Learning Target:  I can define direct and inverse variation and express them through equations and graphs.

Direct and Inverse Variation

Two variables vary directly if there is a nonzero number for which the following is correct:

y = kx
 

In this equation, x and y are the two variables and k is the nonzero number, known as the constant of variation.

Graphs of Direct Variation

The graph of y = kx is a line through the origin, and the slope of said graph is k. From the graph we can see whether the value of k is positive or negative.

Example

Find the constant of variation and the slope of the model y = $\frac{1}{2}$x.
 

For this equation, the constant of variation is k = ¹/₂. To find the slope, we use the slope formula:

We confirm with the answer that k is the slope of the graph.

Writing a Direct Variation Equation

If the variables x and y vary directly when x = 3 and y = 15, then:

a. Write an equation that relates x and y.

b. Find the value of y when x = 6.

Solution

a. Because x and y vary directly, the equation is in the form y = kx. We can then solve for k by using the given values for x and y.

y = kx

15 = k(3)

5 = k
 

The equation that relates x and y is y = 5x. 

b. To find the value of y, we substitute for x.

y = 5x

y = 5(6)

y = 30
 

Then when x = 6, y = 30.

Using a Ratio to Model Direct Variation

The direct variation model can be rewritten in ratio form:

The ratio form lets you know that when x and y have direct variation, the ratio of x to y is alike for all values of x and y. Occasionally, real-life figures can be approximated by a direct variation model, although the data might not fit the model exactly.

Example

The lengths (in centimeters) of 8 Maltese dogs at age 1 and age 10 are shown in the table below.

a. Write a model that relates the length of the dogs at 1 year old to the length at 10 years old.

b. Estimate the body length of a Maltese whose length at 1 year was 26.40 cm.

Solution

a. We need to start by finding the ratio of 1 year to 10 years for each Maltese. Using the ratio form we obtain

b. Use the direct variation model to estimate the length of the Maltese.

Then we can estimate that the Maltese would be 29.33 cm at age 10.

Inverse Variation

For an inverse variation, k is still the constant of variation, but the model differs from direct variation. The variables x and y vary inversely for a constant k.

Example

When x is 7 and y is 2, find an equation that inversely relates y and x.

Solution

The inverse variation that relates x and y is

Graphically it looks like this:

Example

Compare the direct variation model and the inverse variation model for when x = 2 and y = 3.  Do this both numerically and graphically.

a. Numerically:

• Direct Variation:  Because k is positive, y increases as x increases. So as x increases by 1, y increases by 1.5.
• Inverse Variation:  Because k is positive, y decreases as x increases.

We can create a table that will show the comparison:

b. Using the values on the table we can plot the points and then connect them to find the solution by graphing.

• Direct Variation:  The graph for this model has the line passing through the origin with a positive slope.
• Inverse Variation:  The graph for this model is a curve that approaches the x-axis as x increases and approaches the y-axis as x gets closer to zero.

Let's Practice Together

1.  Determine whether the equation y = −4x represents direct or inverse variation, and identify the constant of variation k.

2.  Identify the type of variation shown in the table below.  Find the variation equation.

3.  A car travels 180 miles in 3 hours at a constant speed.  Find a direct variation equation relating distance d and time t, and compute the distance traveled in 5 hours.

4.  The variables x and y vary inversely. If x = 4 when y = 9, find the inverse variation equation.  Then find the value of y when x = 12.

Your Turn

5.  Write the ratio form of the variation equation y = 9x.

6.  The variables x and y vary directly. If x = 5 and y = 35, find the direct variation equation, and compute the value of y when x = 9.

7.  The number of calories burned varies directly with the number of minutes exercised. If 240 calories are burned in 30 minutes, find a direct variation equation.  Then find the number of calories burned in 45 minutes.

8.  A project can be completed in fewer days if more workers are added.  Is this relationship an example of direct or inverse variation?

9.  Complete the table for the inverse variation equation:

$y=\frac{24}{x}$

10.  A rectangle has a fixed area of 48 square inches.  Find an inverse variation equation relating length (l) and width (w).  Then find the width when the length is 12 inches.

Answers

1.  direct, k = −4

2.  direct, y = 5x

3.  d = 60t, 300 miles

4.  y = ³⁶/_x  OR  xy = 36,  y = 3

5.  ^y/_x = 9

6.  y = 7x, y = 63

7.  c = 8m, c = 360

8.  inverse

9.  y = 24, 12, 8, 4, 3

10.  w = ⁴⁸/_l  OR  l = ⁴⁸/_w  OR  lw = 48,  w = 4