1. Section 3.6 Variation

3. 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 that y varies directly as x, or that y is directly proportional to x. The number k is called the variation constant, or constant of proportionality.

4. 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.

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

6. Application Example: Wages. Example: Wages. 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? 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? Solution: We can express the amount of money earned as a function of the amount of hours worked. Solution: We can express the amount of money earned as a function of the amount of hours worked. f(h) = kh f(h) = kh f(18) = k  18 f(18) = k  18 $168.30 = k  18 $168.30 = k  18 $9.35 = k The hourly wage is the variation constant. $9.35 = k The hourly wage is the variation constant. Next, we use the equation to find how much the cashier will Next, we use the equation to find how much the cashier will earn if she works 33 hours. earn if she works 33 hours. f(33) = $9.35(33) f(33) = $9.35(33) = $308.55 = $308.55

8. 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 that y varies inversely as x, or that y is inversely proportional to x. 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 that y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality. The number k is called the variation constant, or constant of proportionality.

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

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

11. Application Example: Road Construction. Example: Road Construction. The time t required to do a job varies inversely as the number of people P who work on the job (assuming that all work at the same rate). If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job? The time t required to do a job varies inversely as the number of people P who work on the job (assuming that all work at the same rate). If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job? Solution: We can express the amount of time required, in days, as a function of the number of people working. Solution: We can express the amount of time required, in days, as a function of the number of people working. t varies inversely as P This is the variation constant.

12. Application 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. It would take 144 days for 15 people to complete the same job.

13. Combined Variation Other kinds of variation: y varies directly as the nth power of x if there is some positive constant k such that. y varies inversely as the nth power of x if there is some positive constant k such that. y varies jointly as x and z if there is some positive constant k such that y = kxz.

14. Example The illuminance 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-cd lamp. Find the illuminance of a 27-cd lamp at a distance of 9 feet. The illuminance 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-cd lamp. Find the illuminance of a 27-cd lamp at a distance of 9 feet. Solve for k. Substitute the second set of data into the equation. The lamp gives an illuminance reading of 2 units. The lamp gives an illuminance reading of 2 units.