4-1 Using Graphs to Relate Two Quantitiesktlmathclass.weebly.com/uploads/2/5/7/6/25760552/hsm12cc...Vocabulary The amount of toothpaste in a tube decreases each time you brush your

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Chapter 4 110

Vocabulary

Review

4-1 Using Graphs to Relate Two Quantities

x

y

O 42

2

4

24N

K

PM

L

2

4

Use the graph at the right. Draw a line from each point in Column A to its coordinates in Column B.

Column A Column B

1. point K (23, 4)

2. point L (22, 21)

3. point M (0, 0)

4. point N (2, 1)

5. point P (3, 22)

Vocabulary Builder

analyze (verb) AN uh lyz

Other Word Forms: analyzed (verb), analysis (noun)

Definition: to examine carefully in detail; to identify the nature and relationship of its parts

What It Means: break down, dissect

Word Origin: from the Greek word analusis, meaning “a dissolving”

Use Your Vocabulary

Complete each statement with the appropriate word from the list.

analyze analysis analyzed

6. The chemist 9 the data to draw a conclusion.

7. Jean needed to 9 the data she gathered in her experiment.

8. An 9 of the traffic at an intersection showed the need for a traffic light.

analyzed

analyze

analysis

HSM11A1MC_0401.indd 110HSM11A1MC_0401.indd 110 2/19/09 1:04:40 PM2/19/09 1:04:40 PM

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d.Problem 1

111 Lesson 4-1

Analyzing a Graph

Got It? What are the variables in the graph? Describe how the variables are related at various points on the graph.

Board Length

Leng

th

Time00

9. Circle the two variables being related in the graph.

time cut board length

10. Show how the variables are related by underlining the correct word or words to complete each sentence.

The length of the board increases / decreases with time.

The length of the board is constant / decreasing while you are actually cutting the board.

During the time shown on the graph, there are three / four cuts.

There is / is not a piece of the board left at the end of the time shown.

Got It? What are the variables in the graph? Describe how the variables are related at various points on the graph.

11. Show how the variables are related by underlining the correct word to complete each sentence.

The cost of the cell phone in June increases / decreases with number of minutes of calls.

The cost of the cell phone in June is constant / increasing for the first part of the month.

12. Use your answers from Exercise 11 to describe how the variables in the graph are related.

June Cell Phone Cost

Cost

Minutes of Calls00

Answers may vary. Sample: For the first part of the month of June,

the cost of the cell phone remains constant. Then the cost

increases steadily for the rest of the month as the number of

minutes of calls increases.


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Chapter 4 112

Problem 3

Problem 2

Time

Hei

ght

Time

Hei

ght

Time

Hei

ght

Number of Uses

Amount of Sunscreen (oz)

Sunscreen

0

5

1

4.8

2

4.6

3

4.4

Matching a Table and a Graph

Got It? The table shows the amount of sunscreen left in a can based on the number of times the sunscreen has been used. Which graph could represent the data shown in the table?

A.

Am

ount

of

Suns

cree

n

00

Number of Uses

B.

Am

ount

of

Suns

cree

n

00

Number of Uses

C.

Am

ount

of

Suns

cree

n

00

Number of Uses

13. Analyze the data in the table. Complete each statement with the correct choice from the list. Use each word only once.

slowly fall decreases

The amount of sunscreen in the container 9 after each use.

The amount of sunscreen in the container changes 9.

The graph should 9 at a slow rate.

14. The graph that could represent the data shown in the table is Graph .

Sketching a Graph

Got It? Suppose you start to swing yourself on a playground swing. You move back and forth and swing higher in the air. Then you slowly swing to a stop. What sketch of a graph could represent how your height from the ground might change over time? Label each section.

15. Multiple Choice The two variables being related are time and 9.

length of distance from your height your height swing top of swing from ground

16. Consider the three cycles during the middle of your time on the swing. Circle the best sketch of your height from the ground during that time.

constant distance start high, swing low, start low, swing from ground end high high, end low

decreases

slowly

fall

C

HSM11A1MC_0401.indd 112HSM11A1MC_0401.indd 112 3/4/09 6:18:20 AM3/4/09 6:18:20 AM

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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

BC

DEA

113 Lesson 4-1

• Do you UNDERSTAND?

Reasoning Describe a real-world relationship that could be represented by the graph sketched at the right.

17. Draw a line from the name of each segment in Column A to its verbal description in Column B.

Column A Column B

A spilling water from a cup

B pouring water into a cup quickly

C stop pouring water into a cup

D water leaking from a hole in a cup

E pouring water into a cup slowly

18. Use the verbal descriptions above to help you write a situation that could be represented by the sketch.

Check off the vocabulary words that you understand.

variable quantities increase decrease

Rate how well you can use graphs.

Answers may vary. Sample: At first I poured water quickly into a

cup, but then I slowed down. I stopped pouring because I saw the

cup wobble. It fell over and much of the water spilled out. The

remaining water leaked out slowly from a hole in the cup.


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Vocabulary

Review

Chapter 4 114

1. A function is a relationship that pairs each input value with exactly one output value. Cross out the relationship below that does NOT show a function.

Input Output1325

4124

Input Output

14

1122

Input Output1212

12

Vocabulary Builder

independent (adjective) in dee PEN dunt

Related Words: dependent, input, output

Definition: An independent variable is a variable whose value determines the value of another variable, called the dependent variable.

Math Usage: In the diagram, the independent variable, x, is called the input of the function. The dependent variable, y, is called the output of the function.

Example: When showing the relationship between amount of sunlight and amount of plant growth, the independent variable is the amount of sunlight.

Use Your Vocabulary

Write I if the first value is independent of the second value. Write D if the first value is dependent on the second value.

2. the growth of a plant and the light the plant receives

3. the speed of a swimmer and the depth of a pool

4. the number of books a shelf holds and the length of the shelf

Patterns and Linear Functions4-2

functionindependent variable (input)

dependent variable (output)

x

y

D

I

D


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d.Problem 1

115 Lesson 4-2

Representing a Geometric Relationship

Got It? In the diagram below, what is the relationship between the number of triangles and the perimeter of the figure they form? Represent this relationship using a table, words, an equation, and a graph.

41 triangle 2 triangles 3 triangles 4 triangles

4

4 4 4 4

4 4 4 4

3 3 3 3 3 3 3 3

5. Define the variables.

Let x 5

. Let y 5

.

6. Complete the table.

HSM11_A1MC_0402_T91107

1 32 4Number of Triangles

Perimeter 2210 14 18

7. Complete the model below.

perimeter

Write

Relate is

6

6times number of triangles plus

• +=

HSM11_A1MC_0402_T14348

y 4

4

x

8. Write an equation to represent the relationship you wrote in Exercise 7.

9. Use the table to list the points you will plot.

(1, ) (2, ) (3, ) (4, )

10. Now plot the points on the graph.

HSM11_A1MC_0402_T91108

x

y

10

3

02 3 4 5

Number of Triangles

Perim

eter

6

9

12

15

18

21

24

the number of triangles the perimeter of the figure

y 5 4x 1 6

10 14 18 22

HSM12A1MC_0402.indd 115 3/19/11 12:12:47 PM

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Problem 2

Input, xOutput, y

0

8

1

10

2

12

3

14

Chapter 4 116

Representing a Linear Function

Got It? Is the relationship in the table at the right a linear function? Describe the relationship using words, an equation, and a graph.

11. Describe the pattern in the table in words.

______________________________________________________________________

12. Multiple Choice Which equation describes the relationship in the table?

y 5 2x 1 8 y 5 2x 2 8 y 5 x 4 2 1 8 y 5 x 4 2 2 8

13. Plot the points from the table on the graph.

HSM11_A1MC_0402_T91109

x

y

10

2

02 3 4 5

4

6

8

10

12

14

16

14. Underline the correct word or words to complete the sentence.

The points lie / do not lie on a line; so, the relationship is / is not a

linear function.

Got It? Reasoning Does the set of ordered pairs (0, 2), (1, 4), (3, 5), and (1, 8) represent a linear function? Explain.

15. Plot the points on the graph.

HSM11_A1MC_0402_T91110

y

10

1

02 3 4 5 6 7 8

2

3

4

5

6

7

8

x

16. Do the ordered pairs represent a linear function? Explain.

______________________________________________________________________

______________________________________________________________________

No. Explanations may vary. Sample: The input value 1 has two output

values, 4 and 8. The points do not all lie on a line.

The value of y is equal to the sum of 2 times x and 8.

HSM12A1MC_0402.indd 116 4/12/11 12:49:35 PM

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Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

Lesson Check

Lesson Check

O

y

x

117 Lesson 4-2


dependent variable independent variable linear function

Rate how well you can describe linear functions.



Reasoning Does the graph at the right represent a linear function? Explain.

19. Draw a line from each word in Column A to its definition in Column B.

Column A Column B

relation function whose graph is a line or part of a line

function pairing of input and output values

linear function relationship that pairs each input value with exactly one output value

20. Use the terms above to explain whether or not the graph represents a linear function.

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

Vocabulary The amount of toothpaste in a tube decreases each time you brush your teeth. Identify the independent and dependent variables in this relationship.

17. Complete each phrase to identify the variables.

Let A 5 the amount of toothpaste in a 9. Let B 5 the number of times you 9 your teeth.

18. Underline the correct word to complete each sentence.

A is the independent / dependent variable. B is the independent / dependent variable.

The graph does not represent a linear function. Explanations may vary. Sample: There

brushtube

is only one output for every input, so the graph represents a function. It is not a

linear function because the graph is curved, not part of a line.

HSM12A1MC_0402.indd 117 4/12/11 1:28:18 PM

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Chapter 4 118

Vocabulary

Review

4-3 Patterns and Nonlinear Functions

x

y

O 2

2

2

2

x

y

O2

2

2

2

x

y

O 2

2

2

2

x

y

O 2

2

2

2

Find the next number in each pattern.

1. 1, 2, 4, 8, 2. 28, 4, 22,

3. 1, 3, 9, 27, 4. 12, 14, 18,

5. The next shape in the pattern below has blocks.

Vocabulary Builder

nonlinear (adjective) nahn LIN ee ur

Related Words: line (noun), linear (adjective)

Definition: Something that is nonlinear is not in a straight line.

Math Usage: A nonlinear function is a function whose graph is not a line or part of a line. A linear function is a function whose graph is a line or part of a line.

Common Usage: A nonlinear narrative is a story where the events are told out of chronological order.

Use Your Vocabulary

6. Circle each graph of a nonlinear function.

116

16 1

81

16


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d.Problem 1

119 Lesson 4-3

Classifying Functions as Linear or Nonlinear

Got It? The table below shows the fraction A of the original area of a piece of paper that remains after the paper has been cut in half n times. Graph the function represented by the table. Is the function linear or nonlinear?

Number of Cuts, n

Fraction of Original Area Remaining, A

Cutting Paper

1 2 3 4

12

14

18

116

7. Complete each ordered pair.

(1, ) (2, ) (3, ) (4, )

8. Graph the ordered pairs from Exercise 7 on the coordinate plane.

n

A

100

2 3 4 5Number of Cuts

Frac

tion

of A

rea

Rem

aini

ng

1

12

14

34

9. Complete the sentence with linear or nonlinear: The function is 9.

Got It? Reasoning Will the area A in Exercise 8 ever reach zero? Explain.

10. If you start with a piece of paper in your hand and repeatedly cut the paper in half, will your hand ever be empty? Yes / No

11. Will the remaining area A ever reach zero? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

No. Explanations may vary. Sample: Each time you cut the paper in

half, the remaining area gets closer to zero, but will never disappear

entirely.

12

14

18

116

nonlinear


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Chapter 4 120

Problem 2

x

y

10

50

02 3 4 5

Figure Number

Num

ber o

f New

Bra

nche

s

100

150

200

250

Problem 3

Representing Patterns and Nonlinear Functions

Got It? The table shows the number of new branches in each figure of the pattern below. What is a pattern you can use to complete the table? Represent the relationship using words, an equation, and a graph.

1 2 3

Number ofFigure, x

Number ofNew Branches, y

3

27

4 51

3

2

9

12. Look for a pattern in the table. Describe it below.

_______________________________________________________________________

_______________________________________________________________________

13. Use the words the figure and new branches to complete the diagram below.

3the number of new branchesthe number of the figure

14. Circle the equation that represents the function.

y 5 2x y 5 x 2 y 5 3x y 5 x3

15. Complete each statement.

When x 5 4, y 5 . When x 5 5, y 5 .

16. Write ordered pairs to represent the data in the table and your results from Exercise 15. Then graph the data.

(1, ) ( , 9) ( , 27)

(4, ) (5, )

You can think of a function as a rule that you apply to the input in order to get the output. You can describe a nonlinear function with words or with an equation, just as you did with linear functions.

Writing a Rule to Describe a Nonlinear Function

Got It? What is a rule for the function represented by the ordered pairs (1, 1), (2, 4), (3, 9), (4, 16), and (5, 25)?

81

Answers may vary. Sample: The number of new branches in each figure

is 3 times the number of new branches in the previous figure.

243

3 2 3

81 243


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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

x y0

1

2

3

4

1

2

5

10

17

121 Lesson 4-3

17. Make a table to organize the x- and y-values.

x

y

1 2

9 25

4

4

3

1

5

16

18. Use words to explain the relationship between the x- and y-values.

_______________________________________________________________________

19. Now use the relationship you described to write an equation that is a rule for the function.


Error Analysis A classmate says that the function in the table at the right can be represented by the rule y 5 x 1 1. Describe and correct your classmate's error.

20. Use the ordered pairs below. Cross out the ordered pairs that are NOT described by the equation y 5 x 1 1.

(0, 1) (1, 2) (2, 5) (3, 10) (4, 17)

21. Explain your classmate’s error in using y 5 x 1 1 to describe the function.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

22. Circle the equation that correctly describes the function.

y 5 2x y 5 x 2 1 1 y 5 2x 1 1


linear function nonlinear function

Rate how well you can describe nonlinear functions.

Explanations may vary. Sample: My classmate chose a rule that

describes only two points in the table. The rule must describe all of

the points.

Sample: The y-value is found by squaring the x-value.

y 5 x2

Explanations may vary.


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Chapter 4 122

Vocabulary

Review

4-4 Graphing a Function Rule

Find each input or output.

1. 2x 13

7

2. x 41

22

3

3.3x 1

11

4

Write T for true or F for false.

4. The inputs of a function are the domain of the function.

5. A function pairs every input with exactly one output.

Vocabulary Builder

discrete (adjective) dih SKREET

Related Words: separate (adjective), distinct (adjective)

Main Idea: Discrete describes something consisting of distinct or unconnected elements.

Example: The set of integers is a discrete set.

Nonexample: The set of real numbers is not a discrete set.

Use Your Vocabulary

6. Circle the word or words that mean the opposite of discrete.

separate continuous infinite countable

7. Circle the situation below that describes a discrete set.

the possible temperatures in Florida the number of oranges sold at a fruit stand each day

T

T


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d.Problem 1

Problem 2

g

W

00

100 200 300Gallons of Water

Spa Weight

Tota

l Wei

ght

1000

2000

3000

x

y

O 42

2

4

24

2

4

123 Lesson 4-4

Graphing a Function Rule

Got It? What is the graph of the function rule y 5 12x 2 1?

8. Complete the table below. Then graph each ordered pair on the coordinate plane at the right. Connect the points with a line.

x (x, y)

y 1 1 1

0

1

2

1.5 ( 1, 1.5)

(1, 0.5)

(0, 1)

(2, 0)

12

y 1 1 0.512

y 0 1 112

y 2 1 012

y x 112

When you graph a real-world function rule, choose appropriate intervals for the units on the axes. Every interval on an axis should represent the same change in value. If all the data are nonnegative, show only the fi rst quadrant.

Graphing a Real-World Function Rule

Got It? The function rule W 5 8g 1 700 represents the total weight W, in pounds, of a spa that contains g gallons of water. What is a reasonable graph of the function rule given that the capacity of the spa is 250 gal?

9. Use the values of g to complete the table.

0

50

150

250

(50, 1100)

(150, 1900)

(250, 2700)

W 8(50) 700 1100

W 8(150) 700 1900

W 8(250) 700 2700

W 8(0) 700 700 (0, 700)

(g, W )g W 8g 700

10. Graph the ordered pairs on the coordinate plane at the right.Connect the points with a line segment.


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Chapter 4 124

A continuous graph is a graph that is unbroken.

Label each graph as discrete or continuous.

11.

Ox

y

A discrete graph is a graph composed of distinct, isolated points.

12. y

xO

Key Concept Continuous and Discrete Graphs

Problem 3

Problem 4

Identifying Continuous and Discrete Graphs

Got It? The amount of water w in a wading pool, in gallons, depends on the amount of time t, in minutes, the wading pool has been filling, as related by the function rule w 5 3t . Graph the function rule. Is the graph continuous or discrete? Justify your answer.

13. Complete the table to find each value of w. 14. Graph each ordered pair on the coordinate plane.

15. Underline the correct word or words to complete each sentence.

While the pool is filling, the water will / will not enter throughout a given minute.

The points on the graph should / should not be connected by a line.

The graph is continuous / discrete .

t

w 6 9

3

3

1 2

12

4

Graphing Nonlinear Function Rules

Got It? What is the graph of the function rule y 5 x3 1 1?

16. Cross out any ordered pair that does not lie on the graph of y 5 x3 1 1.

(22, 29) (21, 0) (0, 21) (1, 2) (2, 9)

x

y

10

3

02 3 4 5

Time (minutes)

Water in Wading Pool

Wat

er (g

allo

ns)

6

9

12

15

continuous discrete


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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

x

y

O 54321

1

2

3

4

12345

2

1

3

4

5

5

O

1

1

y

x2

125 Lesson 4-4

17. Circle the letter of the correct graph of the function.

x

y

84

4

8

48

4

8

x

y

84

4

8

48

4

8


Error Analysis Your friend graphs y 5 x 1 3 at the right. Describe and correct your friend's error.

18. Circle the name for the x-values your friend used.

integers rational numbers real numbers whole numbers

19. Circle the best name for the x-values of y 5 x 1 3.

integers rational numbers real numbers whole numbers

20. Describe your friend’s error.

21. Graph the function correctly on the coordinate plane at the right.


continuous graph discrete graph

Rate how well you can graph function rules.

x

y

84

4

8

48

4

8

Explanations may vary. Sample: The

graph should be a line, since noninteger

values of x satisfy the function.


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Chapter 4 126

Vocabulary

Review

4-5 Writing a Function Rule

In function notation, you read f(x) as “f of x.” You can think of the value “f(x)” as another way of writing “y.”

1. Write how you would read h(g) aloud.

2. Circle the equation that shows function notation.

f(x) 5 2x 1 1 xy 5 f f(x) 2 1 0.8x

3. Carmine wants to buy some peaches. Each peach costs $.25. Circle the function Carmine could use to find the cost of any number of peaches p.

0.25c 5 p(c) c(p) 5 0.25 c(p) 5 0.25p 0.25 5 c ? p(c)

Vocabulary Builder

rule (noun) rool

Main Idea: A mathematical rule is a method or procedure that describes how to solve a problem.

Example: A rule of integer multiplication is that a negative integer multiplied by a negative integer produces a positive integer.

Use Your Vocabulary

Consider the rule ab 4cd 5

ab ?

dc , for b, c, d u 0.

4. Circle the equation that is an example of this rule.

21 4

12 5

25 ?

12 2

5 412 5

25 ?

21 2

5 412 5

52 ?

12

5. According to this rule, 23 469 5 ? .

6. Circle the correct words to complete the sentence.

The reason that this rule states that b, c, d 2 0 is because 9.

you cannot multiply by 0 you cannot divide by 0 the dividend cannot be 0

23

h of g

96


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d.Problem 1

Problem 2

127 Lesson 4-5

Writing a Function Rule

Got It? A landfill has 50,000 tons of waste in it. Each month it accumulates an average of 420 more tons of waste. What is a function rule that represents the total amount of waste after m months?

7. Complete the model below.

total waste

Define

Write

Relate is

420 m

plus420 tons of waste

each monthnumber of

monthstimes

Let T the total waste, and let m the number of months.

T 50,000

50,000 tonsof waste

8. Write an equation to represent the situation.

Writing and Evaluating a Function Rule

Got It? A kennel charges $15 per day to board dogs. Upon arrival, each dog must have a flea bath that costs $12. Write a function rule for the total cost for n days of boarding plus a bath. How much does a 10-day stay cost?

9. Define your variables.

Let T 5

. Let n 5

.

10. Now complete the reasoning model below.

Think Write

I will have to pay $15 per day to board my dog. How much will

that cost for n days?

I also have to pay $12 for the flea bath.

If I put those together, I can write a formula for the total cost, T.

15

12

n

T 15 n 12

11. Now evaluate T for n 5 10.

12. The cost of a 10-day stay is $ .

total cost

T 5 50,000 1 420m

number of days

T 5 15n 1 12T 5 15 ? 10 1 12T 5 150 1 12T 5 162

162


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Chapter 4 128

Problem 3 Writing a Nonlinear Function Rule

Got It? Write a function rule for the area of a triangle whose height is 4 in. more than twice the length of its base. What is the area of the triangle when the length of its base is 16 in.?

13. Use the given information to write an equation for the height of the triangle.

Let h 5 the height of the triangle.

Let b 5 the length of the base of the triangle.

Write

Relate h

4

more than twice the length of its base

h 2b

4 in.

14. Use the justifications at the right to find a function rule for the area of the triangle.

A 5 12 ? b ? h Formula for area of a triangle

A 5 12 ? b ? Substitute for h.

A 5 12 ? 2b

1 1

2 ? Distribute 12 b.

A 5 b

1 Simplify.

15. Now find the area of the triangle when its base is 16 in.

A 5 2 1 2 ? Substitute 16 for b.

A 5 1 Evaluate the exponent and the multiplication.

A 5 Add.

16. The area of the triangle is in.2

Got It? Reasoning Graph the function rule from Exercise 14. How do you know the rule is nonlinear?

17. Complete the table of values.

1

4

7

10

12 2 1

42 2 4

72 2 7

102 2 10

24

63

120

3

2bb AA b2

4b

16

2b

(2b 1 4)

2

2

16

256 32

288

288


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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

129 Lesson 4-5

18. Graph the ordered pairs (b, A) that you found in Exercise 17. Use the points to graph the function rule.

19. How do you know the rule is nonlinear? Explain.

_______________________________________________________________________

_______________________________________________________________________


Reasoning Is the graph of a function rule that relates a square’s area to its side length continuous or discrete? Explain.


A continuous / discrete graph is unbroken.

A continuous / discrete graph is composed of isolated points.

The number of possible values for the length of a side of a square is finite / infinite .

21. Is the graph continuous or discrete? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________


function notation function rule

Rate how well you can write function rules.

A

00

2 4 6 8

30

60

90

120

10b

The graph is continuous. Explanations may vary. Sample: The length

of a side of a square can be any nonnegative value. The graph

includes all values between the whole numbers.

Explanations may vary. Sample: The graph of the function is curved,

which means the function is nonlinear.


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Chapter 4 130

Vocabulary

Review

4-6 Formalizing Relations and Functions

1. Use the words below to label the function machine at the right. Use each word once.

function rule y-values output

x-values input range

Vocabulary Builder

reasonable (adjective) ree zun uh bul

Definition: Something is reasonable if it makes sense or is sensible.

Example: It is reasonable to expect warm weather in Miami in July.

Nonexample: It is not reasonable to expect snow in Miami in July.

Other Word Forms: reasonableness (noun); reasonably (adverb)

Opposite: unreasonable (adjective)

Use Your Vocabulary

Complete each sentence with the appropriate word from the list.

reasonable reasonableness unreasonable

2. The student estimated to check the 9 of her answer.

3. Sales tax of $32 on an $85 item is 9.

4. A price of $14 is 9 for a pizza.

domaininput

x-values

range

x-valuesoutput

y-values

equation

function rule

reasonableness

unreasonable

reasonable


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d.Problem 1

Problem 2

131 Lesson 4-6

Identifying Functions Using Mapping Diagrams

Got It? Identify the domain and range of the following relation:

{(4.2, 1.5), (5, 2.2), (7, 4.8), (4.2, 0)}

Represent the relation with a mapping diagram. Is the relation a function?

5. Use the words domain and range to label the mapping diagram. Then draw arrows to represent the relation.

4.25 7

0 1.52.24.8

domain range

6. Does the relation map each domain value to exactly one range value? Yes / No

7. Is the relation a function? Yes / No

You can use the vertical line test to decide whether a relation is a function. If any vertical line passes through more than one point of the graph, then the relation is not a function.

Identifying Functions Using the Vertical Line Test

Got It? Is the relation {(4, 2), (1, 2), (0, 1), (22, 2), (3, 3)} a function? Use the vertical line test.

8. Begin by graphing the points from the relation on the coordinate plane.

x

y

O 42

2

4

24

2

4

9. Can you draw a vertical line that intersects more than one point? If so, draw it. Yes / No

10. Is the relation a function? Yes / No


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Chapter 4 132

Problem 4

Problem 3

Problem 5

Evaluating a Function

Got It? The function w(x) 5 250x represents the number of words w(x) you can read in x minutes. How many words can you read in 6 min?

11. You should substitute for x.

12. The function is evaluated below. Write the justification for each step.

w(x) 5 250x

w(6) 5 250 ? 6

w(6) 5 1500

13. You can read words in 6 minutes.

Finding the Range of a Function

Got It? The domain of g(x) 5 4x 2 12 is {1, 3, 5, 7}. What is the range?


The domain / range is the set of input values.

The domain / range is the set of output values.

15. Use the function g(x) 5 4x 2 12 with domain {1, 3, 5, 7}. Find each output.

g(1) g(3)

g(5) g(7)

16. The range of g(x) 5 4x 2 12 with domain {1, 3, 5, 7} is

{ , , , }.

Identifying a Reasonable Domain and Range

Got It? You have 7 qt of paint to paint the trim in your house. A quart of paint covers 100 ft2. The function A(q) 5 100q represents the area A(q), in square feet, that q quarts of paint cover. What domain and range are reasonable for the function?

Write the original function.

Substitute 6 for x.

Multiply.

1500

g(5) 5 4(5) 2 125 20 2 125 8

g(3) 5 4(3) 2 12 5 12 2 12 5 0

g(1) 5 4(1) 2 125 4 2 125 28

g(7) 5 4(7) 2 12 5 28 2 12 5 16

–8 0 8 16

6


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Lesson Check

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0 2 4 6 8 10

Math Success

y

xO 2

2

4

133 Lesson 4-6

17. Complete the reasoning model below.

Think Write

The least amount of paint I can use is 0 qt.

So, that is the least domain value.

The greatest amount of paint I can use is 7 qt.

So, that is the greatest domain value.

A( 0 0100 )

A( 0 0)

A( 7 7100 )

A( 7 700)

18. A reasonable domain is # q # . 19. A reasonable range is # A(q) # .


Error Analysis A student drew the dashed line on the graph shown and concluded that the graph represented a function. Is the student correct? Explain.

20. Describe how the vertical line test helps you decide whether a relation is a function.

____________________________________________________________________

____________________________________________________________________

21. Underline the correct word or words to complete each sentence about the graph.

I can draw a vertical line that passes through only one point / more than one point .

Therefore the graph does / does not represent a function.

22. Describe the student’s error.

__________________________________________________________________________________

__________________________________________________________________________________


relation domain range vertical line test function notation

Rate how well you understand functions.

Sample: I know that if any vertical line passes through more than one

0 0

point of a graph, the relation is not a function.

7 700

The student drew one vertical line, but did not check other places on the graph

where a vertical line might pass through more than one point.

Answers may vary.

Answers may vary. Sample:


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Chapter 4 134

Vocabulary

Review

4-7 Sequences and Functions

...

1. Circle the name of the next shape in the pattern at the right.

rectangle circle hexagon octagon

Find the next number in each pattern.

2. 1, 13, 19, 3. 6, 4, 2, 0, 4. 2, 10, 50, 250,

Vocabulary Builder

sequence (noun) SEE kwuns

Definition: A sequence is an ordered list of numbers that often form a pattern. Each number in the list is called a term of the sequence.

Example: The Fibonacci sequence is a sequence of numbers where the first number is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers.

Origin: from the Latin word sequentia, which means “to follow”

Use Your Vocabulary

The following sets of numbers are sequences. Explain each pattern.

5. set of whole numbers greater than or equal to 5: {5, 6, 7, 8, 9, …}

_______________________________________________________________________

_______________________________________________________________________

6. {40, 42, 44, 46, 48, …}

_______________________________________________________________________

_______________________________________________________________________

Fibonacci sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Answers may vary. Sample: Beginning with 5, each whole number is

127 22 1250

1 more than the whole number before it.

Answers may vary. Sample: Beginning with 40, each number is 2 more than

the number before it.


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d.Problem 1

Problem 2

135 Lesson 4-7

Extending Sequences

Got It? Describe a pattern in the sequence 5, 11, 17, 23, …. What are the next two terms of the sequence?

7. Complete the diagram. What number is added to each term?

HSM11_A1MC_0407_T91143

6+ 6+ 6+

5 11 17 23

8. Describe the pattern in the sequence.

_______________________________________________________________________

9. Find the next two terms in the sequence.

5, 11, 17, 23, , , . . .

Identifying an Arithmetic Sequence

Got It? Tell whether the sequence 8, 15, 22, 30, … is arithmetic. If it is, what is the common difference?


HSM11_A1MC_0407_T91144

Consecutive Terms

Difference

8 and 15

7 7 8

15 and 22 22 and 30

11. Do the consecutive terms have a common difference? Yes / No

12. Is the sequence an arithmetic sequence? If so, what is the common difference?

_______________________________________________________________________

_______________________________________________________________________

Got It? Tell whether the sequence 7, 9, 11, 13, … is arithmetic. If it is, what is the common difference?


HSM11_A1MC_0407_T91040

7 and 9Consecutive Terms

Difference 2 2 2

9 and 11 11 and 13

14. Is the sequence an arithmetic sequence? If so, what is the common difference?

_______________________________________________________________________

Add 6 to each term to find the next term.

29 35

No, the sequence is not arithmetic. The difference between terms is

not the same.


Yes, the sequence is arithmetic. The difference between terms is 2.

HSM12A1MC_0407.indd 135 3/2/11 3:41:23 PM

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Chapter 4 136

Problem 3

Problem 4

The nth term of an arithmetic sequence with the first term A(1) and common difference d is given by this rule:

A(n) A(1) (n 1)d c nth term

c �rst term

cterm number

common di�erence

a

17. The equation A(5) 5 3 1 (5 2 1)7 generates the fifth term in a sequence. Draw a line from each number in Column A to its description in Column B.

Column A Column B

7 first term of the sequence

5 term number

3 common difference

Key Concept Rule for an Arithmetic Sequence

Writing a Recursive Formula

Got It? Write the recursive formula for the arithmetic sequence 3, 9, 15, 21,… What is the 9th term in the sequence?

18. A(1) =

A(2) 5 A(1) 5 5

A(3) 5 A(2) 5 5

A(4) 5 A(3) 5 5

19. The recursive formula for the arithmetic sequence is A(n) =

20. A(5) 5 A(4) 5 5

A(6) 5 A(5) 5 5

A(7) 5 A(6) 5 5

A(8) 5 A(7) 5 5

A(9) 5 A(8) 5 5

Writing an Explicit Formula

Got It? A subway pass has a starting value of $100. After one ride, the value of the pass is $98.25. After two rides, its value is $96.50. After three rides, its value is $94.75. Write a rule to represent the remaining value on the card as an arithmetic sequence. What is the value of the pass after 15 rides?

21. Describe how the values for A(n) are found.

_______________________________________________________________________

_______________________________________________________________________

1

1

1

1

1

1

1

1

3 9

15

21

1

1

1

1

1

1

1

1

For each term, you subtract another 1.75 because each time you ride the


3

9

6

A(n 2 1) 1 6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

27

33

39

45

5145

39

33

27

21

15

subway, it reduces the value of your pass by 1.75.

HSM12A1MC_0407.indd 136 4/29/11 3:52:18 PM

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0 2 4 6 8 10

Math Success

137 Lesson 4-7

22. How many times is 1.75 subtracted from 100 when n 5 1? What is the A(1) term?

a. How many times is 1.75 subtracted from 100 when n 5 2?

What is the A(2) term?

b. How many times is 1.75 subtracted from 100 when n 5 3?


c. How many times is 1.75 subtracted from 100 when n 5 4?


d. How many times is 1.75 subtracted from 100 when n 5 n?

What is the A(n) term?

23. What is n for the term after 15 rides are used? Using the formula, what is the value of the pass

after 15 rides?


Reasoning Can you use the rule below to find the nth term of an arithmetic sequence with a first term A(1) and a common difference d? Explain.

A(n) 5 A(1) 1 nd 2 d

24. Use the Distributive Property to write an equivalent formula.

A(n) 5 A(1) 1 nd 2 d

25. Can you use the rule A(n) 5 A(1) 1 nd 2 d to find the nth term of an arithmetic sequence? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________


sequence term of a sequence arithmetic sequence common difference

Rate how well you can understand arithmetic sequences.

Yes. Explanations may vary. Sample: Using the Distributive Property,

d(n 2 1) 5 dn 2 d. Since dn 2 d 5 nd 2 d, the formulas are the same.

A(n) 5 A(1) 1 d ? (n 2 1)

They are just written differently.

0

1

16

2

3

n 2 1

98.25

96.50

73.75

94.75

100 2 (n 2 1) 1.75

100

HSM12A1MC_0407.indd 137 4/28/11 10:07:01 AM

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