4. Example 1
Identify direct and inverse variation
Tell whether the equation represents direct variation,
inverse variation, or neither.
a. xy = 4
y
b.
= x
2
c. y = 2x + 3
SOLUTION
a. xy = 4
y =
4
x
Write original equation.
Divide each side by x.
5. Example 1
Identify direct and inverse variation
Because xy = 4 can be written in the form y =
xy = 4
k
,
x
represents inverse variation. The constant
y
of variation is 4.
b.
= x
2
y = 2x
Write original equation.
Multiply each side by 2.
y
Because = x can be written in the form y = kx ,
2
y
= x represents direct variation.
2
6. Example 1
Identify direct and inverse variation
c. Because y = 2x + 3 cannot be written in the form
k
y = kx, y = 2x + 3
y =
x
or
does not represent
either direct variation or inverse variation.
7. Example 2
Graph y =
Graph an inverse variation equation
4
.
x
SOLUTION
STEP 1 Make a table by choosing several integer
values of x and finding the values of y. Make a
second table to see what happens to the
values of y for values of x close to 0 and far
from 0. Then plot the points.
8. Example 2
Graph an inverse variation equation
x
y
x
y
–4
–1
– 10
– 0.4
–2
–2
–5
– 0.8
–1
–4
– 0.5
–8
0
undefined
– 0.4
– 10
1
4
0.4
10
2
2
0.5
8
4
1
5
0.8
10
0.4
9. Example 2
Graph an inverse variation equation
STEP 2 Connect the points in Quadrant I by drawing a
smooth curve through them. Repeat for the
points in Quadrant III.
10. Example 3
Graph y =
Graph an inverse variation equation
–4
.
x
SOLUTION
–4
4
– 1 • . So,
x =
x
for every nonzero value of x, the
–4
=
value of y in y
is the opposite
x
4
= . You can
of the value of y in y x
–4
=
graph y
by reflecting the
x
4
graph of y =
(see Example 2) in
x
the x-axis.
Notice that y =
11. Example 4
Use an inverse variation equation
The variables x and y vary inversely, and y = 6 when
x = –3.
a. Write an inverse variation equation that relates x and y.
b. Find the value of y when x = 4.
SOLUTION
a. Because y varies inversely with x, the equation has the
k
form y = x .
Use the fact that x = –3 and y = 6 to find the value of k.
12. Example 4
y =
k
x
6=
k
–3
–18 = k
Use an inverse variation equation
Write inverse variation equation.
Substitute
–3 for x and 6 for y.
Multiply each side by –3.
–18
An equation that relates x and y is y =
.
x
–18 – 9
b. When x = 4, y =
=
2
4
13. Example 5
Write an inverse variation equation
Tell whether the table
represents inverse
variation. If so, write the
inverse variation equation.
SOLUTION
Find the products xy for all ordered pairs (x, y):
– 5( 2.4) = –12 –3(4) = –12 4(–3 ) = –12 8(–1.5) = –12
24(– 0.5) = –12
The products are equal to the same number, –12 . So, y
varies inversely with x.
14. Example 5
Write an inverse variation equation
ANSWER
– 12
.
The inverse variation equation is xy = – 12, or y =
x
15. Example 6
Solve a work problem
THEATER
A theater company plans to hire people to build a stage
set. The work time t (in hours per person) varies
inversely with the number p of people hired. The
company estimates that 10 people working for 70 hours
each can complete the job. Find the work time per
person if the company hires 14 people.
SOLUTION
STEP 1 Write the inverse variation equation that
relates p and t.
16. Example 6
t =
Solve a work problem
k
p
k
70 =
10
700 = k
Write inverse variation equation.
Substitute 10 for p and 70 for t.
Multiply each side by 10.
700
.
The inverse variation equation is t =
p
700
700
: t =
STEP 2 Find t when p = 14
=
= 50 .
p
14
17. Example 6
Solve a work problem
ANSWER
If 14 people are hired, the work time per person is
50 hours.
