D1.4 Calculate and interpret average rate of change

Ontario Grade 12 Mathematics

Characteristics of Functions

Calculate and interpret change over an interval

A quantity can change by different amounts at different input values. Average rate of change summarizes the overall change between two chosen inputs. It does not describe every change inside the interval. You can calculate it from function values and see it as the slope of a line through two points on a graph.

What you will learn

Calculate the average rate of change of a function over a specified interval.
Interpret the result as average output change for each one-unit input change.
Connect the numerical rate to the slope of a line through two points on a graph.
Use the sign and units of a rate to describe change.

Prerequisite bridge: change and slope

A function pairs each allowed input with an output. For example, it could pair time with distance travelled. The input is often written as

x

, and the corresponding output as

f(x)

. The notation

f(3)

means the output when the input is

3

.

To find a change, subtract the starting value from the ending value. If an amount goes from

5

to

13

, its change is

13-5=8

. The order matters. Ending value minus starting value tells you the direction and size of the change.

Slope describes how much a line rises or falls for each unit it moves horizontally. For two points, compare the change in vertical position with the change in horizontal position. Average rate of change uses this same calculation for two points on a function's graph.

In a graph, the vertical change is the difference in the points' outputs. The horizontal change is the difference in their inputs. Divide the vertical change by the horizontal change to find the slope.

Output change is ending output minus starting output.
Input change is ending input minus starting input.
Keep the same endpoint order for both changes.

Meaning and representations

The average rate of change over an interval compares the function's outputs at the interval's endpoints. It tells how much the output changes for each one-unit change in input, on average, across that interval. The phrase “on average” matters because the function may not change by the same amount at every input.

Numerically, find the output at each endpoint. Subtract the starting output from the ending output, then divide by the input change. A positive result means the ending output is greater. A negative result means it is smaller. A zero result means the endpoint outputs are equal.

Graphically, locate the two endpoint points on the function's graph. The straight line through these points is called a secant line. Its slope equals the average rate of change over the interval. The secant line summarizes the endpoint change; it does not need to follow the curve between those points.

Suppose a distance function gives

12

metres at

2

seconds and

30

metres at

5

seconds. The distance change is

18

metres and the time change is

3

seconds. The average rate is

6

metres per second. This describes the interval from

2

to

5

seconds, not necessarily the speed at each point in time.

Units help explain the answer. If output is measured in metres and input in seconds, the rate is measured in metres per second. In other settings, describe output units per input unit when those units are known.

\frac{f(b)-f(a)}{b-a}

The chosen interval determines which two endpoint values are compared.
The average rate is the slope of the secant line through the endpoint points.
Include units when they are supplied, and interpret the result as an average.

Calculate with matching endpoint order

For an interval from

a

to

b

, first find the endpoint outputs

f(a)

and

f(b)

. Next calculate the output change and the input change. Divide the output change by the input change.

The subtractions must use matching order. If you use ending minus starting for the outputs, also use ending minus starting for the inputs. Reversing both orders gives the same quotient, but reversing only one changes the sign.

The denominator is the input change, not the output change. It is the actual difference between the input values. A rate compares two changes.

The calculation uses only the endpoint values. A function could rise and then fall between the endpoints. Its average rate over the full interval still describes the net change from the first endpoint to the second.

\frac{f(b)-f(a)}{b-a}

Evaluate the function at both endpoints before calculating.
Divide change in output by change in input.
The result summarizes the full interval, even if the function changes direction within it.

Interpret the result

A numerical answer is more useful when paired with a sentence. State the interval, explain the average change in output for each one-unit input increase, and include units if they are known.

For a positive result, say that the output increases on average over the interval. For a negative result, say that it decreases on average. Do not claim the output changes by that amount at every input. The calculation gives an average across the interval only.

The graph gives a visual check. When the second endpoint is to the right and higher than the first, the secant line rises from left to right and the average rate is positive. If the second endpoint is lower, the line falls and the rate is negative.

Interpret the sign and size of the rate.
Use “on average” to avoid describing the rate at every input.
Check that the graph's direction agrees with the sign.

Endpoint values in the worked example

Input	Output
$1$	$f(1)=2$
$4$	$f(4)=11$
Change from $1$ to $4$	$11-2=9$

Worked example

Average change in a function

Let

f(x)=x^2-2x+3

. Calculate and interpret the average rate of change on the interval

[1,4]

.

Find the endpoint outputs
The interval endpoints are $1$ and $4$ . Substitute each input into the function. These values identify the two points on the graph that determine the interval's average rate.
$f(1)=1^2-2(1)+3=2,\quad f(4)=4^2-2(4)+3=11$
Calculate the changes
Subtract the starting output from the ending output. Use the same endpoint order for the inputs, so both changes describe movement from $1$ to $4$ .
$\Delta y=11-2=9,\quad \Delta x=4-1=3$
Divide output change by input change
Average rate of change is the output change divided by the input change. The positive result agrees with the fact that the ending output is greater than the starting output.
$\frac{\Delta y}{\Delta x}=\frac{9}{3}=3$
Interpret the graph and result
The endpoint points are $(1,2)$ and $(4,11)$ . The secant line through them has slope $3$ . Since no physical units are given, describe the result in output units per input unit.
$\frac{f(4)-f(1)}{4-1}=3$

Answer: The average rate of change on

[1,4]

is

3

output units per input unit. The function's output increases by an average of

3

for each one-unit increase in input over this interval.

Check: The output changes by

9

while the input changes by

3

. Since

9

divided by

3

is

3

, the calculation is consistent. The positive result also agrees with the outputs increasing from

2

to

11

.

Common mistakes and how to avoid them

Dividing the input change by the output change.

Correction: Divide change in output by change in input. This gives output units for each input unit.

Subtracting the endpoint values in different orders.

Correction: Use ending minus starting for both outputs and inputs. Inconsistent order can change the sign.

Treating the average rate as the change at every input.

Correction: The result summarizes the endpoint change. The function may change by different amounts between the endpoints.

Leaving out the units or meaning.

Correction: State what the rate says about output change per input unit, and include the given units.

Lesson summary

Choose the interval endpoints and find the function output at each endpoint.
Subtract outputs and inputs in matching order.
Divide change in output by change in input.
Interpret the sign, size, and units as an average across the interval.
On a graph, the average rate is the slope of the secant line through the endpoint points.

Check your understanding

Question 1

A function has

g(2)=7

and

g(6)=19

. What is its average rate of change from

x=2

to

x=6

?

$3$
$-3$
$\frac{1}{3}$
$12$

Show answer and explanation

3

The output change is

19-7=12

, and the input change is

6-2=4

. The average rate is

12/4=3

output units per input unit.

Question 2

A graph's endpoint outputs are

14

at the start of an interval and

8

at the end. What can you say about the average rate of change, assuming the input increases?

It is negative because the ending output is lower.
It is positive because both outputs are positive.
It is zero because both outputs are numbers.
It cannot be found without knowing the curve between the endpoints.

Show answer and explanation

It is negative because the ending output is lower.

The output change is ending output minus starting output, or

8-14=-6

. Dividing by a positive input change gives a negative average rate. The curve between the endpoints is not needed.

Question 3

On a graph, which line's slope gives the average rate of change over an interval?

The secant line through the two endpoint points
A horizontal line through the starting point
The vertical distance between the endpoints only
A line through any two points, whether or not they are endpoints

Show answer and explanation

The secant line through the two endpoint points

The secant line through the interval's endpoint points has slope equal to the output change divided by the input change between those points.

Key terms

Average rate of change: The change in a function's output divided by the change in its input between two endpoints.
Endpoint: One of the two boundary input values of an interval.
Secant line: A straight line through two points on a graph. Its slope compares the output and input changes between those points.
Slope: The change in vertical position divided by the change in horizontal position between two points.

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D1.4 · Calculate and interpret average rate of change

Ontario Grade 12 Mathematics

Calculate and interpret change over an interval

What you will learn

Prerequisite bridge: change and slope

Meaning and representations

Calculate with matching endpoint order

Interpret the result

Endpoint values in the worked example

Average change in a function

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review