13-1 Inverse Variation
Name
Date
Graph the inverse variation xy 20. Describe the asymptotes.
Remember: An inverse variation can be written as an equation
in the form xy k, where k 0.
Make a function table using both positive and negative values of x.
x
y
1
2
4
5
10
20
1
2
4
5
10
20
20
10
5
4
2
1
20
10
5
4
2
1
y
Graph the ordered pairs in the table on a coordinate plane.
Draw smooth curves through them.
20
15
10
5
The graph of xy 20 is shown at the right. Notice that the hyperbola
gets very close to the lines x 0 and y 0 but do not touch them.
Therefore, the asymptotes of the graph are the x- and y-axes.
20
5 0
5
x
5 10 15 20
15
20
Remember: Asymptotes are lines that a graph
approaches but never intersects.
Each table or graph represents a variation. Is it a direct or an inverse
variation? Explain.
Copyright © by William H. Sadlier, Inc. All rights reserved.
1.
x
y
2
4
6
2
4
6
6
3
2
6
3
2
2.
inverse variation; y varies inversely
as x: as the value of x increases, the
value of y decreases.
x
y
0
1
2
1
2
0
1.5
3
1.5
3
3.
y
x
direct variation; for each
ordered pair, y varies
directly with x.
inverse variation; the graph is a hyperbola
and the graph does not intersect the axes.
Make a function table for each equation. Tell whether the equation represents a direct or an inverse
variation. Graph the relation on a separate sheet of paper. Check students’ graphs.
4. xy 8
x
y
1
8
2
4
4
2
1
8
2
4
4
2
inverse variation
5. xy 9
x
y
1
9
3
3
9
1
9
1
3
3
1
9
inverse variation
6. xy 12
x
y
1
0.5
2
0.25
4
0.125
1
0.5
2
0.25
4 0.125
inverse variation
Lesson 13-1, pages 330–331.
7. y 1.6x
x
y
0
0
1
1.6
2
3.2
1.6
1
3.2
2
direct variation
Chapter 13 331
For More Practice Go To:
Solve each problem using the equation xy k.
8. y varies inversely as x, and y 12
when x 5. Find x when y 6.
xy k
5(12) k
60 k
xy 60
6x 60
x 10
10. y varies inversely as x, and y 12.3
when x 2. Find y when x 3.
(2)12.3 k
24.6 k
3y 24.6
y 8.2
9. y varies inversely as x, and y 7
when x 6. Find x when y 21.
6(7) k
42 k
21x 42
x2
11. y varies inversely as x, and y 15.25
when x 8. Find y when x 5.
8(15.25) k
122 k
5y 122
y 24.4
Solve each problem using a proportion.
12. y varies inversely as x, and y 9
when x 2. Find x when y 3.
2
3
x 9
3x 18
x 6
14. y varies inversely as x, and y 9.2
when x 4. Find y when x 0.2.
4(9.2) 0.2y
36.8 0.2x
y 184
13. y varies inversely as x, and y 15
when x 4. Find x when y 5.
4(15) 5x
60 5x
x –12
15. y varies inversely as x, and y 16.4
when x 7. Find y when x 0.4.
7(16.4) 0.4y
114.8 0.4y
y 287
16. Travel Mr. Wu commutes 40 mi to work.
How many minutes does it take Mr. Wu to
get to work if his average speed is 50 mph?
What happens to the time if Mr. Wu’s
speed decreases?
rt 40; 50t 40; t 0.8; 0.8 • 60 48.
It takes Mr. Wu 48 min; As his speed
decreases, the time to work increases.
17. Construction A factory building is 80 m
by 10 m. How wide is another building with
an equal floor area and length that is twice
its width?
A 800; Let x width; then 2x length
x(2x) 800; 2x2 800; x2 400; Width can
not be negative, so x 20;
The width of the other building is 20 m.
18. If y varies inversely with the square of x, the inverse variation can
be written as x2y k. Make a function table and graph the equation
x2y 4 on grid paper. What happens to y as x increases? decreases? Check students’ graphs.
Check that students graph hyperbolas in quadrants I and II with these asymptotes: the x-axis and the
positive y-axis. As x approaches 0, y rapidly approaches infinity. As x approaches `, y approaches 0.
332 Chapter 13
Copyright © by William H. Sadlier, Inc. All rights reserved.
Solve. Show your work.