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5. Rational Functions
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Algebra 2

5.02 Direct and inverse variation

Lesson
Practice

Direct variation

A direct variation represents a proportional relationship between two quantities. This means that one variable is always a constant multiple of the other.

For example, if you earn \$18 per hour, your earnings are directly proportional to the number of hours worked because \text{earnings}=18 \cdot \text{hours worked}. We would say hours worked and money earned are directly proportional.

The constant of proportionality \left(k\right), is the ratio of the dependent variable to the independent variable. So k=\dfrac{y}{x}.

We can also refer to this as the constant of variation. In the hourly earnings example, the constant would be 18.

A direct variation can be written in the form:

\displaystyle y=kx
\bm{k}
constant of proportionality \left(k \neq 0 \right)

A very common misconception is that two variables are directly proportional if one increases as the other increases. This is not the case. We can only say that two variables are directly proportional if the ratio between the variables stays constant. In other words, both variables increase or decrease at a constant rate.

If we graph a direct variation, we will see a linear graph (straight line) that passes through the origin , \left(0,\,0\right).

The image shows a linear graph.

\\\\ \, \\\\\\\\ The graph shows a direct variation. We can see this creates a linear graph, where A is directly proportional to B.

A linear graph with two points plotted and labeled as 2:2=1:1 and 1:1. Ask your teacher for more information.

\\\\ \, \\\\\\\\ The ratios explain why A and B are directly proportional.

  • First, we can see that this is a straight line that passes through the origin.
  • Next, looking at the point \left(1,1\right), we can see the ratio of A to B, is constant for both points.
  • If we continued this for the point \left(3, 3\right), \left(4, 4\right) or even \left(4.5, 4.5\right) we will see the ratio is always equivalent to 1:1.

The value of k can be any real number, including fractions and negative numbers.

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y = 2 \cdot x
-\frac{5}{3}
-\frac{4}{3}
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-\frac{1}{3}
\frac{1}{3}
\frac{2}{3}
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-\frac{5}{3}
-\frac{4}{3}
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\frac{5}{3}
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y=\dfrac{1}{3} \cdot x
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y = -\dfrac{5}{2} \cdot x

Examples

Example 1

Consider P=90t

a

Find the constant of proportionality.

Worked Solution
Create a strategy

Use the equations of the form y=kx, where k is the constant of proportionality.

Apply the idea

The constant of proportionality in P=90t is 90.

b

Find the value of P when t=2.

Worked Solution
Create a strategy

Substitute the given value into the equation.

Apply the idea
\displaystyle P\displaystyle =\displaystyle 90 \cdot 2Substitute t=2
\displaystyle =\displaystyle 180Evaluate

Example 2

If I pay \$6 for 12 eggs and \$10 for 20 eggs, are these rates directly proportional?

Worked Solution
Create a strategy

The constant of proportionality is represented by the cost per egg. To find the constant of proportionality, divide each amount paid by the number of eggs. The rates are directly proportional if they have the same cost per egg.

Apply the idea

Constant of proportionality for the \$6 carton of eggs:

\$6 \div 12 = \$0.50 \text{ per egg}

Constant of proportionality for the \$10 carton of eggs:

\$10 \div 20 = \$0.50 \text{ per egg}

Since both variables show a constant cost of \$ 0.50 per egg, the prices are directly proportional.

Example 3

Consider the values in each table. Which one of them could represent a directly proportional relationship between x and y?

A
The image shows a tabular data with values of x and y
B
The image shows a tabular data with values of x and y. Ask your teacher for more information.
C
The image shows a tabular data with values of x and y. Ask your teacher for more information.
D
The image shows a tabular data with values of x and y. Ask your teacher for more information.
Worked Solution
Create a strategy

Calculate each value of k using k=\dfrac{y}{x}. If the table is directly proportional, k will be constant for each coordinate pair in the table.

Apply the idea

Option A:

\displaystyle \frac{50}{-5}\displaystyle =\displaystyle -10So, k=-10
\displaystyle \frac{40}{-4}\displaystyle =\displaystyle -10So, k=-10
\displaystyle \frac{30}{-3}\displaystyle =\displaystyle -10So, k=-10
\displaystyle \frac{20}{-2}\displaystyle =\displaystyle -10So, k=-10

Therefore, option A is directly proportional since the value of k is constant.

Option B:

\displaystyle \frac{5}{1}\displaystyle =\displaystyle 5So, k=5
\displaystyle \frac{20}{2}\displaystyle =\displaystyle 10So, k=10

Therefore, option B is not directly proportional since the value of k is not constant.

Option C:

\displaystyle \frac{5}{1}\displaystyle =\displaystyle 5So, k=5
\displaystyle \frac{10}{2}\displaystyle =\displaystyle 5So, k=5
\displaystyle \frac{15}{3}\displaystyle =\displaystyle 5So, k=5
\displaystyle \frac{20}{4}\displaystyle =\displaystyle 5So, k=5

Therefore, option C is directly proportional since the value of k is constant.

Option D:

\displaystyle \frac{100}{1}\displaystyle =\displaystyle 100So, k=100
\displaystyle \frac{75}{5}\displaystyle =\displaystyle 15So, k=15

Therefore, option D is not directly proportional since the value of k is not constant.

Reflect and check

Notice that we did not calculate the value of \dfrac{y}{x} for every pair of values in all of the tables. As soon as we identify a k value that is different from the others we have enough information to determine that the relationship is not directly proportional.

Example 4

Ivan paints 10 plates every 6 hours.

a

Write an equation to represent this situation.

Worked Solution
Create a strategy

Remember that for every 6 hours that passes10 additional plates will be painted. To define the variables let y represent the total number of plates painted and x represent the total number of hours.

Apply the idea

First, we can find the constant of proportionality, k, by finding the number of plates painted every hour.

k = \frac{10 \text{ plates}}{6\text{ hours}} = \frac{5}{3} \text{ plates per hour}

Since proportional equations can be represented by the equation y = kx, we have:

y = \frac{5}{3}x

b

Use your equation to plot this situation on a coordinate plane.

Worked Solution
Create a strategy

Remember the constant of proportionality represents the ratio \dfrac{y}{x}. So, for every 3 units over, we go 5 units up.

Apply the idea
5
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\text{hours}
5
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\text{number of plates}
Idea summary

A direct variation can be written in the form:

\displaystyle y = kx
\bm{k}
is the constant of proportionality, or constant of variation (and k \neq 0)

The graph of all points describing a direct variation is a straight line passing through the origin.

Inverse variation

Now that we know about direct variation we will look at inverse variation. Inverse variation means that as one amount increases the other amount decreases.

For example, speed and travel time are inversely proportional because the faster you go, the shorter your travel time.

We express these kinds of inversely proportional relationships generally in the form:

\displaystyle y=\dfrac{k}{x}
\bm{k}
constant of proportionality \left(k \neq 0\right)

While the graph of a direct variation is linear, we can see the graph of an inverse variation is nonlinear.

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The graph of direct variation is linear.
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The graph of inverse variation is nonlinear.

Inverse variation is a subset of the rational function family with parent function y = \dfrac{1}{x}. Different constants of proportionality represent dilations. A negative constant of proportionality applys a reflection.

Examples

Example 5

Consider s=\dfrac{375}{t}

a

Find the constant of proportionality.

Worked Solution
Create a strategy

Use the equations of the form y=\dfrac{k}{x}, where k is the constant of proportionality.

Apply the idea

The constant of proportionality in s=\dfrac{375}{t} is 375.

b

Find the value of s when t=6. Give your answer as an exact value.

Worked Solution
Create a strategy

Substitute the given value into the equation.

Apply the idea
\displaystyle s\displaystyle =\displaystyle \frac{375}{6}Substitute t=6
\displaystyle =\displaystyle \frac{125}{2}Simplify
c

Find the value of s when t=12. Give your answer as an exact value.

Worked Solution
Create a strategy

Substitute the given value into the equation.

Apply the idea
\displaystyle s\displaystyle =\displaystyle \frac{375}{12}Substitute t=12
\displaystyle =\displaystyle \frac{125}{4}Simplify

Example 6

Consider the table of values.

x12345
y12060403024
a

Determine whether the table of values could represent an inverse variation between x and y.

Worked Solution
Create a strategy

We determine whether the two variable quantities can be represented by the equation for the inverse variation.

If the product of the two variables is always equal to the constant of variation, then the table of values represents an inverse variation.

Apply the idea

Note that the two quantities have inverse variation if they can be represented by the equation y=\dfrac{k}{x} where k is the constant of variation and the variable quantities are x and y.

Observe that y=\dfrac{k}{x} implies k=xy. So we multiply the values of x and y to find k.

If the product of the two variables is always equal to the constant of variation k, then the table of values represents an inverse variation.

Now,

\qquadIf x=1 and y=120, then xy=1 \cdot 120=120.

\qquadIf x=2 and y=60, then xy=2 \cdot 60=120.

\qquadIf x=3 and y=40, then xy=3 \cdot 40=120.

\qquadIf x=4 and y=30, then xy=4 \cdot 30=120.

\qquadIf x=5 and y=24, then xy=5 \cdot 24=120.

This means that k=120 since xy=120 for all \left(x,y\right).

Therefore, the table of values represents an inverse variation between x and y.

b

Write a function relating y and x, given the table of values.

Worked Solution
Create a strategy

We simply substitute the obtained value of the constant variation in part (a) to the equation for the inverse of variation.

Apply the idea

From part (a), we obtained the constant variation k=120.

Substituting k=120 to the equation y=\dfrac{k}{x}, we get y=\dfrac{120}{x}

Therefore, the table of values is represented by the equation y=\dfrac{120}{x}.

c

Describe the behavior of the function as x \to 0 from the right, and describe the behavior of the function as x \to \infty.

Worked Solution
Create a strategy

Use the table of values and graph the function using technology to determine the behavior of the function.

A screenshot of the GeoGebra graphing calculator showing the graph of y equals 120 over x. Speak to your teacher for more details.
Apply the idea

As x \to 0 from the right, the function goes to larger and larger values.

The rational function is undefined at 0 so 0 is excluded from the domain.

As x \to \infty, the function gets closer to 0.

Example 7

Is the variation relating the distance between two locations on a map and the actual distance between the two locations an example of a direct variation or an inverse variation?

Worked Solution
Create a strategy

The scale of a map is defined as the ratio of a single unit of distance on the map to the corresponding distance in real life. Maps are created with a constant multiplier between real life distances and map distances.

Think about whether the distance between two locations on the map increases or decreases when the actual distance between the two locations increases.

Apply the idea

Since the ratio between the map distance and real life distance is constant, the relationship between map and real life distances represents direct variation.

Reflect and check

There are also real life examples of inverse variation. The distance between two locations and the time taken to travel between those locations at a constant speed represents an inverse variation. The constant of proportionality, or ratio of distance to time, represents the speed of travel.

Idea summary

We express inversely proportional relationships in the form:

\displaystyle y=\dfrac{k}{x}
\bm{k}
is the constant of proportionality \left(k \neq 0\right)

Outcomes

A2.F.1

The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.F.1d

Determine when two variables are directly proportional, inversely proportional, or neither, given a table of values. Write an equation and create a graph to represent a direct or inverse variation, including situations in context.

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