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Holt CA Course 1
7-9 Direct Variation
Warm UpWarm Up
California StandardsCalifornia Standards
Lesson PresentationLesson Presentation
PreviewPreview
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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
?
?
?
?
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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
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Holt CA Course 1
7-9 Direct Variation
Vocabulary
direct variationconstant of proportionality
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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.
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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
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Holt CA Course 1
7-9 Direct Variation
Method 1: Make a graph.
Additional Example 1A Continued
The graph is not linear.
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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
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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
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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.
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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.
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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
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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.
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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
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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
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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.
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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
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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
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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.
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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
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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