Variation Direct and Inverse

7/9/2013 Variation 2 Direct Variation A variable y varies directly as variable x if y = kx for some constant k The constant k is called the constant of variation K is also known as the constant of proportionality

7/9/2013 Variation 3 Direct Variation Example State sales tax t varies directly as the amount of sale s, i.e. t = ks For tax of $200 on a $12.50 sale, what is the constant of variation ? s t k = t s = = k =.0625 Question: Does this look like y = mx + b ?

7/9/2013Variation4 Direct Variation Output varies directly with input Example: y = kx Newtons Second Law The resultant force acting on a mass m is directly proportional to the acceleration a of the mass: Direct and Inverse Variation k is the constant of variation k y x = OR F = ma OR = m F a

7/9/2013 Variation 5 Inverse Variation Variable y varies inversely as variable x if for constant of variation k k is also known as the constant of inverse proportionality Variation x y y = k x

7/9/2013 Variation 6 Inverse Variation Functions Output varies inversely with x n Example: y = kx –n k is the constant of variation The Inverse Square Law The earths gravitational force F acting on an object of mass m is inversely proportional to the square of the distance r between the mass and the center of the earth Direct and Inverse Variation yx n = k OR OR Fr 2 = GMm = GMm r2r2 F

7/9/2013 Variation 7 Inverse Variation Example At constant temperature the pressure P of a gas in a balloon is inversely proportional to its volume V so that V P P = k V

7/9/2013Variation8 Direct Variation Output varies directly with input Example: y = kx Inverse Variation Output varies inversely with input Example: y = kx –1 Inverse Variation Functions Output varies inversely with x n Example: y = kx –n Variation Review constant of variation is k k y x = OR yx = k OR yx n = k OR

7/9/2013 Variation 9 Think about it !