1. Variation
Unit 13

2. Direct Variation
The variable y varies directly as x if there is a nonzero constant, k, such that y = kx. The equation y = kx is called a direct variation equation and the number k is called the constant of variation.

3. Steps: Plug in the given information to find k.
Once you have found k, rewrite the direct variation equation including this value.
Use the found k value to evaluate the equation at another value (x or y).

4. Examples
Find the constant of variation, k, and the direct variation equation if y varies directly as x and y = -24 when x = 4.

5. Examples
Find the constant of variation, k, and the direct variation equation if y varies directly as x and y = 15 when x = 3.

6. Examples
A varies directly as b. If a is 2.8 when b is 7, find a when b is -4.

7. Examples
A varies directly as b. If a is -5 when b is 2.5, find b when a is 6.

8. Examples
Ohm’s law states that the voltage, V, measured in volts varies directly as the electric current, I, according to the equation V= IR. The constant of variation is the electrical resistance of the circuit, R. Ex: An iron is plugged into a 110-volt electrical outlet, creating a current of 5.5 amperes in the iron. Find the electrical resistance to the iron.

9. Determine whether y varies directly as x. If so, find k.

10. Example
If y varies directly as x and y = 8 when x = -4, find x when y = 7. If y varies directly as x and y = 3 when x = 5, find y when x = 15.

11. Example
If y varies directly as and y = 10 when x = 5, find y when x = 2.

12. Homework
Book Page #14-28 even and #32-36 all

13. Inverse Variation
Two variables, x and y, have an inverse variation relationship if there is a nonzero number, k, such that y = k/x

14. Joint Variation
If y=kxz, then y varies jointly as x and z, and the constant of variation is k

15. Example
The volume of a rectangular prism is V=lwh. Therefore, volume varies jointly as the length and the width.

16. Example
The variable y varies inversely as x, and y = 132 when x = 15. Find the constant of variation and write an equation for the relationship. Then find y when x is 1.5.

17. Example
Y varies jointly as x and z. Write the appropriate joint-variation equation and find y for the given values of x and z. Ex: y = -108 when x = -4 and z = 3. Find y when x =6 and z = -2

18. Example
Z varies jointly as x and y and inversely as w. Write the appropriate combined variation equation and find z for the given values of x, y and w. Ex: z = 3 when x = 3, y = - 2, and w = -4. Find z when x = 6, y = 7, and w = -4

19. Homework
Book Page 486 # odd

20. Word Problems
The heat loss through a glass window varies jointly as the area of the window and the difference between inside and outside temperature. The heat loss through a window with an area of 4 square meters is 820 BTU when the temperature difference is 10 degrees. Write the joint variation equation and find the constant.