hitting the target.

P(not hitting the target) = 1 - P(hitting the target)

= 1 - area of the triangle

__ area of square wall

= 1 - 1 _ 2 · 4 · 3.5

_ 72


= 1 - 7 _ 49

Multiply and square 7.

= 1 - 1 _ 7 Simplify the fraction.

= 6 _ 7 Subtract.

The probability of not hitting the target is 6 _ 7 .

Example 4 Application: Zoning

A new school is being built. All students live within a 4-mile radius of the school, and homes are evenly distributed throughout the area. Planners think that students who live within 0.5 mile of the school would walk to school. Students who live between 0.5 and 2 miles would ride a bike, a city bus, or a private car to school. Students who live between 2 and 4 miles from the school would ride the school bus. What is the probability of a student not walking to school?

SOLUTION

Find the complement of a student walking to school.

P(not walking to school) = 1 - P(walking to school)

= 1 - area of walking circle

__ total area

= 1 - π (0.5)2

_ π (4)2


= 1 - 0.25π

_ 16π

Simplify the powers.

= 1 - 1 _

64 Simplify the fraction.

= 63 _ 64

Subtract.

The probability of not walking to school is 63_64

.

bike, city bus, car

school bus

walk

4 miles

2 miles1_2

mile

bike, city bus, car

school bus

walk

4 miles

2 miles1_2

mile

Math Reasoning

Analyze When calculating with the area of circles, why are some answers exact and some approximate?

Sample: When π cancels, the answer is exact, but when πis multiplied or divided, the answer is approximated.


Lesson 120 819

TEACHER TIPTreat these problems as multi-step rather than putting the geometric formulas in a probability formula. Have students fi rst calculate the area using geometric formulas. Then substitute those values into a probability formula.

Example 4

Students apply geometric probability to solve a problem involving zones.

Extend the Example

What is the probability that a student rides the school bus? 3 __ 4


A tile border has a pattern of three triangles that repeats inside a rectangle. When paint splatters on the wall, what is the probability that it will not hit one of the triangles?

18 in.

11 in.

9 in.

3 in.

6 in.

4 in.

8 in.

4 in.

313 ____ 396

Error Alert Students may try to estimate the probability by just looking at the picture. Explain that the pictures are labeled correctly but may not be drawn to scale.



Saxon Algebra 1820

Lesson Practice

a. A rectangular swimming pool is 15 feet by 30 feet. A raft in the pool is 2 feet by 3 feet. A beach ball is thrown randomly into the pool. What is the probability that the ball hits the raft? 1_

75

b. A child’s crown is made by cutting the 6-inch center out of a 10-inch paper plate. The plate is then decorated by sprinkling it with glitter. What is the probability that a piece of glitter misses the crown? 9_

25

c. A round piece of cardboard with radius 4 inches is used to make a mask. A two-inch square hole is cut for a mouth piece. If a piece of popcorn is tossed at the mask, what is the probability that it will miss the mouth hole? 1 - 4_

16π ≈ 0.92

d. A rectangular yard is 10 feet by 15 feet. Tulips are planted in a triangular area that has a 5-foot base and a height of 6 feet. A bird lands in the yard. What is the probability that it does not land in the tulip garden? 1 - 15_

150= 0.9

e. Puzzles A rug in the kindergarten class has various shapes on it. One student steps onto the rug. What is the probability that he is not standing on the square? 11_

12

(Ex 1)(Ex 1)30 feet

15 feet 2 feet

3 feet

30 feet

15 feet 2 feet

3 feet

(Ex 2)(Ex 2)10 inches

6 inches

10 inches

6 inches

(Ex 3)(Ex 3)

2 inches

4 inches

2 inches

4 inches

(Ex 3)(Ex 3)

15 feet

10 feet

5 feet

6 feet

15 feet

10 feet

5 feet

6 feet

(Ex 4)(Ex 4) 2 ft

2 ft

2 ft

2 ft

6 ft

8 ft

1 ft

2 ft

2 ft

2 ft

2 ft

6 ft

8 ft

1 ft

*1. Find the probability of landing in the shaded area. 15_64

5 in.

6 in.

8 in.

8 in.

2. Write Explain what geometric probability is. Sample: using geometric formulas to calculate the favorable and total outcomes.

(120)(120)

(120)(120)


820 Saxon Algebra 1

Lesson Practice

Problem b

Scaff olding Remember that a 10-inch plate has a 10-inch diameter. First fi nd the area of the plate and the center. Then write the probability.

Problem c

Error Alert Students may look at the picture and assume they are fi nding the probability of the popcorn hitting the mouth. Have students read each problem carefully and shade the area the words describe before working the problem.



“What is geometric probability?” Sample: It is using geometry formulas to help solve probability problems.

“How do you fi nd the probability of something not landing in the given area?” Sample: Find the probability of its landing there and subtract the probability from 1.


Lesson 120 821

3. Generalize A system of equations consists of a quadratic and a linear equation. If the graphs of the two equations do not intersect, what can you conclude about the solution to the system? The system has no solution.

*4. Verify Show that the probability of landing in the shaded region is 1 _ 2 .

8 inches

8 inches

5. Find 7C2 21

6. Manufacturing A number cube is made with side lengths of 5 centimeters. Use the function V = s3 to find the volume of plastic that is contained in the number cube. 125 cubic centimeters

*7. Multiple Choice A parachutist will land in a rectangular field with a circular landing area as her target. What is the probability that she will land on target? B

56 yd

45 yd

15 yd

A ≈ 0.09 B ≈ 0.28 C ≈ 0.72 D ≈ 0.92

8. Jewelry To make a friendship bracelet, 8 beads are used. How many different combinations of beads could be on the bracelet if there are 20 different beads? 125,970

*9. Puzzles A children’s stacking puzzle teaches shapes. A child randomly points to the puzzle. What is the probability that the child’s finger lands on the shaded part of the square?

4 in.

5 in.

8 in.

6 in.

10. Model Draw right ΔABC with right angle C so that sin A = 3 _ 5 and cos A = 4 _

5 .

11. What is the domain of f(x) = 3 √ � x - 5? x ≥ 0

12. Find the 4th term in the geometric series that has a common ratio of -1.1 and a first term of 7. -9.317

(112)(112)

(120)(120)

Sample:

1 - 1_ 2 · 8 · 8_ 8 · 8

= 1 - 32_ 64

= 32_ 64

= 1_ 2

Sample:

1 - 1_ 2 · 8 · 8_ 8 · 8

= 1 - 32_ 64

= 32_ 64

= 1_ 2

(118)(118)

(115)(115)

(120)(120)

(118)(118)

(120)(120)

36 - 1_2

(4)(5) _

64π ≈ 0.13

36 - 1_2

(4)(5) _

64π ≈ 0.13

(117)(117)

10. Any right triangle with sides that are similar to a 3-4-5 right triangle is valid where the shorter leg is opposite angle A.

10. Any right triangle with sides that are similar to a 3-4-5 right triangle is valid where the shorter leg is opposite angle A.

(114)(114)

(105)(105)


Lesson 120 821

Practice3


Problem 3

The solution to the system of equations tells where the graphs intersect.

Problem 6

Extend the Problem

If you double the sides of the cube, how much more plastic will it contain? 8 times

Problem 8

Extend the Problem

How many different bracelets could be made? 5,079,110,400

Problem 10


“What ratio does sine represent?” opposite over hypotenuse

“What ratio does cosine represent?” adjacent over hypotenuse

“What is the length of the hypotenuse?” 5

“What is the length of the leg opposite A?” 3

“What is the length of the leg adjacent to A?” 4

Problem 12

Suggest that students either list the fi rst four terms or use the formula Sn = 7(-1.1 ) 3 .


Saxon Algebra 1822

*13. Baking After rolling the dough into a 9-inch by 12-inch rectangle, a boy cuts as many biscuits with a 3-inch diameter as possible. His little sister comes and touches the dough. What is the probability that she did not touch a biscuit? ≈ 0.21

*14. Error Analysis Two students find the probability of landing in the shaded region. Which student is correct? Explain the error.

Student A

A = 1 _ 2

(10)(7) = 35

A = 132 = 169

P(shaded) = 35 _ 169

Student B

A = 1 _ 2 (10)(7) = 35

A = 132 = 169

P(shaded) = 1 - 35 _

169 =

134 _ 169

*15. Justify Find the probability of landing in the shaded region and explain your steps. Sample: Using A = s2, the area of the square is 49 square centimeters. The radius of the circle is half the diameter or 3 centimeters. Using A = πr2, the area of the circle is 9π. Find the probability of not landing in the circle by fi nding the complement of the probability of landing in the circle. The formula is 1 - 9π

_ 49

which is approximately 0.42.

*16. Geometry A spinner is divided into equal sectors. The red sector is 120°. What is the probability of not landing on a red sector? 2_3

17. Find the probability of landing in the shaded square. 4_15

*18. Multi-Step A game uses a large square game board that is divided into congruent smaller squares. The probability of landing on one of those smaller squares is 1 _

4 .

Each of the smaller squares has area 49 square millimeters. a. What is the probability of not landing on that square the next time? 3_

4

b. What is the area of the larger square? 196 square millimeters

19. Error Analysis Students were asked to sketch an example of a linear function. Which student is correct? Explain the error.

Student A

x

y

4

6

2

2-2-4

Student B

x

y

2

-4

2 4-4O

(120)(120)

(120)(120)13 in.

10 in.

7 in.

13 in.

10 in.

7 in.

14. Student B; Sample: Student A found the probability of landing on the triangle.

14. Student B; Sample: Student A found the probability of landing on the triangle.

(120)(120)

6 cm.

7 cm.

6 cm.

7 cm.

(120)(120)

4 in.8 in.

15 in.

4 in.8 in.

15 in.

(120)(120)

(120)(120)

(119)(119)19. Student B; Sample: A linear function must have a constant rate of change. Student A’s graph does not have a constant rate of change; it gets steeper as xincreases.

19. Student B; Sample: A linear function must have a constant rate of change. Student A’s graph does not have a constant rate of change; it gets steeper as xincreases.


For Problem 30, Explain the meaning of the word unscramble.

Say:

“To scramble something means mix something up. Eggs are scrambled when the clear and yellow parts are mixed together. However, to unscramble means to sort or order a set of objects.”

Have students write the letters A, C, S, and T on equal sized sheets of paper. Have them

ENGLISH LEARNERS

scramble the papers. Ask the students if the scrambled papers form a word. If not, have the students unscramble the letters to form a word.

Ask:

“Why can the letters be unscrambled, but eggs cannot?” because once the eggs are mixed the parts are permanently combined

822 Saxon Algebra 1

Problem 17


“What is the formula for fi nding the geometric probability of an event?” favorable area divided by the total area

“What is the area of the shaded region?” 16 square inches

“What is the area of the triangle?” 60 square inches


Lesson 120 823

*20. Multi-Step The graph shows the height in feet, h, of a tennis ball hit into the air with initial velocity of 30 feet per second at time t seconds. a. What type of function is this? How do you know?

b. Use the graph to approximate the maximum height that the ball reaches. Round to the nearest foot. 14 feet

21. Identify the function family: y = 4x2 + 2. quadratic

22. Error Analysis Two students are asked to list all the possible combinations choosing 2 letters from A, B, and C. Which student is correct? Explain the error. Student A; Sample: Student B made order count.

Student A

AB, AC, BC

Student B

AB, AC, BA, BC, CA, CB

23. Use the discriminant of the equation -x2 + 2x + 1 = 0 to find the number of real solutions. d = 8; two real solutions

24. Find the probability of a randomly tossed dart landing in the shaded area. ≈ 0.42

25. An investment of $2600 is made for 7 years at 8% simple interest. How much will be in the account after 7 years? $4056

26. Multi-Step Graph the function f(x) = ⎢x + 2�, then translate the function up by 3. What function does this graph now represent? f(x) = |x + 2| + 3

27. Multiple Choice Find ∠A if cos A = 5 _ 7 . A

A 44.4° B 45.6° C 0.71° D 1.00°

28. Solve 3x2 + 9x = 5.25 by completing the square. x = 0.5 or x = -3.5

29. Find the probability of a randomly tossed bean bag not landing in the shaded area. ≈ 0.99

30. Multi-Step To work a word jumble the letters P, S, A, C, M, and H need to be unscrambled. a. How many different arrangements of the letters are possible? 720

b. What is the probability of writing the word CHAMPS on the first try if each letter is equally likely to be chosen? 1_

720

(119)(119)

x

y

4

8

12

1-2 -1O x

y

4

8

12

1-2 -1O

20a. quadratic; Sample: The graph is a parabola.

20a. quadratic; Sample: The graph is a parabola.

(119)(119)

(118)(118)

(113)(113)

(120)(120)

4 in.

11 in.

4 in.

11 in.

(116)(116)

(107)(107)

(117)(117)

(104)(104)

(120)(120) 2 in.

10 in.

2 in.

10 in.

(111)(111)


LOOKING FORWARD

Using geometric formulas to fi nd the probability of an event will be further developed in other Saxon Secondary Mathematics courses.

CHALLENGE

A rectangular garden has a length twice as long as its width. In the middle of the garden is a triangular fountain with height and base equal to the garden’s width. What is the probability that a bird would land on the fountain?

w

2w

w 1 __ 4

Lesson 120 823

Problem 21

Choose from linear, quadratic, or exponential.

Problem 25

Error Alert The simple interest formula results in just that, the interest. To fi nd the total amount in the account, the interest must be added to the principal.

Problem 26

To shift a function up 3, add 3 to f(x).

Problem 27

Use cos-1 5 __ 7 .

Problem 28

Divide all terms by 3 before completing the square.


Saxon Algebra 1824

11L AB

Matrix Operations

Graphing Calculator Lab (Use with Investigation 12)

You can enter and store one or more matrices in a graphing calculator. Then you can use the calculator to perform matrix operations such as addition, subtraction, and scalar multiplication.

Let A = ⎡ ⎢

⎣ 4 2

-6

3 ⎤ �

⎦ and B =

⎡ ⎢

⎣ -3

5

7

-9 ⎤ �

⎦ . Use a graphing calculator to find

A + B, A - B, and 2A.

1. Press to highlight the EDIT

menu.

2. Select 1: [A].

3. Matrix A has two rows and two columns.

Press 2 2 .

4. Enter each element of matrix A, starting with the element in the first row, first column.

Press 4 6 2 3 .

5. Press to highlight the EDIT

menu.

6. Select 2: [B].

7. Matrix B has two rows and two columns.

Press 2 2 .

8. Enter each element of matrix B, starting with the element in the first row, first column. Press

after keying in each value.

9. Press to return to the home screen.

10. Find the sum of A + B.

Press 1 2 .

The sum is A + B = ⎡ ⎢

⎣ 1 7

1

-6 ⎤ �

⎦ .


824 Saxon Algebra 1

11LAB

Materials


Discuss

Addition and subtraction of matrices can only be done with matrices of the same dimensions. Have students verify this by attempting to add or subtract matrices with different dimensions on a graphing calculator.

When multiplying by a scalar, the dimensions of the matrix do not matter as each element of the matrix is multiplied by the scalar.

When multiplying matrices, the number of columns in the fi rst matrix must match the number of rows in the second matrix.

Multiplying matrices will be studied in future mathematics courses.

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Lab 11 825

11. Find the difference of A - B.

Press 1 2 .

The difference is A - B = ⎡ ⎢

⎣

7 -3

-13

12

⎤ �

⎦ .

12. Find 2A.

Press 2 1 .

So, 2A = ⎡ ⎢

⎣ 8 4

-12

6 ⎤ �

⎦ .

Lab Practice

Let A = ⎡ ⎢

⎣

3 -1

2

4

4

0 ⎤ �

⎦ and B =

⎡ ⎢

⎣ 2 5

8

-9

-3

2 ⎤ �

⎦ . Use a graphing calculator

to find each value. a. A + B,

b. A - B

c. 3B

d. B - A

e. -2A

a.a. b.b.

c.c. d.d. e.e.


Lab 11 825

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12INVESTIGATION

Saxon Algebra 1826

Investigating Matrices

The table shows the price of tickets sold at a local movie theater.

Type Matinee Regular

Child $5 $7

Student $6 $8

Adult $9 $11

Use a matrix to organize data, such as the price of movie tickets. A matrix is a rectangular array of numbers in horizontal rows and vertical columns. Each number in a matrix is called an element. An element is any individual object or member belonging to a set.

Below is a matrix for the ticket prices.

⎡

⎢

⎣

5

6 9

7

8 11

⎤

�

⎦

Notice that the entries in the table above correspond to the elements in the matrix.

Fundraiser The table below shows the number of items sold during the first day of a school fundraiser.

Items Sold on First Day

Item Medium Large

T-shirts 10 7

Hats 3 2

Sweatshirts 4 9

1. Model Use the data in the table to create a matrix that represents the situation.

2. Explain how to use the matrix to calculate the total number of sweatshirts sold during the first day of the fundraiser. How many sweatshirts were sold? Sample: Add the numbers in the third row.; 13 sweatshirts

The dimensions of a matrix with m rows and n columns are m × n. Each element has a specific position in the matrix that is relative to its row and column.

Complete the statements. Refer to the matrix of the number of items sold during the school fundraiser’s first day.

3. The matrix has rows and columns. 3, 2

4. The dimensions of the matrix are . 3 × 2

1.

⎡

⎢ ⎣

1034

729

⎤

� ⎦

1.

⎡

⎢ ⎣

1034

729

⎤

� ⎦

Online Connection


Materials

• graph paper


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Matrices are used to organize data in a clear, concise format. Operations with matrices are used to solve problems. For example, matrices and diagrams are used to represent communication networks and grids such as those that model the delivery of electricity.

Matrices are also used for transformations of geometric fi gures in a coordinate plane. Addition translates, or shifts, the shape on the coordinate plane without changing its shape and size. Scalar multiplication dilates, or shrinks or enlarges, a shape on a coordinate plane while preserving its shape.

INVESTIGATION RESOURCES

Reteaching Master Investigation 12

826 Saxon Algebra 1

12INVESTIGATION

Materials

• graphing calculator• graph paper

Alternate Materials

Function-Graphing Program for a computer

Discuss

In this investigation, students explore the characteristics of matrices. They see how matrices can be used to solve real-world problems and transform geometric shapes.

Defi ne matrix.

Fundraiser

Extend the Problem

Have students write a matrix that could be used to show how much money the school made from each item sold. Sample:

Item Medium Large

T-shirts $12 $15

Hats $8 $10

Sweat shirts $15 $18

⎡

⎢

⎣

12

8 15

15

10 18

⎤

�

⎦

MATH BACKGROUND

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Investigation 12 827

5. The element in the third row, second column is . 9

6. Describe the dimensions of a matrix representing data about a school fundraiser with five types of items available in four different sizes. The matrix would have five rows and four columns.

Since matrices are a method of organizing information, sometimes they are added or subtracted to get new information. Matrices that have the same dimensions can be added subtracted.

The table shows the number of items sold during the second day of the fundraiser.

Items Sold on Second Day

Item Medium Large

T-shirts 8 3

Hats 1 5

Sweatshirts 4 7

7. Generalize Explain how to find the total number of large hats sold during the first two days of the fundraiser.

8. Formulate Explain how to use matrices to find the total number of large hats sold during the first two days of the fundraiser.

9. Use matrices to model the sum of the items sold during the first two days. Then add the matrices using a graphing calculator.

10. Find the difference between the two matrices. How many more medium hats were sold on the first day than on the second day? 2 hats

Matrix addition also has a geometric application.

Exploration Exploration Using Matrix Addition to Transform Geometric Figures

11. Complete the following table by finding the coordinates of the vertices of �ABC.

Point A Point B Point C

x-coordinate -1 -2 2

y-coordinate 2 -1 -2

12. Use the data in the table to create matrix A.See Additional Answers.

13. Using a graphing calculator find the sum A + B, when B = ⎡ ⎢

⎣ 3 2

3 2

3 2

⎤ �

⎦ .

See Additional Answers. 14. Use the elements of A + B as the coordinates of the vertices of �A'B'C'.

Plot the vertices of �A' B' C' to create a graph of the triangle.

15. Write Describe the relationship between �ABC and �A'B'C'.

16. What matrix would translate �ABC one unit to the left and three units down? ⎡ ⎢

⎣ -1-3

-1-3

-1-3

⎤ �

⎦

7. Sample: Add the number of large hats sold on the first day, 2, to the number of large hats sold on the second day, which is 5.

7. Sample: Add the number of large hats sold on the first day, 2, to the number of large hats sold on the second day, which is 5.

8. Put the data from each table into a matrix and then find the sum of the values in the second row and second column of each matrix.

8. Put the data from each table into a matrix and then find the sum of the values in the second row and second column of each matrix.


10. See Additional Answers.10. See Additional Answers.

x

y

O

4

2

2 4

-2

-4

-4

B

A

C

x

y

O

4

2

2 4

-2

-4

-4

B

A

C

14. See Additional Answers.14. See Additional Answers.

15. Sample: The matrix addition A + Btranslates �ABC three units to the right and two units up.

15. Sample: The matrix addition A + Btranslates �ABC three units to the right and two units up.

Graphing

Calculator Tip

For help with adding matrices, see the Graphing Calculator Lab 11 on page 824.

Caution

Matrices can be added or subtracted only if they have the same number of rows and the same number of columns.

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For the exploration, explain the meaning of the word transform. Say:

“Transform means to change the form or appearance of something. The job of a make-up artist is to transform the face.”

Discuss how moving furniture in a room can transform the appearance of the room. Have volunteers use the word transform in a sentence. Sample: We will transform the room with decorations.


Problems 11–16

Error AlertWhen writing a matrix that transforms a shape, students may write a 2 × 1, where the fi rst row shows what happens to the x-coordinate and the second row shows the change in the y-coordinate. In order for matrices to be added, they must have exactly the same number of rows and columns, even if each column is the same.

ENGLISH LEARNERS


Saxon Algebra 1828

17. What matrix was used to translate �ABC to �A'B'C ' using matrix addition? ⎡

⎢ ⎣ 52

52

52

⎤ �

⎦

x

yA´

B´

C´

AB

C

You can also transform a geometric figure through scalar multiplication. Scalar multiplication multiplies a matrix A by a scalar c. To find the resulting matrix cA, multiply each element of A by c.

Exploration Exploration Using Scalar Multiplication to Transform

Geometric Figures

18. Predict Use matrix A from the previous Exploration. What would be the effect of multiplying each element in matrix A by 2?

19. Find the matrix 2A using a graphing calculator.

20. Use the elements of 2A as the coordinates of the vertices of �A''B''C''. Plot the vertices of �A''B''C'' to create a graph of the triangle.

21. Write Describe the relationship between �ABC and �A''B''C''.

22. What matrix would create a triangle one-fourth the size of �ABC ?

23. Describe the relationship between �ABC and �A'B'C '.Sample: �ABC was transformed by scalar multiplication using scalar 3.

xy

A´ B´

C´

A B

C

18. Sample: The coordinates of the vertices would double so the transformed triangle would be four times as big as �ABC.

18. Sample: The coordinates of the vertices would double so the transformed triangle would be four times as big as �ABC. See Additional Answers.See Additional Answers.


21. Sample: Scalar multiplication 2Aquadrupled the size of �ABC.

21. Sample: Scalar multiplication 2Aquadrupled the size of �ABC.


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828 Saxon Algebra 1

Problems 18–22

Error AlertTo enlarge or reduce a fi gure, students may try to add or subtract, respectively, a value to each element in the matrix. Discuss the phrase “How many times as big …” to show that multiplication is the correct operation to be used. To shrink a fi gure, division or multiplication by a fraction can be used.



Investigation Practice

The first table shows how much money Abram and two friends earned doing chores last summer. The second table shows how much money they made this summer.

Money Earned Last Summer

Mowing Lawns Washing Cars Babysitting

Abram $15 $55 $0

Paul $75 $10 $30

Leila $20 $40 $35

Money Earned This Summer

Mowing Lawns Washing Cars Babysitting

Abram $25 $60 $10

Paul $85 $20 $35

Leila $30 $60 $40

a. How can matrices be used to calculate how much more money each person made during the second summer?

b. Use a matrix operation to display the additional money each person made during the second summer. How much more money did Paul make during the second summer than the first? $25

c. Use scalar multiplication to find the vertices of a quadrilateral whose area is one-fourth of the quadrilateral shown.

a. Put the data in each table above into a matrix. Label the first table matrix A and the second matrix B.Subtract the data in matrix A from matrix B.

a. Put the data in each table above into a matrix. Label the first table matrix A and the second matrix B.Subtract the data in matrix A from matrix B.

x

y

O

4

2

2 4

-2

-2

-4

B

A

C

D

x

y

O

4

2

2 4

-2

-2

-4

B

A

C

D

b.

⎡

⎢ ⎣

258530

602060

103540

⎤

� ⎦

-

⎡

⎢ ⎣

157520

551040

03035

⎤

� ⎦

=

⎡

⎢ ⎣

101010

51020

1055

⎤

� ⎦

b.

⎡

⎢ ⎣

258530

602060

103540

⎤

� ⎦

-

⎡

⎢ ⎣

157520

551040

03035

⎤

� ⎦

=

⎡

⎢ ⎣

101010

51020

1055

⎤

� ⎦

1_2A =

⎡ ⎢

⎣ -11.5

-2-0.5

2-1

1.51.5

⎤ �

⎦ 1_

2A =

⎡ ⎢

⎣ -11.5

-2-0.5

2-1

1.51.5

⎤ �

⎦

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Using matrices will be further developed in other Saxon Secondary Mathematics courses.


Investigation Practice


Problem a


“What is used to represent the data from the tables?” matrices

“What dimensions will each matrix have?” 3 × 3

“What operation should be used to fi nd how much more money was made?” subtraction

“Which matrix would come fi rst in the subtraction problem?” matrix B

Problem b

Error AlertStudents may forget to add each element in Paul’s row. Remind them that Paul earned money in three different ways.

Problem c

Scaff olding First form a matrix where each column represents a point, and each row represents one of the coordinates of those points. Then multiply each element in the matrix by 1 __ 2 .

LOOKING FORWARD


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