Direct Variation We say that two positive quantities x and y vary directly if an increase in one causes a proportional increase in the other. In this case, there is a constant k such that y = kx. Similarly, we have that y varies directly as the power x n for some positive n when y = kx n. The constant k is called the constant of variation. Example. The circumference of a circle C varies directly as the radius r. The constant of variation is 2 . Example. The area of a circle A varies directly as the square of the radius r. The constant of variation is .

Inverse Variation We say that two positive quantities x and y vary inversely if an increase in x causes a proportional decrease in y. In this case, there is a constant k such that y = k/x. Similarly, we may have that y varies inversely as the power x n for some positive n when y = k/(x n ) Example. Suppose that y varies inversely as x 2 and that y = 10 when x = 10. Write the appropriate equation.

Joint Variation An equation of variation can involve more than two variables. We say that a quantity varies jointly as two or more other quantities if it varies directly as their product. Example. The intelligence quotient (IQ) varies directly as a person’s mental age and varies inversely as a person’s chronological age. If a child with an IQ of 125 has a mental age of 15 and a chronological age of 12, write the appropriate equation.

Summary of Direct and Inverse Variation; We discussed Direct variation as a power of x Inverse variation as a power of x Joint variation