Direct Variation

Direct VariationThe value of y varies directly with the value of x if there exists a nonzero constant k such that

y = kxOr equivalently: The constant k is called the constant of variation

Direct Variation A relationship between

two variables in which one is a constant multiple of the other.

Constant of Variation – it is the constant (unchanged) ratio of two variable quantities

Direct VariationThe graph of a direct variation equation (y = kx) is a line with constant variation k as the slope and y-intercept (0,0)

Direct Variation

Direct VariationIf y varies directly as x and one set of values, we can use the proportion to find the other sets of corresponding values

Direct VariationExample no. 1:A worker’s pay check (P) varies directly as the number of hours (h) worked. For working 20 hours, the payment is P1,000.00. Find the payment for 45 hours of week

Direct VariationBecause h and P are in direct proportion, P = kh, where k is a constant.

When h = 20, P = 1,000 1000 = k x 20 k = k = 50Thus, P = 50h

Direct Variation as a Power

The value of y varies directly as the power of x if there exists a nonzero real number k such that y = kxn

Direct VariationExample no. 2:The distance of a body falls from rest is directly proportional to the square of the time it falls. (a) If an object falls 20 meters in 2 seconds, how far will it fall in 8 seconds? (b) If an object was dropped from a height of 405 meters, find the time it took for the object to hit the ground

Direct VariationSolution for letter a:The distance of d varies directly as the square of time t.Then, d = kt2 Direct Variation as a power 20 = k(2)2 Replace d with 20 and t with 2 20 = k(4) Simplify k = 5 Divide both sides by 4

Direct VariationSolution for letter b:The constant of variation equation is equal to 5Thus, d = 5t2

When t = 8, d = 5(8)2

d = 320