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

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

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

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

Direct Variation Direct variation uses the following formula: 5

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

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

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

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

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

Inverse Variation With 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 12

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

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 =0.3 (k = 2.4) L = 16

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.

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