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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