1. Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

3. The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together.
Helpful Hint

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

5. Example 1 Continued
Make a graph that shows the relationship between Adam’s age and his length.

6. Example 1 Continued
You can also compare ratios to see if a direct variation occurs. 81
81 ≠ 264
The ratios are not proportional. 22 3 27 12 = ?
264
The relationship of the data is not a direct variation.

7. Example 2: Determining Whether a Data Set Varies Directly
Determine whether the data set shows direct variation. B.

8. Example 2 Continued
Make a graph that shows the relationship between the number of minutes and the distance the train travels.
Plot the points.
The points lie in a straight line.
(0, 0) is included.

9. Example 2 Continued
You can also compare ratios to see if a direct variation occurs. 25 10 50 20 75 30
100 40
Compare ratios. = = =
The ratios are proportional. The relationship is a direct variation.

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

11. Example 3 Continued
Make a graph that shows the relationship between number of baskets and distance. 5 4 3
Number of Baskets 2 1 20 30 40
Distance (ft)

12. Example 3 Continued
You can also compare ratios to see if a direct variation occurs. 60
150  60.
The ratios are not proportional. 5 20 3 30 = ?
150
The relationship of the data is not a direct variation.

13. Determine whether the data set shows direct variation. B.
Example 4
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

14. Make a graph that shows the relationship between ounces and cups.
Example 4 Continued
Make a graph that shows the relationship between ounces and cups.
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.

15. Example 4 Continued
You can also compare ratios to see if a direct variation occurs. = 1 8 2 16 3 24 4 32
Compare ratios.
The ratios are proportional. The relationship is a direct variation.

16. Example 3: Finding Equations of Direct Variation
Find each equation of direct variation, given that y varies directly with x.
3. y is 54 when x is 6

17. Example 4: Finding Equations of Direct Variation
4. x is 12 when y is 15
y = kx .

18. Example 5: Finding Equations of Direct Variation
5. y is 8 when x is 5

19. Example 6
Find each equation of direct variation, given that y varies directly with x.
6. y is 24 when x is 4

20. Example 9
B. x is 28 when y is 14

21. Example 10
C. y is 7 when x is 3

22. Lesson Review: Part 1
Find each equation of direct variation, given that y varies directly with x.
1. y is 78 when x is 3.
2. x is 45 when y is 5.
3. y is 6 when x is 5.
y = 26x
y = x 1 9
y = x 6 5