Chapter 2 More on Functions Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Sec 2.6 Variation and Applications

Direct Variation If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say that we have direct variation, or y varies directly as x, or y is directly proportional to x. The number k is called the variation constant, or constant of proportionality.

Direct Variation The graph of y = kx (k > 0), always goes through the origin and rises from left to right. As x increases, y increases; that is, the function is increasing on the interval (0,  ). The constant k is also the slope of the line.

Example Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3. We know that (3, 42) is a solution of y = kx. y = kx so, k = 14 The variation constant 14, is the rate of change of y with respect (WRT) to x. The equation of variation is: y = 14x.

Direct Variation Example A cashier earns an hourly wage. If the cashier worked 18 hours and earned $168.30, how much will the cashier earn if she works 33 hours? We can express the amount of money earned as a function of the amount of hours worked. I(h) = kh $ = k  18 k=$9.35/hr The hourly wage is the variation constant. Next, we use the equation to find how much the cashier will earn if she works 33 hours. I(33) = $9.35(33) = $308.55

Inverse Variation If a situation gives rise to a function f(x) = k/x, or y = k/x, where k is a positive constant, we say that we have inverse variation, or: y varies inversely as x, or y is inversely proportional to x. k is called the variation constant, or constant of proportionality.

Inverse Variation For the graph y = k/x, as x increases, y decreases; that is, the function is decreasing on the interval (0,  ).

Inverse Variation Example Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4. The variation constant is 8.8. The equation of variation is y = 8.8/x.

Example Road Construction. The time “t” (days) required to do a job varies inversely as the number of people P who work on the job. If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job? Find “k”: (Worker-days)

Example continued The equation of variation is t(P) = 2160/P. Next we compute t(15). It would take 144 days for 15 people to complete the same job.

Combined Variation Other kinds of variation: y varies directly as the nth power of x y varies inversely as the nth power of x y varies jointly as x and z y = kxz.

Example Find the equation of variation in which y varies directly as the square of x, and y = 12 when x = 2. Thus: y = 3x 2.

Example Find the equation of variation in which y varies jointly as x and z and inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2.

Example The luminance of a light (E) varies directly with the intensity (I) of the light and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50 watt lamp. Find the luminance of a 27 watt lamp at a distance of 9 feet. Therefore: the lamp gives an luminance reading of 2 units at 9 ft.