Lesson 2.6 Direct Variation Goal: Write and graph direct variation equations.

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Lesson 2.6 Direct Variation Goal: Write and graph direct variation equations. Key Words: direct variation constant of variation. Direct Variation: Two variables x and y show direct variation , or vary directly , if y = kx

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## Lesson 2.6 Direct Variation Goal: Write and graph direct variation equations.

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Lesson 2.6 Direct Variation

Goal: Write and graph direct

variation equations.

Key Words:

direct variation

constant of variation

Direct Variation:

Two variables x and y show direct variation, or vary directly,

if y = kx

where k is a nonzero constant called the constant of variation.

Example 1

y

=

x

SOLUTION

7

7

Plot a point at the origin.

4

4

Write equation.

y

=

x

7

(

)

Substitute 4 for x.

4

y

=

4

Graph a Direct Variation Equation

Graph .

Find a second point by substituting

a convenient value for x.

Example 1

A second point is .

(

4, 7

)

y

=

7

Graph a Direct Variation Equation

Simplify.

Plot the second point. Then draw

a line through the two points as

shown.

Checkpoint

1.

x

y

=

2.

2

y

x

=

Graph a Direct Variation Equation

Graph the equation.

Checkpoint

3.

y

4x

=

2

4.

y

x

=

3

Graph a Direct Variation Equation

Graph the equation.

Example 2

Write a Direct Variation Equation

Recipes:To make mango salsa, replace the tomatoes in the recipe above with mangoes, using 3 mangoes for 2 recipes. Write an equation relating the number of servings y to the number of mangoes x given so that y varies directly with x. How many servings of salsa can you make if you have 5 mangoes?

SOLUTION

In 2 recipes there are servings, so 3 mangoes make 8 servings. This means when . Write an equation using these values of y and x to solve for k.

(

)

2

4

8

=

y

8

x

3

=

=

Write a direct variation equation.

y

kx

=

(

)

Substitute 8 for y and 3 for x.

k

3

8

=

8

Simplify.

k

=

3

8

An equation is . To find how many servings of salsa you can make using 5 mangoes, substitute 5 for x.

y

x

=

3

Example 2

Write a Direct Variation Equation

Example 2

Five mangoes will make about 13 servings.

Write a Direct Variation Equation

40

1

8

8

(

)

13

y

x

5

=

=

=

=

3

3

3

3

Checkpoint

6.

1

y

x

=

5

x

10,

y

2

=

=

x

3,

y

y

9

3x

=

=

=

Write a Direct Variation Equation

The variables x and y vary directly. Write an equation that relates x and y.

5.

Executive chefs supervise kitchen operations and food service staff.

Lines, x

5

10

15

20

25

Words, y

153

305

454

602

757

SOLUTION

Find the ratio of y to x for each data pair.

454

757

30.28

30.27

153

305

602

15

25

30.6

30.1

30.5

=

=

=

5

20

10

Example 3

Identify Direct Variation

Tell whether the data, the numbers of words y in the first x lines of the Declaration of Independence, show direct variation. If so, write an equation relating x and y.

Example 3

Identify Direct Variation

The ratio of y to x in each pair is nearly the same, so the data show direct variation. Substituting 30.3 for k in ykx gives the direct variation equation y30.3x.

=

=

Checkpoint

7. Tell whether the data show direct variation. If so, write an equation relating x and y.

not direct variation

(

),

(

),

(

),

(

),

(

)

1, 2

2, 4

3, 8

4, 14

5, 22

Identify Direct Variation