2. CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise
Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry and Transformations
2.5 Variation and Applications
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3. 2.5 Variation and Applications
Find equations of direct, inverse, and combined variation given values of the variables.
Solve applied problems involving variation.
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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.
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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.
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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
14 = k
The variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x.
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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.
I(h) = kh
I(18) = k  18
$ = 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.
I(33) = $9.35(33)
= $308.55
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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. The number k is called the variation constant, or constant of proportionality.
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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, ).
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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.
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Application
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 they 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.
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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.
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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.
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Example
Example: Find the equation of variation in which y varies directly as the square of x, and y = 12 when x = 2.
Solution:
Thus y = 3x2.
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Example
Example: Find the equation of variation in which y varies jointly as x and z, and y = 42 when x = 2 and z = 3.
Solution:
Thus y = 7xz.
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Example
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
Solution:
Thus
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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-cd lamp. Find the luminance 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 luminance reading of 2 units.
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