306 HotTopics 6• 9 Direct Variation When the ratio of two variable quantities is constant, their relationship is called a direct variation. The constant ratio is called the constant of variation. In a direct variation equation, the constant rate of change, or slope, is assigned a special variable, k. A direct variation is a relationship in which the ratio of y to x is a constant, k. We say y varies directly with x. k = y _ x or y = kx, where k ≠ 0. Consider the graph of gas mileage below. y x Distance (miles) 120 60 0 90 30 150 1 2 3 4 5 Gallons Gas Mileage Since the graph of the data forms a line, the rate of change is constant. Use the graph to find the constant ratio. k = miles (y) _ gallon (x) → 60 _ 2 or 30 _ 1 , 90 _ 3 or 30 _ 1 , 120 _ 4 or 30 _ 1 , 150 _ 5 or 30 _ 1 Therefore, the slope (k ) = 30 _ 1 . In this example, the ratio of miles traveled to gallons of gas used remains constant. The car travels 30 miles for every gallon of gas. 6•9 DIRECT VARIATION
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306 HotTopics
6•9 Direct VariationWhen the ratio of two variable quantities is constant, their relationship is called a direct variation. The constant ratio is called the constant of variation. In a direct variation equation, the constant rate of change, or slope, is assigned a special variable, k.
A direct variation is a relationship in which the ratio of y to x is a constant, k. We say y varies directly with x.
k = y _ x or y = kx, where k ≠ 0.
Consider the graph of gas mileage below.
y
x
Dist
ance
(mile
s)
120
60
0
90
30
150
1 2 3 4 5Gallons
Gas Mileage
Since the graph of the data forms a line, the rate of change is constant. Use the graph to find the constant ratio.
k = miles (y)
_ gallon (x) → 60 _ 2 or 30 _ 1 , 90 _ 3 or 30 _ 1 , 120 _ 4 or 30 _ 1 , 150 _ 5 or 30 _ 1
Therefore, the slope (k) = 30 _ 1 .
In this example, the ratio of miles traveled to gallons of gas used remains constant. The car travels 30 miles for every gallon of gas.
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Direct Variation 307
Determining Direct VariationDetermine whether the linear function is a direct variation. If so, state the constant of variation.
Hours, x 2 4 6 8
Earnings, y 16 32 48 64
Compare the ratio of y to x.
k = earnings
_ hours → 16 _ 2 or 8 _ 1 , 32 _ 4 or 8 _ 1 , 48 _ 6 or 8 _ 1 , 64 _ 8 or 8 _ 1 Because the ratios are the same, the function is a direct variation. So, the constant of variation, k, is 8 _ 1 .
EXAMPLE
Check It OutSolve.
1 The Shelby Super Car (SSC) can travel 13.77 kilometers in 2 minutes and 41.31 kilometers in 6 minutes. If the distance varies directly with time, how many kilometers per hour can the SSC travel?
2 At a farm in Georgia, you can pick 4 peaches for $1.75. How much would it cost to pick 9 peaches?
3 Determine whether the linear function is a direct variation. If so, state the constant of variation.
Minutes, x 20 40 60 80
Profit, y 35 55 75 95
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308 HotTopics
6•9 ExercisesDetermine whether each linear function is a direct variation. If so, state the constant of variation.1. x 75 90 105 120
y 5 6 7 8
2. x 4 6 8 10
y 32 48 64 80
3. x 10 15 20 25
y 20 25 30 35
4. x 3 6 9 12
y 12 24 36 48
If y varies directly with x, write an equation for the direct variation. Then find each value. 5. If y = 45 when x = 15, find y when x = 30. 6. Find y when x = 20 if y = 4 when x = 40.
7. A cupcake recipe requires 2 1 _ 4 cups of flour to make 24 cupcakes. How much flour is required to make 36 cupcakes?
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