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D1.1 · Interpret rates of change in multiple representations

Learn to interpret rates of change in multiple representations through clear examples and targeted practice.

Ontario Grade 12 Mathematics

Characteristics of Functions

Connecting meaning, tables, graphs, and calculations

A function connects an input to an output. For example, a cyclist’s distance can depend on time. A rate of change describes how much the distance changes compared with a change in time. We can represent a rate with words, a table, a graph, or a calculation. The situation and units help us explain what the rate means.

What you will learn

Explain what a rate of change compares.
Calculate an average rate of change from numerical values.
Connect an average rate to the slope of a line through two graph points.
Interpret the sign, size, units, and meaning of a rate in context.

1. Prerequisite bridge: changes and slope

An input is a value supplied to a function. An output is the value that corresponds to that input. For example, time can be an input and distance can be an output. In a table or graph, the input is often called

x

and the output is often called

y

A change tells how much a quantity increases or decreases between two values. Find a change by subtracting the starting value from the ending value. Keep the order consistent when comparing changes.

Slope describes how much a straight line rises or falls as it moves from left to right. It compares vertical change with horizontal change. On a graph of output against input, slope compares output change with input change.

A rate’s units come from dividing the output units by the input units. If output is measured in kilometres and input in hours, the rate is measured in kilometres per hour. Units are part of the interpretation, not extra decoration.

In words, a rate of change is the output change divided by the matching input change. This can be written as a fraction with output change on top and input change below.

A rate compares an output change with an input change.
Use the same endpoint order for both changes.
Rate units are output units per input unit.

2. Average rate in numerical and graphical representations

An average rate of change describes what happens across an interval. An interval is the range from one input value to another. The average rate answers this question: for each one-unit increase in the input, how much does the output change on average over this interval?

For a function

f

, choose starting and ending inputs,

a

and

b

. Their outputs are

f(a)

and

f(b)

. Subtract the starting output from the ending output. Divide by the ending input minus the starting input. The subtraction order must match in both parts.

A table gives the values needed for this calculation. Find the row for each endpoint, then compare the outputs and inputs. A graph shows the same information as two points. The straight line through those points is called a secant line. Its slope is the average rate of change between the points.

A curved graph does not have to change at a constant rate. The secant line still gives a useful average for its two selected endpoints. It does not claim that the output changed by that amount during every part of the interval.

The sign gives direction. A positive rate means the ending output is greater than the starting output as the input increases. A negative rate means the ending output is smaller. A rate of zero means the two endpoint outputs are equal.

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

Choose the interval before calculating.
Use the outputs and inputs at the same two endpoints.
On a graph, the secant line’s slope represents the interval average.

3. Interpreting and comparing rates

A numerical answer is only part of an interpretation. State what is changing, whether it is increasing or decreasing, and how much it changes per unit of input. Include units. For example, a rate of negative 2 metres per second for a water level means the level decreases by an average of 2 metres for each second over the stated interval.

Compare rates only after checking their units and intervals. A rate of 5 kilometres per hour is greater than a rate of 3 kilometres per hour in magnitude. But a rate over a short interval and a rate over a long interval may describe different average behaviour.

On a graph, a steeper secant line has a greater slope magnitude than a flatter one. The sign still matters: a steep rising line has a large positive slope, while a steep falling line has a large negative slope. Read the direction of the graph as the input increases to determine the sign.

A function value and a rate are different. A value such as 17 centimetres describes an output at one input. A rate such as 3 centimetres per week describes how the output changes across an interval. Keep these meanings separate.

Interpret sign, size, units, and context together.
A positive or negative rate describes the direction of endpoint change.
Graph steepness relates to the size of the slope, while graph direction determines its sign.

4. A dependable interpretation routine

First name the input and output, with their units. Next identify the interval named in the question. Select the corresponding endpoint values from the table, graph, or function.

Calculate output change divided by input change, keeping the endpoint order consistent. If you use a graph, identify the two points and consider the slope of the line through them. Check whether the result is positive, negative, or zero.

Finally, write a sentence in context. A complete sentence might say that the output increased by a stated amount per input unit on average over the interval. Avoid saying that the rate stayed constant unless the representation supports that claim.

Identify quantities and interval before calculating.
Keep subtraction order consistent.
Finish with a contextual interpretation and units.

Cyclist distance data

Time (h)	Distance (km)
1.0	12
1.9	20.3
2.1	21.5
3.0	30

Worked example

Comparing cyclist distance rates

A cyclist’s distance from the starting point is recorded in the table. Find the average rate of change from 1 to 3 hours. Then compare it with the average rate from 1.9 to 2.1 hours.

Identify the quantities
Time is the input, measured in hours. Distance is the output, measured in kilometres. Therefore, each rate will be in kilometres per hour.
Calculate the full-interval rate
From 1 to 3 hours, the distance changes from 12 km to 30 km. Divide the 18 km increase by the 2-hour increase. This gives the average rate across that interval.
$\frac{30-12}{3-1}=9\text{ km/h}$
Calculate the shorter-interval rate
From 1.9 to 2.1 hours, the distance changes from 20.3 km to 21.5 km. The output change is 1.2 km and the input change is 0.2 hours.
$\frac{21.5-20.3}{2.1-1.9}=6\text{ km/h}$
Interpret the comparison
The average rate is 9 km/h from 1 to 3 hours and 6 km/h from 1.9 to 2.1 hours. The different averages show that the cyclist’s distance did not increase at one constant rate across both intervals.

Answer: The average rate from 1 to 3 hours is 9 km/h. From 1.9 to 2.1 hours, it is 6 km/h.

Check: The first calculation uses the endpoints 1 and 3. The second uses 1.9 and 2.1. Both divide a distance change by the matching time change.

Common mistakes and how to avoid them

Dividing an output by an input instead of comparing changes.

Correction: Subtract the endpoint outputs, then divide by the difference between the endpoint inputs.

Reversing the subtraction order in only one part of the rate.

Correction: Use the same endpoint order for output change and input change.

Leaving out units or context.

Correction: State what changes, by how much per input unit, and over which interval.

Assuming an average rate applies at every input in the interval.

Correction: An average summarizes the two endpoints. It does not guarantee a constant rate throughout the interval.

Using the curve’s bending direction to decide whether the rate is positive or negative.

Correction: Read whether the graph rises or falls as the input increases. That direction determines the sign of the rate between the selected endpoints.

Lesson summary

A rate of change compares output change with input change.
An average rate of change uses two endpoints and describes an interval.
On a graph, the slope of the secant line through two points gives the average rate of change between them.
Interpret a rate using its sign, size, units, and context.

Check your understanding

Question 1

A temperature rises from 14°C at 2 p.m. to 20°C at 5 p.m. What is its average rate of change?

2°C per hour
6°C per hour
3°C per hour
−2°C per hour

Show answer and explanation

2°C per hour

The temperature increases by 6°C over 3 hours. Dividing gives an average increase of 2°C per hour.

Question 2

On a graph of output against input, what does a negative average rate over an interval mean?

The ending output is lower than the starting output as the input increases.
The output is negative at both endpoints.
The input decreases as the output increases.
The output has the same value at both endpoints.

Show answer and explanation

The ending output is lower than the starting output as the input increases.

A negative rate means the output change is negative when measured from the first endpoint to the second as the input increases.

Question 3

A function’s output increases by 18 units while its input increases by 6 units over an interval. What is the average rate?

3 output units per input unit
12 output units per input unit
108 output units per input unit
−3 output units per input unit

Show answer and explanation

3 output units per input unit

Divide the positive output change, 18 units, by the positive input change, 6 units. The rate is 3 output units per input unit.

Key terms

Rate of change: A comparison of how much an output changes for a change in input.
Average rate of change: The output change divided by the input change between two endpoints.
Slope: A measure of how much a graph rises or falls compared with its movement from left to right.
Secant line: A straight line through two points on a graph. Its slope gives the average rate of change between those points.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D1.1. It is a study resource, not an official curriculum publication.

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