Algebra 1 Glencoe McGraw-Hill JoAnn Evans

Math 8H

11-1Inverse Variation

The direct variation equation

states that as y increases, x increases.

We say that y varies directly as x.

kxy

You learned in chapter 4 that a direct variation equation represents a constant rate of change.

You learned that every direct variation equation will graph as a straight line that passes through the

origin. (0, 0)

Direct variation equations are linear equations.

x

Today you’ll learn about inverse variation equations. The relationship

between x and y is different in an inverse variation.

If the two variables x and y always have the same product, this will be true:

as y increases, x decreases.

xy k

Two quantities that vary inversely always have the same product.

The product, k, is called the constant of variation. The constant of variation cannot be equal to zero.

Let’s say that k, the constant of variation, is 36.

The two quantities x and y would always have a product of

36.

x y k

1 36 36

2 18 36

3 12 36

4 9 36

6 6 36

8 4.5 36

9 4 36

10 3.6 36

12 3 36

16 2.25 36

18 2 36

36 1 36

As you look down the column of products, as x

increases, y decreases in order for the product to

remain the same.xy 36

is the inverse variation equation to describe this

relationship.

xy k

2 4 6 -2 -4 -6

Graph the inverse variation equation xy = 36.

-4-3

-9-12

-4

-9-12

-6-6

-3

x y

-2 -18

2 183 124 96 69 4

12 3

-18 -2

• ••••••

•••••••

The graph of an inverse variation isn’t a straight line like the graph of a direct variation. Instead,

it’s a sweeping curve that will approach the x- and y- axes, but never touch them.

Grandma’s house is 100 miles away. The speed you use to drive there will vary inversely with the time it takes to get there. In other words, as the speed increases, the

time decreases.The equation rt = 100

can be used to represent you driving there. At 50 miles per hour, the trip will take

2 hours.

rt = k

50 ∙ 2 = 100At 40 mph, how long will the trip

take?

40 ∙ t = 100At 80 mph, how long will the trip

take?

80 ∙ t = 100

At 20 mph, how long will the trip

take?

20 ∙ t = 100At 10 mph, how long will the trip

take?

10 ∙ t = 100

speed and time vary inversely,

but their product (the distance) is

always 100

2.5 hrs

1.25 hrs

5 hrs

10 hours

5 10

5 10 ••••

•

50

105

10

2

520

20

r t 2 50

80 1.25

•Why is this graph only a single curve, unlike the previous

graph?Your rate of speed can’t be negative, so

it’s only logical to use positive values

for r.

r

t

Graph the inverse variation if y varies inversely as x. y = 4 when x = 5.

Inverse variation equation: xy = k (5)(4) = k

20 = k

xy = 20

2

-4-5

-5

10-10

-2

-4

x y -10 -2

4 55 4

10 2

The Product Rule for inverse variations:

If (x1, y1) and (x2, y2) are solutions to an

inverse variation, then x1y1 = k and x2y2

= k. If the products of x1y1 and x2y2 are both equal to the same constant, then

they would also be equal to each other.

1 1 2 2x y x y

If y = 12.5 when x = 4, find x when y = 10.

Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then

solve.

1 1 2 2x y x y

4 12.5 x 10

5 x

50 10x

Product Rule for inverse variations

Substitute known values.

Divide each side by 10.

x is 5 when y is 10

y = -7.5 when x = 25 What is y when x = 5?

If y varies inversely as x and y = -7.5 when x = 25, find y when x = 5.

1 1 2 2x y x y

25 7.5 5 y

37.5 y

187.5 5y

Product Rule for inverse variations

Substitute known values.

Divide each side by 10.

y is -37.5 when x is 5

y = 12 when x = 4 What is x when y = 18?

If y varies inversely as x and y = 12 when x = 4, find x when y = 18.

1 1 2 2x y x y

4 12 x 18

8x

3

48 18x

Product Rule for inverse variations

Substitute known values.

Divide each side by 18.

8x is when y is 18

3

The Malahowski family can drive from Lisle, Illinois, to Oshkosh, Wisconsin in 4 hours if they

drive an average of 45 miles per hour. How long would it take them if they increased their average

speed to 50 mph?

1 1 2 2x y x y 4 45 x 50

180 50x

3.6 x

Trip time would decrease to 3.6 hours if the average

speed was increased to 50 mph.

The manager of Stringer’s Hardware scheduled 8 employees to start and finish taking inventory in

an 8 hour period. Assume, like the manager, that all employees work at the same rate.

Unfortunately, 3 employees called in sick that day. How long will it take 5 employees to complete the

inventory?1 1 2 2x y x y

8 8 5 y

64 5y

12.8 y

The 5 employees will need 12.8 hours for the inventory.

The sound produced by a string inside a piano depends on the length of the string. The

frequency of a vibrating string varies inversely as its length.

frequency length = k

If a 2 ft string vibrates 300 cycles per second, what would be the frequency of a string that is 5 ft long?

1 1 2 2x y x y 300 2 x 5

600 5y

120 y

A 5 ft string will vibrate 120 cycles per second.