The average rate of change of a function over an interval is the change in value of the dependent variable per unit change in the independent variable.

The average rate of change of a function can be calculated by dividing the change in function values between the start and the end of the interval by the length of the interval.

For the function shown, the function values increase from 3 to 7 over the interval 1 \leq x \leq 5.

This means that over the interval 1 \leq x \leq 5, the function has an average rate of change of \dfrac{7 - 3}{5 - 1} = 1

We can see this graphically as well: the dashed lines show an increase of 1 unit in the y-values per unit increase in the x-values over this interval.

Worked examples

Example 1

A flock of birds migrate to a new island. The population of birds on that island over the next six years is shown on the graph.

Determine the average rate of change of the bird population over the six year period.

Approach

To find the average rate of change, we want to divide the difference between the initial and final populations by the length of the time period.

Solution

Looking at the graph we can see that the initial population is 15 birds, and after 6 years the population is 60 birds.

So we can calculate the average rate of change as \frac{60 - 15}{6} = 7.5

Therefore, the average rate of change is 7.5 birds per year.

Example 2

Consider the function f\left(x\right) = 2x^2 - 1.

Calculate the average rate of change over the interval -4 \leq x \leq 0.

Approach

The average rate of change can be thought of as:\dfrac{\text{Change in }f\left(x\right)}{\text{Change in }x}

To determine the change in f\left(x\right), we want to identify the values of f\left(x\right) when x=-4 and when x=0.

Solution

We know thatf\left(x\right) = 2x^2 - 1So we have that

\displaystyle f\left(-4\right)	\displaystyle =	\displaystyle 2\left(-4\right)^2 - 1
	\displaystyle =	\displaystyle 2 \cdot 16 - 1
	\displaystyle =	\displaystyle 31

and

\displaystyle f\left(0\right)	\displaystyle =	\displaystyle 2\left(0\right)^2 - 1
	\displaystyle =	\displaystyle -1

Going back to the idea that the average rate of change can be thought of as \dfrac{\text{Change in }f\left(x\right)}{\text{Change in }x}, we have:

\displaystyle \text{Average rate of change}	\displaystyle =	\displaystyle \frac{\text{Change in }f\left(x\right)}{\text{Change in }x}
	\displaystyle =	\displaystyle \dfrac{-1 - 31}{0 -(-4)}
	\displaystyle =	\displaystyle \dfrac{-32}{4}
	\displaystyle =	\displaystyle -8

Outcomes

MA.912.F.1.2

Given a function represented in function notation, evaluate the function for an input in its domain. For a real-world context, interpret the output.

MA.912.F.1.3

Calculate and interpret the average rate of change of a real-world situation represented graphically, algebraically or in a table over a specified interval.

4.03 Average rate of change

Concept summary

Worked examples

Example 1

Approach

Solution

Example 2

Approach

Solution

Outcomes

MA.912.F.1.2

MA.912.F.1.3

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