How to Find the Rate of Change Given a Graph of a Linear Function

Finding the Rate of Change From a Graph

Step 1: Pick two points on a graph, {eq}(x_1, y_1) {/eq} and {eq}(x_2, y_2) {/eq}. Any two will work.

Step 2: Find the change in y, {eq}\Delta y {/eq}. We want to figure out how many y units it takes to go from one labeled y to the other.

Step 3: Find the change in x, {eq}\Delta x {/eq}. We want to figure out how many x units it takes to go from one labeled x to the other.

Step 4: Plug into {eq}\frac{\Delta y}{\Delta x} {/eq} and simplify as much as possible. The resulting rate of change can be positive, negative, zero, or undefined if the denominator is zero.

Finding the Rate of Change From a Graph Equations and Vocabulary

The average rate of change of a function is given by this expression:

rate of change = {eq}\frac{vertical change}{horizontal change} = \frac{rise}{run} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = m {/eq} = slope

Note: finding the rate of change of a graph and finding the slope of a graph mean the same thing.

There is a special circumstance that happens when working with straight lines (linear functions). The rate of change (the slope) is always constant. No matter where you check the slope on a straight line, you will get the same answer.

The diagram below describes the rate of change of a line. The rise of the line is represented in purple, and the run is represented in green.

Finding rate of change

You can calculate the slope from any two coordinates on the line by measuring the change on the y-axis (rise) over the change on the x-axis (run).

So, let's try using the formula mentioned above to find the rate of change of a linear function in four example problems that result in positive, negative, zero, and undefined rates of change.

Finding the Rate of Change From a Graph: Positive Rate of Change Example

Use the graph below to determine the rate of change of the linear function.

Positive rate of change

Step 1: Let's pick two points on the graph, (0, -4) and (2, 2)

Step 2: Now, find the change in y. Looking at our y-values, -4 and 2, if we were to count the units from -4 to 2, we would have 6 y-units. We notice that we need to go up, so our y is positive. Therefore, {eq}\Delta y = 6 {/eq}

Step 3: Now, find the change in x. Looking at our x-values, 0 and 2, if we were to count the units from 0 to 2, we would have 2 x-units. Hence, {eq}\Delta x = 2 {/eq}

Step 4: Plugging into the rate of change formula:

{eq}\frac{\Delta y}{\Delta x} = \frac{6}{2} = 3 {/eq}

Therefore, the rate of change of this linear function is 3.

Finding the Rate of Change From a Graph: Negative Rate of Change Example

Use the graph below to determine the rate of change of the linear function.

Negative rate of change

Step 1: Let's pick two points on the graph, (-2, 3) and (4, 1)

Step 2: Our y-values are 3 and 1. If we were to count the units from 3 to 1, we would have 2 y-units, but we notice that we need to go down, so our y is negative. Therefore, {eq}\Delta y = -2 {/eq}

Step 3: Our x-values are -2 and 4. If we were to count the units from -2 to 4, we would have 6 x-units. Hence, {eq}\Delta x = 6 {/eq}

Step 4: Plugging into the rate of change formula:

{eq}\frac{\Delta y}{\Delta x} = \frac{-2}{6} = \frac{-1}{3} {/eq}

Therefore, the rate of change of this linear function is -1/3.

Finding the Rate of Change From a Graph: Zero Rate of Change Example

Use the graph below to determine the rate of change of the linear function.

Zero rate of change

Step 1: Let's pick two points on the graph, (-2, 2) and (2, 2)

Step 2: Our y-values are 2 and 2. Notice that they are the same, so we would move 0 y-units. Therefore, {eq}\Delta y = 0 {/eq}

Step 3: Our x-values are -2 and 2. If we were to count the units from -2 to 2, we would have 4 x-units. Hence, {eq}\Delta x = 4 {/eq}

Step 4: Plugging into the rate of change formula

{eq}\frac{\Delta y}{\Delta x} = \frac{0}{4} = 0 {/eq}

Therefore, the rate of change of this linear function is 0.

Finding the Rate of Change From a Graph: Undefined Rate of Change Example

Use the graph below to determine the rate of change of the linear function.

Undefined rate of change

Step 1: Let's pick two points on the graph, (1, 1) and (1, 4)

Step 2: Our y-values are 1 and 4. If we were to count the units from 1 to 4, we would have 3 y-units. Therefore, {eq}\Delta y = 6 {/eq}

Step 3: Our x-values are 1 and 1. Notice that they are the same, so we would move 0 x-units. Hence, {eq}\Delta x = o {/eq}

Step 4: Plugging into the rate of change formula:

{eq}\frac{\Delta y}{\Delta x} = \frac{3}{0} = undefined {/eq}

Because we cannot divide by zero, the rate of change of this linear function is undefined.