11-6
Inverse Variation
Vocabulary
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.
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 increases, the total cost of the downloads increases / decreases .
The relationship between the number of downloaded
songs and the total cost of the downloads
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.
Chapter 11
334 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
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.
Key Concept Inverse Variation
k
An equation of the form xy 5 k or y 5 x , 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
26
y5 x x 5 3y Problem 1 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.
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
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
6. An equation for the inverse variation is xy 5 .
Problem 2 Using HSM11_A1MC_1106_T91408
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.
120 lb
lb
ft
x ft
8. Circle the equation that models the inverse variation.
120
80
5 5 x 5 1 x 5 120 1 80
120 ? 5 5 80 ? x
HSM11_A1MC_1106_T91409
335
Lesson 11-6
9. Complete the steps to solve the equation.
120 ? 5 5 Simplify.
5 x
Solve for x.
? x
Write the equation.
10. The 80-lb weight should be placed ft from the fulcrum.
Problem 3 Graphing an Inverse Variation
Got It? What is the graph of y 5 28
x ?
11. Complete the table of values. x
12.Plot the points from the table.
Connect the points with smooth curves.
y
8
2
y
4
−2
x
–8
−1
–4
O
4
8
–4
0
undefined
–8
2
4
HSM11_A1MC_1106_T91412
Concept Summary Direct and Inverse Variations
HSM11_A1MC_1106_T91411
Direct Variation
y varies directly with x.
y is directly proportional to x.
Inverse Variation
y varies inversely with x.
y is inversely proportional to x.
y
y
x
The product xy is constant.
y
y
x
x
The ratio x is constant.
y
x
y  kx, k  0
Chapter 11
y  kx, k  0
k
y  x ,k 0
336 k
y  x ,k 0
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
1
Problem 4 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 expression for the data.
y
x
xy
–48
x
y
4
–12
6
–18
8
–24
–3
HSM11_A1MC_1106_T91413
14. Circle the expression that is constant for this data.
y
x xy
HSM11_A1MC_1106_T91414
15. The data represent a(n) direct / inverse
variation.
16. The equation models the data.
Lesson Check • Do you UNDERSTAND?
Does the graph of an inverse variation always, sometimes, or never pass through the
origin? Explain.
y
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
17. At x 5 0, what will the value of x be? Explain.
________________________________________________________________________________
18. Does the graph of an inverse variation always, sometimes, or never pass through
the origin? Explain.
_______________________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
inverse variation
constant of variation
Rate how well you can write and graph inverse variations.
Need to
review
0
2
4
6
8
Now I
get it!
10
337
Lesson 11-6