1. Section 2.3
Direct Variation

2. Direct Variation
Definition 1: If a relationship can be expressed in the form , the two variables are said to vary directly or to be proportional. The value of k shows the constant rate of change of the dependent variable and is the constant of variation.

3. Direct Variation
Definition 2: A direct variation is a linear function that can be written in the form y = kx, where k ≠ 0.

4. Example 1
For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y 2 8 3 12 5 20

5. Example 2
For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y 1 4 2 7 5 16

6. Example 3
For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y -6 -2 3 1 12 4

7. Example 4
For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y -9 5 3
-5/3 6
29/8

8. Examples 5 – 7
For each function, determine whether y varies directly with x. If so, find the constant of variation.
3y = 2x
y = 2x + 3
5/6x = 1/3y

9. TOTD
For the following function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y 2 -6 4
-12 5
-15

10. Example 8: Water Conservation
Suppose a dripping faucet wastes a cup of water if it drips for three minutes. The amount of water wasted varies directly with the amount of time the faucet drips. Find the constant of variation k. Then write an equation to model the direct variation.

11. Examples 9 – 13 Using a proportion.
Suppose y varies directly with x, and x = 27 when y = -15. Find x when y = -17.
If y = 4 when x = 3, find y when x = 6.

12. Examples 9 – 13 If y = 7 when x = 2, find y when x = 8.
If y = 10 when x = -3, find x when y = 2.
If y = 1 when x = 10, find y when x = 2.

13. TOTD In the following, y varies directly with x.
If y = 7 when x = 3, find x when y = 21.
If y = -20 when x = 2, find y when x = 14.