1. Warm up
Determine the asymptotes for:
1. x=-2, x=0, y=1

2. Lesson 3-8 Direct, Inverse & Joint Variation
Objective: To recognize and use direct variation to solve problems

3. Another way of writing this is k =
Definition:
Y varies directly as x means that y = kx where k is the constant of variation.
Another way of writing this is k =
In other words:
* As x increases in value, y increases or
* As x decreases in value, y decreases. 3

4. Examples of Direct Variation:
Note: X decreases,
30, 15, 9
And Y decreases.
10, 5, 3
What is the constant of variation of the table above?
Since y = kx we can say Therefore:
10/30=k or k = 1/3 5/15=k or k = 1/3
3/9=k or k =1/3 Note k stays constant.
y = 1/3x is the equation! 4

5. Direct Variation
Direct variation uses the following formula: 5

6. Direct Variation example:
if y varies directly as x and y = 10 as x = 2.4, find x when y =15.
what x and y go together? 6

7. Direct Variation
if y varies directly as x and y = 10 as x = 2.4, find x when y =15
x=3.6 7

8. Direct Variation Example:
If y varies directly as the square of x and y = 30 when x = 4, find x when y=270.
y=kx2
30=k42
k=1.875
270=1.875x2
x=12 8

9. Inverse Variation
Inverse is very similar to direct, but in an inverse relationship as one value goes up, the other goes down. 9

10. Inverse Variation
If y varies inversely as x, then for some constant k.
k is still called the constant of variation.

11. Inverse Variation With Direct variation we Divide our x’s and y’s.
In Inverse variation we will Multiply them.
x1y1 = x2y2

12. Inverse Variation
If y varies inversely with x and y = 12 when x = 2, find y when x = 8.
x1y1 = x2y2
2(12) = 8y
24 = 8y
y = 3 12

13. Inverse Variation
If y varies inversely as x and x = 18 when y = 6, find y when x = 8.
18(6) = 8y
108 = 8y
y = 13.5 13

14. Practice
If t varies inversely as q. If t = 240 when q = 0.01, then find the value of t when q = 8
Two rectangles have the same area. The length of a rectangle varies inversely as the width. If the length of a rectangle is 20 ft when the width is 8 ft, find the length of the rectangle when the width is 10 ft.?
t = (k = 2.4)
L = 16

15. Joint and Combined Variation
Joint variation is like direct variation but it involves more than one quantity.
Combined variation combines both direct and inverse variation in the same problem.

16. Joint and Combined Variation
For example: if z varies jointly with x & y, then z=kxy.
Ex: if y varies inversely with the square of x, then y=k/x2.
Ex: if z varies directly with y and inversely with x, then z=ky/x.

17. Example
y varies jointly as x and w and inversely as the square of z.  Find the equation of variation when y = 100, x = 2, w = 4, and z = 20. Then find k.

18. Example
If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = 15.
z=9/10