1. Chapter 1.9
Direct Variation

2. Vocab 𝑦 𝑥 =𝑘 𝐎𝐑 𝑦=𝑘𝑥 𝑘≠0 direct variation constant of variation
When two variable quantities have a constant ratio, their relationship is called a _____________________.
The constant ratio is called the ______________________.
Constant of variation = constant of _________________.
𝑦 𝑥 =𝑘 𝐎𝐑 𝑦=𝑘𝑥
𝑘≠0
direct variation
constant of variation
proportionality

3. The pool fills at a rate of 0.4 inches per minute
Example 1
The height of the water as a pool is being filled is shown in the graph. Determine the rate in inches per minute.
The pool fills at a rate of 0.4 inches per minute
ℎ𝑒𝑖𝑔ℎ𝑡 𝑡𝑖𝑚𝑒 =
𝑦 𝑥 =
2 5
4 10
6 15
8 20
0.4
0.4
0.4
0.4

4. Two minutes after a diver enters the water, he descends 52 feet
Two minutes after a diver enters the water, he descends 52 feet. After 5 minutes, he has descended 130 feet. At what rate is the scuba diver descending?
𝑓𝑒𝑒𝑡 𝑚𝑖𝑛𝑢𝑡𝑒 =
52 2
130 5
𝑦 𝑥 =
OR 26
OR 26
The scuba diver is descending 26 feet per minute

5. The constant of proportionality is 10. So, Julie earns $10 per hour.
Example 2
The equation 𝑦=10𝑥 represents the amount of money 𝑦 that Julie earns for 𝑥 hours of work. Identify the constant of proportionality. Explain what is represents in this situation.
𝑦=𝑘𝑥
𝑦=10𝑥
The constant of proportionality is 10. So, Julie earns $10 per hour.

6. Got it?
The distance 𝑦 traveled in miles by the Chang family in 𝑥 hours is represented by the equation 𝑦=55𝑥. Identify the constant of proportionality. Then explain what it represents in this situation. 𝑦=𝑘𝑥 𝑦=55𝑥 The constant of proportionality is 55. So, The Chang family traveled 55 miles per hour.

7. Determine Direct Variation
Not all situations with a constant rate of change are proportional relationships.
Not all linear functions are direct variations.

8. Example 3
There is no constant ratio and the line does not go through the origin. So, there is no direct variation.
Pizza costs $8 plus a $3 delivery charge. Show the cost of 1, 2, 3, and 4 pizzas. Is there a direct variation?
Number of pizzas 1 2 3 4
Cost ($)
𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑧𝑧𝑎𝑠 =
$11
$19
$27
$35
11 1
19 2
27 3
35 4
=11
=9.5 =9
=8.75

9. Two pounds of cheese cost $8. 40
Two pounds of cheese cost $8.40. Show the cost of 1, 2, 3, and 4 pounds of cheese. Is there a direct variation? Explain.
Pounds of cheese 1 2 3 4
Cost ($)
𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑢𝑛𝑑𝑠 =

10. Example 4
Since the ratios are the same, the relationship is a direct variation. The constant of proportionality is 12.
Determine whether the linear relationship is a direct variation. If so, state the constant of proportionality.
𝑤𝑎𝑔𝑒𝑠 𝑡𝑖𝑚𝑒 =
12 1
24 2
36 3
48 4
=12
=12
=12
=12

11. EXIT TICKET

12. Chapter 1.9
Direct Variation

13. Vocab
When two variable quantities have a constant ratio, their relationship is called a _____________________.
The constant ratio is called the ______________________.
Constant of variation = constant of _________________.
𝑦 𝑥 =𝑘 𝐎𝐑 𝑦=𝑘𝑥
𝑘≠0

14. Example 1
The height of the water as a pool is being filled is shown in the graph. Determine the rate in inches per minute.
ℎ𝑒𝑖𝑔ℎ𝑡 𝑡𝑖𝑚𝑒 =
𝑦 𝑥 =

15. Two minutes after a diver enters the water, he descends 52 feet
Two minutes after a diver enters the water, he descends 52 feet. After 5 minutes, he has descended 130 feet. At what rate is the scuba diver descending?
𝑓𝑒𝑒𝑡 𝑚𝑖𝑛𝑢𝑡𝑒 =
𝑦 𝑥 =
The scuba diver is descending ____________________.

16. Example 2
The equation 𝑦=10𝑥 represents the amount of money 𝑦 that Julie earns for 𝑥 hours of work. Identify the constant of proportionality. Explain what is represents in this situation.
𝑦=𝑘𝑥
The constant of proportionality is ______. So, Julie earns ______________________.

17. Got it?
The distance 𝑦 traveled in miles by the Chang family in 𝑥 hours is represented by the equation 𝑦=55𝑥. Identify the constant of proportionality. Then explain what it represents in this situation. 𝑦=𝑘𝑥 The constant of proportionality is ________. So, The Chang family traveled ________________.

18. Determine Direct Variation
Not all situations with a constant rate of change are proportional relationships.
Not all linear functions are direct variations.

19. Example 3
Pizza costs $8 plus a $3 delivery charge. Show the cost of 1, 2, 3, and 4 pizzas. Is there a direct variation?
Number of pizzas 1 2 3 4
Cost ($)
𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑧𝑧𝑎𝑠 =

20. Two pounds of cheese cost $8. 40
Two pounds of cheese cost $8.40. Show the cost of 1, 2, 3, and 4 pounds of cheese. Is there a direct variation? Explain.
Pounds of cheese 1 2 3 4
Cost ($)
𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑢𝑛𝑑𝑠 =

21. Example 4
Determine whether the linear relationship is a direct variation. If so, state the constant of proportionality.
𝑤𝑎𝑔𝑒𝑠 𝑡𝑖𝑚𝑒 =

22. EXIT TICKET

23. EXIT TICKET