Algebra IAlgebra I4.6 4.6

Model Direct VariationModel Direct Variation

VocabularyVocabulary

• Constant of Variation:Constant of Variation: the number the number by which x and y are relatedby which x and y are related

• Direct Variation:Direct Variation: When y = kx and When y = kx and k does not equal zerok does not equal zero

EXAMPLE 1

Tell whether the equation represents direct variation. If so, identify the constant of variation.

2x – 3y = 0a. – x + y = 4b.

Yes; y = 2/3x so k = 2/3 NO

GUIDED PRACTICE

Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. – x + y = 1

2. 2x + y = 0

3. 4x – 5y = 0

No; it does not vary directly.

Yes; it does vary directly.y = -2xConstant = -2

Yes; it does vary directly.y = 4/5xConstant = 4/5

EXAMPLE 2

Graph the direct variation equation.

a. y = x2 3

y = – 3xb.y

x

10

10

-10-10

y

x

10

10

-10-10

EXAMPLE 3

The graph of a direct variation equation is shown.

y = ax

2 = a (– 1)

Write the direct variation equation.a.Find the value of y when x = 30.b.

– 2 = a

So, the equation is y = -2x

Plug in 30 for xy = -2xy = -2 (30)

y = -60

GUIDED PRACTICE

4. Graph the direct variation equation.

y = 2x y

x

10

10

-10-10

Properties of Graphs of Direct Variation Equations

• The graph of a direct variation equation is a line through the origin.• The slope of the graph of y = ax is a.

GUIDED PRACTICE

y = ax

6 = a (4)

64a = 3

2=

5. The graph of a direct variation on equation passes through the point (4,6). Write the direct variation equation and find the value of y when x =24.

So, y = 3/2x

y = 3/2x

y = 3/2 (24)

y = 36

SALTWATER AQUARIUM

EXAMPLE 4

• Write a direct variation equation that relates w and s.• How many tablespoons of salt should be added

to a 30 gallon saltwater fish tank?

The number of tablespoons, s, of sea salt needed in a saltwater fish tank varies directly with the number of gallons of water, w, in the tank. A pet shop owner recommends adding 100 tablespoons of sea salt to a 20 gallon tank.

EXAMPLE 4

s = aw

100 = a(20)

5 = a

So, s = 5w

s = 5ws = 5 (30)s = 150

Therefore, 150 tablespoons of salt

are needed.

a. b.

EXAMPLE 5

a. Explain why C varies directlywith s.

b. Write a direct variation equation that relates s and C.

ONLINE MUSICThe table shows the cost C of downloading s songs at an Internet music site.

SOLUTION

EXAMPLE 5

Because the ratios all equal 0.99, C varies directly with s.

2.97 3 =

4.95 5

6.93 7= = 0.99.

Cs

To explain why C varies directly with s, compare the

ratios for all data pairs (s, C ):

a.

b. A direct variation equation is C = 0.99s.