Cop

yrig

ht ©

by

Pear

son

Educ

atio

n, In

c. o

r its

aff

iliat

es. A

ll Ri

ghts

Res

erve

d.

Vocabulary

Chapter 11 334

11-6 Inverse Variation

Review

1. Circle the constant term in each equation.

y 5 25x 1 7 x 5 5 y 5 4x 2 1 5x 2 11

Vocabulary Builder

inverse variation (noun) in vurs vehr ee ay shun

Definition: An inverse variation is a relationship between two quantities where one quantity increases as the other decreases.

Main Idea: “y varies inversely as x” means that when x increases, y decreases by the same factor.

Example: The amount of time a car travels increases as the car’s speed decreases. This relationship between time and speed is an inverse variation.

Nonexample: One ticket to a play costs $10. The total cost of the tickets increases as the number of tickets bought increases. The relationship between total cost and number of tickets is a direct variation, not an inverse variation.

Use Your Vocabulary

Consider each of the following situations. Then underline the correct word to complete each sentence.

2. A student downloads several songs at $2 each.

As the number of downloaded songs The relationship between the number of downloaded increases, the total cost of the songs and the total cost of the downloads

downloads increases / decreases . represents a(n) direct / inverse variation.

3. The job of building a patio is split evenly among several workers.

As the number of workers on the job The relationship between the number of workers on

increases, the workload of each person the job and the workload of each person

increases / decreases . represents a(n) direct / inverse variation.

HSM12A1MC_1106_NL.indd 334 3/9/11 4:25:03 PM

Cop

yrig

ht ©

by

Pear

son

Educ

atio

n, In

c. o

r its

aff

iliat

es. A

ll Ri

ghts

Res

erve

d.

Problem 1

Problem 2

Key Concept Inverse Variation

335 Lesson 11-6

Writing an Equation Given a Point

Got It? Suppose y varies inversely with x, and y 5 9 when x 5 6. What is an equation for the inverse variation?

5. Complete the reasoning model below.

HSM11_A1MC_1106_T91408

Think Write

First I write the general form of an inverse variation.

To find the value of k, I substitute 6 for x and 9 for y.

Then I simplify to find k.

xy =

k=

=∙

6. An equation for the inverse variation is xy 5 .

Using Inverse Variation

Got It? The weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. A 120-lb weight is placed on a lever, 5 ft from the fulcrum. How far from the fulcrum should an 80-lb weight be placed to balance the lever?

7. Let x be the distance of the 80-lb weight from the fulcrum. Complete the diagram.

ft

120 lb lb

x ft

HSM11_A1MC_1106_T91409

8. Circle the equation that models the inverse variation.

1205 5

80x 5 1 x 5 120 1 80 120 ? 5 5 80 ? x

An equation of the form xy 5 k or y 5 kx , where k 2 0, is an inverse variation. The

constant of variation is k, which is the product x ? y for an ordered pair (x, y) that satisfies the inverse variation.

4. Cross out the equations that do NOT represent an inverse variation.

y 5 5 xy 5 3 y 5 26x x 5 3y

HSM12A1MC_1106_NL.indd 335 3/9/11 4:25:00 PM

Cop

yrig

ht ©

by

Pear

son

Educ

atio

n, In

c. o

r its

aff

iliat

es. A

ll Ri

ghts

Res

erve

d.

Problem 3

Chapter 11 336

9. Complete the steps to solve the equation.

120 ? 5 ? x Write the equation.

5 Simplify.

5 x Solve for x.

10. The 80-lb weight should be placed ft from the fulcrum.

Graphing an Inverse Variation

Got It? What is the graph of y 5 28x ?

11. Complete the table of values. 12. Plot the points from the table. Connect the points with smooth curves.

HSM11_A1MC_1106_T91411

y

2

−2

−1

1

2

4

0 undefined

x

HSM11_A1MC_1106_T91412

x

y

O 84

4

8

–4–8

–8

–4

Concept Summary Direct and Inverse Variations

Direct Variation Inverse Variationy varies directly with x. y varies inversely with x.y is directly proportional to x. y is inversely proportional to x.

The ratio yx is constant. The product xy is constant.

y

y kx, k 0

x

y

y kx, k 0

x

y

x

y , k 0kx

y

x

y , k 0kx

Cop

yrig

ht ©

by

Pear

son

Educ

atio

n, In

c. o

r its

aff

iliat

es. A

ll Ri

ghts

Res

erve

d.

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

Lesson Check

Problem 4

HSM11_A1MC_1106_T91413

–18

–24

–124

8

6

yx

337 Lesson 11-6

Check off the vocabulary words that you understand.

inverse variation constant of variation

Rate how well you can write and graph inverse variations.

• Do you UNDERSTAND?

Does the graph of an inverse variation always, sometimes, or never pass through the origin? Explain.

17. At x 5 0, what will the value of yx be? Explain.

_______________________________________________________________________________

18. Does the graph of an inverse variation always, sometimes, or never pass through the origin? Explain.

______________________________________________________________________________

Determining Direct or Inverse Variation

Got It? Do the data in the table represent a direct variation or an inverse variation? Write an equation to model the data.

13. Find the value of each

HSM11_A1MC_1106_T91414

–3

–48

xy yx

expression for the data.

14. Circle the expression that is constant for this data.

xy yx

15. The data represent a(n) direct / inverse variation.

16. The equation models the data.

HSM12A1MC_1106_NL.indd 337 3/9/11 4:24:50 PM