1. Lesson 12 Solving inverse variation problems

2. Inverse variation If the product of 2 variables is a constant, then the equation is an inverse variation. xy = k or y = k x Both x and y are variables and k is a nonzero constant. k is called the constant of variation.

3. Differences between direct and inverse variations Direct Inverse As x increases, y increases As x increases, y decreases As x decrease, y decreases As x decreases, y increases The ratio y/x is a constant The product yx is a constant

4. Modeling inverse variation X 1 2 4 Y 2 1.5 These values represent the equation y = 2/x Graph the equation Is it an inverse variation? Find the constant of variation by multiplying xy constant of variation = 2 What will the y value be when x is 8? xy = 2 8y = 2 y = 1/4

5. example y = 8/x x 1 2 4 y 8 4 2 Is this an inverse variation? If so, find the constant of variation What will the y value be when x is 16?

6. Testing for an inverse variation By finding the constant of variation, it is possible to find an equation for the inverse variation. You must test every ordered pair. x 1 2 3 4 y 3 1.5 1.75 xy 3 3 3 3 Constant of variation is 3 for all ordered pairs so xy = 3 or y = 3/x

7. example x 2 3 4 5 y 6 4 3 2.4 Is this an inverse variation? If so, what is the constant of variation?

8. Direct or inverse variation? Determine if the data set represents a direct or inverse variation, find the constant of variation and the equation. x 1 2 3 4 y 25 10 3 1 Look at the relationship between x and y- as x increases, y decreases, so it could be an inverse variation- it cannot be a direct variation. There is no constant of variation, so it is not an inverse variation

9. example Determine if the data set represents an inverse or direct variation. If so, find the constant of variation and the equation. x 3 4 5 6 y 6 8 10 12

10. Joint variation Joint variation involves 3 variables, one of which depends on changes in the other 2 variables. A joint variation can be written as shown: z = k xy Variables are x, y, z and k is the constant of variation.

11. Joint variation The force (F) on an object of mass m that causes it to move with acceleration a can be found using the equation F= ma. Is this an example of a joint variation? Explain. The equation F = ma can be written F = 1 ma Since 1 is a constant, the equation is a joint variation

12. example Is the equation d = rt an example of a joint variation?

13. Using a table to look for joint variation Kinetic energy (KE) of a moving object is based on its mass (m) and the square of its velocity( v 2 ). Use the data points to see if it is a joint variation. KE 2 9 24 50 m 1 2 3 4 v 2 4 9 16 25 Calculate the ratio KE mv 2.5.5.5.5 constant of variation is.5, so it is a joint variation

14. example Does the data set represent a joint variation? x 4 16 36 64 y 1 2 3 4 t 2 4 6 8

15. Joint variation? In an electric circuit, the electric power (P) is the product of the square of the current (I) and the electrical resistance (R). Write an equation that represents this relationship. Is it a joint variation? P = I 2 R P = 1 I 2 R Yes joint variation, with constant of variation 1

16. Determining type of variation The area of a triangle is given by the formula A = 1/2 bh What kind of variation is this? rewrite it! A = 1 bh 2 joint variation!

17. example Isaac Newton found that gravitational force is proportional to the product of the 2 masses. What kind of variation is this. Since there are 3 variables, this is a joint variation Write the equation F g =k m 1 m 2 or F g = k m 1 m 2

18. example The total cost of x number of shirts at $20 is an example of what kind of variation? Write the equation. T = 20x direct variation