11.1

VocabularyDirect Variation: (and )

Inverse Variation: (and )

Constant of Variation – The nonzero number

k.

kxy 0k

x

ky 0k

Graphs of Direct Variation and Inverse Variation Equations

Example 1 Identify direct and inverse variation

Tell whether the equation represents direct variation, inverse variation, or neither.

a.

xy = 4 xb.

=2

yc.

y = 2x + 3

SOLUTION

a.

xy = 4 Write original equation.

=x

4y Divide each side by x.

Example 1 Identify direct and inverse variation

2b.

x=y

Write original equation.

= 2xy Multiply each side by 2.

Because can be written in the form ,

represents inverse variation. The constant

of variation is 4.

xy = 4

xy = 4

y

= xk

Because can be written in the form ,

represents direct variation.

x=2

y= kxy

x=2

y

Example 1 Identify direct and inverse variation

c. y = 2x + 3

y

= xk y

= kx, y = 2x + 3

Because cannot be written in the form

or does not represent

either direct variation or inverse variation.

Example 2 Graph an inverse variation equation

Graph y .x4

=

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.

Example 2 Graph an inverse variation equation

x y

0 undefined

1

2

4

4–

2–

1–

1–

2–

4–

4

2

1

x y

0.4

0.5

5

10–

5–

0.5–

0.4–

0.8–

8–

10

8

0.8

0.4–

10

10–

0.4

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.

Example 3 Graph an inverse variation equation

Graph y .=x4–

SOLUTION

Notice that y

for every nonzero value of x, the

value of y in y is the opposite

of the value of y in y

graph y by reflecting the

graph of y (see Example 2) in

the x-axis.

=x4–

x4

=

=x4–

= x4–

= x4

•1– . So,

x4

= . You can

Example 4 Use an inverse variation equation

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

form y .xk

=

Use the fact that x = =and y 6 to find the value of k.–3

The variables x and y vary inversely, ==x

and y 6 when –3.

Example 4 Use an inverse variation equation

Write inverse variation equation.xk

=y

63k

=–

Substitute 3 for x and 6 for y.––

Multiply each side by 3.–k=–18

b. When = 4,x y4

18=

–

29

=–

An equation that relates x and y is yx

18=

–.

Example 5 Write an inverse variation equation

Tell whether the table represents inverse variation. If so, write the inverse variation equation.

The products are equal to the same number, . So, y varies inversely with x.

12–

5– )2.4( = 12– 3(4)– = 12– 4( )3– = 12– 8 )1.5(– = 12–

24 )0.5–( = 12–

Find the products xy for all ordered pairs (x, y):

SOLUTION

Example 5 Write an inverse variation equation

ANSWER

The inverse variation equation is xy = 12,– or y =12–

x.

Example 6 Solve a work problem

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.

THEATER

Write the inverse variation equation that relates p and t.

STEP 1

SOLUTION

Example 6 Solve a work problem

=tpk

The inverse variation equation is =tp

700.

Write inverse variation equation.

=7010k

Substitute 10 for p and 70 for t.

=700 k Multiply each side by 10.

STEP 2 Find t when =p 14: =t p700

=14

700= 50.

Example 6 Solve a work problem

If 14 people are hired, the work time per person is 50 hours.

ANSWER

11.1 Warm-Up