1. 3.4-1 Variation

2. Many natural (physical) phenomena exhibit variation = one quantity (quantities) changing on account of another (or several) Principle is some kind of dependence What things can we think about that depend on another action/object?

3. Direct Variation Direct Variation = as one variable changes, the other changes at some constant rate Y varies directly with the n th power of x (y is proportional to the n th power of x) if: – y = kx n – K is a constant; n is a real number D = rt is an example of direct variation

4. The constant In most applications, we have to determine the constant value k, given information about y and x Example. Hooke’s Law says the force exerted by a spring on a spring scale varies directly with the distance the spring is stretched. If a 15 pound mass suspended on a string stretches the spring 6 inches, how far will a 20 pound mass stretch it?

5. y = kx

6. Example. Write the mathematical model for the following statement. A) S varies directly as the product of 4 and x. B) Z varies directly with y-cubed. C) J(x) varies directly with the nth-root of x.

7. Inverse Variation Inverse Variation = as one quantity increases, a second quantity decreases y varies inversely with the n th power of x (or, y is inversely proportional to the n th power of x) if there is a constant k such that y =

8. Example. Supper y is inversely proportional to the 2 nd power of x, and y = 9 when x = 3. What is y when x = 10? Example. Supper y is inversely proportional to the square of x, and that y = 5 when x = 2. What is y when x = 10?

9. Joint Variation More than 2 variables Z varies jointly as x and y (proportional to x and y) if there is a constant k such that – Z = kxy Z varies jointly as the n th power of x and the m th power of y is there is a constant k such that Z = kx n y m

10. Example. Suppose z is jointly proportional to x and y, and that z = 200 when x = 10 and y = 6. What is z when x = -5 and y = 3? Example. Suppose z is jointly proportional to the square of x and the cube of y, and that z = 500 when x = 2 and y = 27. Find z when x = 16 and y = 32.

11. Assignment Pg. 238 1-18 all

12. Solutions