Holt CA Course 1

7-9 Direct Variation

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California StandardsCalifornia Standards

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Holt CA Course 1

7-9 Direct Variation

Warm UpTell whether the ratios are proportional.

1. =

2. =

3. =

4. =

69

2436

yes

5668

1417

1213

6078

45 6

30 4

yes

no

yes

?

?

?

?

Holt CA Course 1

7-9 Direct Variation

AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.Also covered: AF3.3, AF3.4

California Standards

Holt CA Course 1

7-9 Direct Variation

Vocabulary

direct variationconstant of proportionality

Holt CA Course 1

7-9 Direct Variation

A direct variation is a linear function that can be written as y = kx, where k is a nonzero constant called the constant of variation.

Holt CA Course 1

7-9 Direct Variation

Determine whether the data set shows direct variation.

A.

Additional Example 1: Determining Whether a Data Set Varies Directly

Holt CA Course 1

7-9 Direct Variation

Method 1: Make a graph.

Additional Example 1A Continued

The graph is not linear.

Holt CA Course 1

7-9 Direct Variation

Method 2: Compare ratios.

223

2712=

?81

264

81 ≠ 264

The ratios are not equivalent.

Both methods show the relationship is not a direct variation.

Additional Example 1A Continued

Holt CA Course 1

7-9 Direct Variation

Determine whether the data set shows direct variation.

B.

Additional Example 1: Determining Whether a Data Set Varies Directly

Holt CA Course 1

7-9 Direct Variation

Method 1: Make a graph.

Additional Example 1B Continued

Plot the points.

The points lie in a straight line.

(0, 0) is included.

Holt CA Course 1

7-9 Direct Variation

Method 2: Compare ratios.

Both methods show the relationship is a direct variation.

2510

5020

7530

10040= = =

Additional Example 1B Continued

The ratio is constant.

Holt CA Course 1

7-9 Direct Variation

Determine whether the data set shows direct variation.

A.

Check It Out! Example 1

Kyle's Basketball Shots 

Distance (ft) 20 30 40

Number of Baskets 5 3 0

Holt CA Course 1

7-9 Direct Variation

Method 1: Make a graph.

Check It Out! Example 1A Continued

Num

ber

of

Bask

ets

Distance (ft)

2

3

4

20 30 40

1

5

Holt CA Course 1

7-9 Direct Variation

Method 2: Compare ratios.

Check It Out! Example 1A Continued

520

330=

?60

150

150 60.

The ratios are not equivalent.

Both methods show the relationship is not a direct variation.

Holt CA Course 1

7-9 Direct Variation

Determine whether the data set shows direct variation.

B.

Check It Out! Example 1

Ounces in a Cup

Ounces (oz) 8 16 24 32

Cup (c) 1 2 3 4

Holt CA Course 1

7-9 Direct Variation

Method 1: Make a graph.

Check It Out! Example 1B Continued

Num

ber

of

Cup

s

Number of Ounces

2

3

4

8 16 24

1

32

Plot the points.

The points lie in a straight line.

(0, 0) is included.

Holt CA Course 1

7-9 Direct Variation

Method 2: compare ratios.

Check It Out! Example 1B Continued

Both methods show the relationship is a direct variation.

The ratio is constant. = 1 8 = =2

163

24 432

Holt CA Course 1

7-9 Direct Variation

Rachel rents space in a salon to cut and style hair. She paid the salon owner $24 for 3 cut and styles. Write a direct variation function for this situation. If Rachel does 7 cut and styles, how much will she pay the salon owner?

Additional Example 2: Finding Equations of Direct Variation

y = kx

24 = k 3

8 = k

y = 8x

Think: The amount owed varies directly with the amount of cuts given.

Substitute 24 for y and 3 for x.

Solve for k.

Substitute 8 for k in the original equation.

x = 3 and y = 24

Step 1 Write the direct variation function.

Holt CA Course 1

7-9 Direct Variation

Step 2 Find how much Rachel will pay the salon owner for 7 cut and styles.

Additional Example 2 Continued

y = 8(7)

y = 56

Substitute 7 for x in the direct variation function.

Multiply.

Rachel will pay the salon owner $56 for 7 cut and styles.

Holt CA Course 1

7-9 Direct Variation

Rinny cuts and styles hair in a salon. She earns $120 for 4 cut and styles. Write a direct variation function for this situation. If Rinny does 9 cut and styles, how much will she earn?

Check It Out! Example 2

y = kx

120 = k 4

30 = k

y = 30x

Think: The amount owed varies directly with the amount of cuts given.

Substitute 120 for y and 4 for x.

Solve for k.

Substitute 30 for k in the original equation.

x = 4 and y = 120

Step 1 Write the direct variation function.

Holt CA Course 1

7-9 Direct Variation

Step 2 Find how much Rinny will earn for 9 cut and styles.

Check It Out! Example 2 Continued

y = 30(9)

y = 270

Substitute 9 for x in the direct variation function.

Multiply.

Rinny will earn $270 for 9 cut and styles.

Holt CA Course 1

7-9 Direct Variation

Mrs. Perez has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Additional Example 3: Money Application

Holt CA Course 1

7-9 Direct Variation

Additional Example 3 ContinuedA. interest from CD and time

interest from CDtime = 17

1 = = 17interest from CDtime

342

The second and third pairs of data result in a common ratio. In fact, all of the nonzero interest from CD to time ratios are equivalent to 17.

The variables are related by a constant ratio of 17 to 1.

= = = 17interest from CDtime = = 17

1342

513

684

Holt CA Course 1

7-9 Direct VariationAdditional Example 3 Continued

B. interest from money market and time

interest from money markettime = = 19

191

interest from money markettime = =18.5

372

19 ≠ 18.5

If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Holt CA Course 1

7-9 Direct Variation

Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.

Check It Out! Example 3

  Interest Interest from

Time (mo) from CD ($) Money Market ($)

0 0 0

1 12 15

2 30 40

3 40 45

4 50 50

Holt CA Course 1

7-9 Direct Variation

Check It Out! Example 3 Continued

interest from CDtime = 12

1interest from CD time = = 1530

2

The second and third pairs of data do not result in a common ratio.

If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

A. interest from CD and time

Holt CA Course 1

7-9 Direct Variation

Check It Out! Example 3 Continued

B. interest from money market and time

interest from money markettime = = 1515

1interest from money market

time = =20 402

15 ≠ 20

If any of the ratios are not equal, then there is no direct variation. It is not necessary to compute additional ratios or to determine whether (0, 0) is included.

Holt CA Course 1

7-9 Direct VariationLesson Quiz: Part I

Determine whether the data sets show direct variation.

1.

2.

direct variation

Amount of Water in a Rain Gauge

Time (h) 1 2 3 4 5

Rain (in) 2 4 6 8 10

Driving Time

Speed (mi/h) 30 40 50 60 80

Time (h) 10 7.5 6 5 3.75

no direct variation

Holt CA Course 1

7-9 Direct Variation

Lesson Quiz: Part II

3. Roy’s income varies directly with the number

of dogs that he walks. He earned $8.50 for

walking 2 dogs. Write a direct variation

function for this situation. If Roy walks 5 dogs,

how much will he earn?

y = 4.25x; $21.25