NAME ____________________________________________ DATE ____________________________ PERIOD _____________
11-1 Study Guide and Intervention
Inverse Variation
!
Identify and Use Inverse Variations An inverse variation is an equation in the form of y = ! or xy = k. If two points
(𝑥! , 𝑦! ) and (𝑥! , 𝑦! ) are solutions of an inverse variation, then 𝑥! ⋅ 𝑦! = k and 𝑥! ⋅ 𝑦! = k.
Product Rule for Inverse Variation
𝑥! ⋅ 𝑦! = 𝑥! ⋅ 𝑦!
From the product rule, you can form the proportion
!!
!!
=
!!
!!
.
Example: If y varies inversely as x and y = 12 when x = 4, find x when y = 18.
Method 1 Use the product rule.
Method 2 Use a proportion.
𝑥! ⋅ 𝑦! = 𝑥! ⋅ 𝑦!
!!
Product rule for inverse variation
4 ⋅ 12 = 𝑥! ⋅ 18
!"
!"
!
!
!!
𝑥! = 4, 𝑦! = 12, 𝑦! = 18
!
= 𝑥!
Divide each side by 18.
!!
= 𝑥!
Simplify.
!
=
=
!!
!!
!"
!"
48 = 18𝑥!
!
= 𝑥!
Proportion for inverse variation
𝑥! = 4, 𝑦! = 12, 𝑦! = 18
Cross multiply.
Simplify.
!
Both methods show that 𝑥! = when y = 18.
!
Exercises
Determine whether each table or equation represents an inverse or a direct variation. Explain.
1.
x
y
3
6
5
10
8
16
12
24
2. y = 6x
3. xy = 15
Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then solve.
4. If y = 10 when x = 5,
find y when x = 2.
5. If y = 8 when x = –2,
find y when x = 4.
6. GEOMETRY For a rectangle with given area, the width of the rectangle varies inversely as the length. If the width of
the rectangle is 40 meters when the length is 5 meters, find the width of the rectangle when the length is 20 meters.
Chapter 11
5
Glencoe Algebra 1
NAME ____________________________________________ DATE ____________________________ PERIOD _____________
11-1 Study Guide and Intervention (continued)
Inverse Variation
Graph Inverse Variations Situations in which the values of y decrease as the values of x increase are examples of
inverse variation. We say that y varies inversely as x, or y is inversely proportional to x.
Inverse Variation Equation
an equation of the form xy = k, where k ≠ 0
Example 1: Suppose you drive 200 miles without
Example 2: Graph an inverse variation in which y
stopping. The time it takes to travel a distance varies
inversely as the rate at which you travel. Let x = speed
in miles per hour and y = time in hours. Graph the
variation.
varies inversely as x and y = 3 when x = 12.
Solve for k.
xy = k
Inverse variation equation
12(3) = k
x = 12 and y = 3
36 = k
Simplify.
Choose values for x and y, which have a product of 36.
The equation xy = 200 can be used to represent the
situation. Use various speeds to make a table.
x
y
10
20
20
10
30
6.7
40
5
50
4
60
3.3
x
y
–6
–6
–3
–12
–2
–18
2
18
3
12
6
6
Exercises
Graph each variation if y varies inversely as x.
1. y = 9 when x = –3
Chapter 11
2. y = 12 when x = 4
6
3. y = –25 when x = 5
Glencoe Algebra 1