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Section 3.6

Section 3.6. Variation. 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 .

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Section 3.6

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  1. Section 3.6 Variation

  2. 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 yis directly proportional to x. • The number k is called the variation constant, or constant of proportionality.

  3. 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.

  4. Direct Variation Example: 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. y = kx 42 = k  3 = k 14 = k The variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x.

  5. Application 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? Solution: We can express the amount of money earned as a function of the amount of hours worked. f(h) = kh f(18) = k 18 $168.30 = k  18 $9.35 = k 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. f(33) = $9.35(33) = $308.55

  6. 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 tox. • The number k is called the variation constant, or constant of proportionality.

  7. 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, ).

  8. 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. Solution: The variation constant is 8.8. The equation of variation is y = 8.8/x.

  9. Application Example: RoadConstruction. 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. t varies inversely as P This is the variation constant.

  10. 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.

  11. 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 ofx 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.

  12. 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. Solve for k. Substitute the second set of data into the equation. The lamp gives an illuminance reading of 2 units.

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