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Variation Direct and Inverse Variation 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 2 Variation 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 ? t k = s t 12.50 = 200.00 = .0625 Question: Does this look like y = mx + b ? 7/9/2013 Variation s 3 Direct and Inverse Variation Direct Variation Output varies directly with input Example: y = kx OR x =k k is the constant of variation Newton’s Second Law The resultant force acting on a mass m is directly proportional to the acceleration a of the mass: F = ma OR 7/9/2013 y Variation F a =m 4 Variation Inverse Variation Variable y varies inversely as variable x if y = k x for constant of variation k y k is also known as the constant of inverse proportionality x 7/9/2013 Variation 5 Direct and Inverse Variation Inverse Variation Functions n Output varies inversely with x Example: y = kx–n OR k is the constant of variation The Inverse Square Law The earth’s 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 F = 7/9/2013 yxn = k GMm r2 OR Fr2 = GMm Variation 6 Variation Inverse Variation Example At constant temperature the pressure P of a gas in a balloon is inversely proportional to its volume V so that k P = V 7/9/2013 Variation P V 7 Variation Review Direct Variation 7/9/2013 Output varies directly with input Example: y = kx OR y x =k Inverse Variation Output varies inversely with input Example: y = kx–1 OR yx = k constant of variation is k Inverse Variation Functions Output varies inversely with xn Example: y = kx–n OR yxn = k Variation 8 Think about it ! 7/9/2013 Variation 9