1. Teacher Most Extraordinaire
Direct Variation
Taught very directly and directed to you by:
Mr. Richard
Teacher Most Extraordinaire

2. Tell whether the ratios are proportional.
Warm Up
Tell whether the ratios are proportional.
1. =
2. =
3. =
4. = 6 9 ? 24 36
yes 56 68 ? 14 17
yes 12 13 ? 60 78 no 45 6 ? 30 4
yes

3. Vocabulary Direct Variation Constant of Proportionality
Varies Directly

4. 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.

5. Additional Example 1: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation. A.

6. Additional Example 1A Continued
Method 1: Make a graph.
The graph is not linear.

7. Additional Example 1A Continued
Method 2: Compare ratios. 81
81 ≠ 264
The ratios are not equivalent. 22 3 27 12 = ?
264
Both methods show the relationship is not a direct variation.

8. Additional Example 1: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation.
Are ratios equivalent?

9. Additional Example 1B Continued
Method 1: Make a graph.
Plot the points.
The points lie in a straight line.
(0, 0) is included.

10. Additional Example 1B Continued
Method 2: Compare ratios. 25 10 50 20 75 30
100 40
The ratio is constant. = = =
Both methods show the relationship is a direct variation.

11. Kyle's Basketball Shots
Check It Out! Example 1
Determine whether the data set shows direct variation. A.
Kyle's Basketball Shots 
Distance (ft) 20 30 40
Number of Baskets 5 3

12. Check It Out! Example 1A Continued
Method 1: Make a graph. 5 4 3
Number of Baskets 2 1 20 30 40
Distance (ft)

13. Check It Out! Example 1A Continued
Method 2: Compare ratios. 60
150  60.
The ratios are not equivalent. 5 20 3 30 = ?
150
Both methods show the relationship is not a direct variation.

14. Check It Out! Example 1
Determine whether the data set shows direct variation. B.
Ounces in a Cup
Ounces (oz) 8 16 24 32
Cup (c) 1 2 3 4

15. Check It Out! Example 1B Continued
Method 1: Make a graph.
Number of Cups
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.

16. Check It Out! Example 1B Continued
Method 2: compare ratios. = 1 8 2 16 3 24 4 32
The ratio is constant.
Both methods show the relationship is a direct variation.

17. Additional Example 2: Finding Equations of 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?
Step 1 Write the direct variation function.
Think: The amount owed varies directly with the amount of cuts given.
x = 3 and y = 24
y = kx
24 = k  3
Substitute 24 for y and 3 for x.
8 = k
Solve for k.
y = 8x
Substitute 8 for k in the original equation.

18. Additional Example 2 Continued
Step 2 Find how much Rachel will pay the salon owner for 7 cut and styles.
Substitute 7 for x in the direct variation function.
y = 8(7)
y = 56
Multiply.
Rachel will pay the salon owner $56 for 7 cut and styles.

19. Check It Out! Example 2
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?
Step 1 Write the direct variation function.
Think: The amount owed varies directly with the amount of cuts given.
x = 4 and y = 120
y = kx
120 = k  4
Substitute 120 for y and 4 for x.
30 = k
Solve for k.
y = 30x
Substitute 30 for k in the original equation.

20. Check It Out! Example 2 Continued
Step 2 Find how much Rinny will earn for 9 cut and styles.
Substitute 9 for x in the direct variation function.
y = 30(9)
y = 270
Multiply.
Rinny will earn $270 for 9 cut and styles.

21. Additional Example 3: Money Application
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.

22. Additional Example 3 Continued
A. interest from CD and time
interest from CD
time = 17 1
= = 17
interest from CD
time 34 2
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.
= = = 17
interest from CD
time
= = 17 1 34 2 51 3 68 4
The variables are related by a constant ratio of 17 to 1.

23. Additional Example 3 Continued
B. interest from money market and time
interest from money market
time
= = 19 19 1
interest from money market
time
= =18.5 37 2
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.

24. Check It Out! Example 3
Mr. Ortega has $2000 in a CD and $2000 in a money market account. The amount of interest he has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either of the data sets and time. If so, find the equation of direct variation.
Interest
Interest from
Time (mo)
from CD ($)
Money Market ($) 1 12 15 2 30 40 3 45 4 50

25. Check It Out! Example 3 Continued
A. interest from CD and time
interest from CD
time = 12 1
interest from CD
time
= = 15 30 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.

26. Check It Out! Example 3 Continued
B. interest from money market and time
interest from money market
time
= = 15 15 1
interest from money market
time
= =20 40 2
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.

27. Amount of Water in a Rain Gauge
Try These Out!
Determine whether the data sets show direct variation. 1. 2.
Amount of Water in a Rain Gauge
Time (h) 1 2 3 4 5
Rain (in) 6 8 10
direct variation
Driving Time
Speed (mi/h) 30 40 50 60 80
Time (h) 10
7.5 6 5
3.75
no direct variation

28. Try This Out!
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

29. Directed Quiz Time! Quiz: Page 74/75 # 2, 4, 6, 24, 32
Homework # 1, 3, 7, 25, 41