Inverse Variation

A relationship that can be written in the form y = , where k is a nonzero constant and x ≠ 0, is an inverse variation. The constant k is the constant of variation. Inverse variation implies that one quantity will increase while the other quantity will decrease (the inverse, or opposite, of increase).

Multiplying both sides of y = by x gives xy = k. So, for any inverse variation, the product of x and y is a nonzero constant.

There are two methods to determine whether a

relationship between data is an inverse variation.

You can write a function rule in y = form, or you

can check whether xy is a constant for each

ordered pair.

Example 1: Identifying an Inverse Variation

Tell whether each relationship is an inverse variation. Explain.

Method 1 Write a function rule.

Can write in y = form.

The relationship is an inverse variation.

Method 2 Find xy for each ordered pair.

1(30) = 30, 2(15) = 30, 3(10) = 30

The product xy is constant, so the relationship is an inverse variation.

Example 2: Identifying an Inverse Variation

Tell whether each relationship is an inverse variation. Explain.

Method 1 Write a function rule.

Cannot write in y = form.

The relationship is not an inverse variation.

y = 5x

Method 2 Find xy for each ordered pair.

1(5) = 5, 2(10) = 20, 4(20) = 80

The product xy is not constant, so the relationship is not an inverse variation.

Example 3: Identifying an Inverse Variation

Tell whether each relationship is an inverse variation. Explain.

2xy = 28

Find xy. Since xy is multiplied by 2, divide both sides by 2 to undo the multiplication.

xy = 14 Simplify.

xy equals the constant 14, so the relationship is an inverse variation.

Tell whether each relationship is an inverse variation. Explain.

Method 1 Write a function rule.

Cannot write in y = form.

The relationship is not an inverse variation.

y = –2x

Method 2 Find xy for each ordered pair. –12 (24) = –228 , 1(–2) = –2, 8(–16) = –128

The product xy is not constant, so the relationship is not an inverse variation.

Example 4

Tell whether each relationship is an inverse variation. Explain.

Example 5

Method 1 Write a function rule.

Can write in y = form.

The relationship is an inverse variation.

Method 2 Find xy for each ordered pair.

3(3) = 9, 9(1) = 9, 18(0.5) = 9

The product xy is constant, so the relationship is an inverse variation.

2x + y = 10

Tell whether each relationship is an inverse variation. Explain.

Example 6

Cannot write in y = form.

The relationship is not an inverse variation.

An inverse variation can also be identified by its graph. Some inverse variation graphs are shown. Notice that each graph has two parts that are not connected.

Also notice that none of the graphs contain (0, 0). This is because (0, 0) can never be a solution of an inverse variation equation.

Example 7: Graphing an Inverse Variation

Write and graph the inverse variation in which y = 0.5 when x = –12.

Step 1 Find k.

k = xy

= –12(0.5)

Write the rule for constant of variation.

Substitute –12 for x and 0.5 for y.

= –6 Step 2 Use the value of k to write an inverse variation equation.

Write the rule for inverse variation.

Substitute –6 for k.

Example 7 Continued

Write and graph the inverse variation in which y = 0.5 when x = –12.

Step 3 Use the equation to make a table of values.

y

–2–4x –1 0 1 2 4

1.5 3 6 undef. –6 –3 –1.5

Example 7 Continued

Write and graph the inverse variation in which y = 0.5 when x = –12.

Step 4 Plot the points and connect them with smooth curves.

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Example 8

Write and graph the inverse variation in which y = when x = 10.

Step 1 Find k.

k = xy Write the rule for constant of variation.

= 5 Substitute 10 for x and for y.= 10

Step 2 Use the value of k to write an inverse variation equation.

Write the rule for inverse variation.

Substitute 5 for k.

Write and graph the inverse variation in which y = when x = 10.

Step 3 Use the equation to make a table of values.

Example 8 Continued

x –4 –2 –1 0 1 2 4

y –1.25 –2.5 –5 undef. 5 2.5 1.25

Example 8 Continued

Step 4 Plot the points and connect them with smooth curves.

Write and graph the inverse variation in which y = when x = 10.

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Example 9: Transportation Application

The inverse variation xy = 350 relates the constant speed x in mi/h to the time y in hours that it takes to travel 350 miles. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate how long it will take to travel 350 miles driving 55 mi/h.

Step 1 Solve the function for y so you can graph it.

xy = 350

Divide both sides by x.

Example 9 Continued

Step 2 Decide on a reasonable domain and range.

x > 0

y > 0

Length is never negative and x ≠ 0

Because x and xy are both positive, y is also positive.

Step 3 Use values of the domain to generate reasonable ordered pairs.

4.385.838.7517.5y

80604020x

Example 9 Continued

Step 4 Plot the points. Connect them with a smooth curve.

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Step 5 Find the y-value where x = 55. When the speed is 55 mi/h, the travel time is about 6 hours.

Example 10

The inverse variation xy = 100 represents the relationship between the pressure x in atmospheres (atm) and the volume y in mm³ of a certain gas. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the volume of the gas when the pressure is 40 atmospheric units.

Step 1 Solve the function for y so you can graph it.

xy = 100

Divide both sides by x.

Step 2 Decide on a reasonable domain and range.

x > 0

y > 0

Pressure is never negative and x ≠ 0

Because x and xy are both positive, y is also positive.

Step 3 Use values of the domain to generate reasonable pairs.

2.53.34510y

40302010x

Example 10 Continued

Step 4 Plot the points. Connect them with a smooth curve.

Example 10 Continued

Step 5 Find the y-value where x = 40. When the pressure is 40 atm, the volume of gas is about 2.5 mm3.

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Example 11: Using the Product Rule

Let and Let y vary inverselyas x. Find

Write the Product Rule for Inverse Variation.

Substitute 5 for 3 for and 10 for .

Simplify.

Solve for by dividing both sides by 5.

Simplify.

Example 12

Write the Product Rule for Inverse Variation.

Simplify.

Simplify.

Substitute 2 for –4 for and –6 for

Let and Let y vary inversely as x. Find

Solve for by dividing both sides by –4.

Example 13: Physics Application

Boyle’s law states that the pressure of a quantity of gas x varies inversely as the volume of the gas y. The volume of gas inside a container is 400 in3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in3?

Use the Product Rule for Inverse Variation.

Substitute 400 for 125 for and 25 for

Simplify.

Solve for by dividing both sides by 125.

(400)(25) = (125)y2

Example 13 Continued

Boyle’s law states that the pressure of a quantity of gas x varies inversely as the volume of the gas y. The volume of gas inside a container is 400 in3 and the pressure is 25 psi. What is the pressure when the volume is compressed to 125 in3?

When the gas is compressed to 125 in3, the pressure increases to 80 psi.

Example 14

On a balanced lever, weight varies inversely as the distance from the fulcrum to the weight. The diagram shows a balanced lever. How much does the child weigh?

Example 14 Continued

Use the Product Rule for Inverse Variation.

Substitute 3.2 for , 60 for and 4.3 for

Simplify.

Solve for by dividing both sides by 3.2.

Simplify.

The child weighs 80.625 lb.

Lesson Quiz: Part I

1. Write and graph the inverse variation in which y = 0.25 when x = 12.

Lesson Quiz: Part II

2. The inverse variation xy = 210 relates the length y in cm to the width x in cm of a rectangle with an area of 210 cm2. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the length when the width is 14 cm.

Lesson Quiz: Part III

3. Let x1= 12, y1 = –4, and y2 = 6, and let y vary inversely as x. Find x2. –8