1. 9.1: Inverse and Joint Variation Objectives: Students will be able to… Write and use inverse variation models Write and use joint variation models

2. Review of Direct Variation:  Linear function whose graph passes through the origin  y= kx, where k is the constant of variation  “y varies directly with x” (y = kx)   If every is the same for a set of data pairs, then it is direct variation EXAMPLE: x246810 y12345 k = y/x = ½ for every data pair y varies directly with x y = (1/2) x

3. Inverse Variation:  “y varies inversely with x”  k = x∙y  If every x∙y is the same for a set of data pairs, then it is inverse variation EXAMPLE: x1.52.545 y20127.56 k = xy= 30 for each data pair Inverse variation model:

4. Tell whether x and y show direct variation, inverse variation, or neither.  Solve for y. Does it look like a direct or inverse variation model?? 1. xy = 4.8 2. x = 3. y –x = 2 Inverse variation Direct variation neither

5. Does the following show inverse variation, direct variation, or neither? x-5-4-3-2 y10864 Direct: y = -2x

6. x and y vary inversely and y = 6 when x = 1.5: a.) write and equation that relates x and y b.) Find y when x = 4/3 a.) Find the constant, k: k = xy k = (1.5)(6) = 9 Write equation using k: b.) Find y when x = 4/3:

7. YOU TRY: x and y vary inversely and x = 7 when y = 1. Write and equation relating x and y and find y when x = 3.

8. The volume of gas in a container varies inversely with the amount of pressure. A gas has volume 75 in 3 at a pressure of 25 lb/in 2. Write a model relating volume and pressure. “Volume varies inversely with pressure” “ y varies inversely with x “ k = V∙P = 75∙25 k= 1875 V =

9. Joint Variation:  A quantity varies directly as the product of two or more quantities.

10. Examples of Variation: Example:Equation: z varies jointly with x and y y varies inversely with the square of x z varies directly with y and inversely with x.

11. Write and equation for the following: 1. y varies directly with x and inversely with z 2 2. y varies inversely with x 3 3. y varies directly with x 2 and inversely with z 4. z varies jointly with x 2 and y 5. y varies inversely with x and z

12. The volume of a geometric figure varies jointly with the square of the radius of the base and the height. a.) Write an equation for the volume. b.) Estimate the constant of variation of V = 63.33 in 3, r = 2.4 in, and h = 10.5 in.