Warm up

Determine the asymptotes for:

)2(

)3)(2()(

xx

xxxf

Lesson 3-8 Direct, Inverse & Joint Variation

Objective: To recognize and use direct variation to solve problems

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.

y

x

X Y 30 10 15 5 9 3

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!

yk

x

Examples of Direct Variation:

Direct Variation

y1

x1

y2

x2

Direct variation uses the following formula:

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?

Direct Variation

if y varies directly as x and y = 10 as x = 2.4, find x when y =15

10

2.4

15

x

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=1.875270=1.875x2

x=12 12

270

4

3022

xx

Inverse Variation

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

Inverse Variation

If y varies inversely as x, then

for some constant k.x

ky

k is still called the constant of variation.

Inverse VariationWith Direct variation we

Divide our x’s and y’s. In Inverse variation we will

Multiply them.x1y1 = x2y2

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

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

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.?

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.

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.

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.

2z

kxwy

5000k

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.