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D1.9 · Solve rate-of-change applications

Learn to solve rate-of-change applications through clear examples and targeted practice.

Ontario Grade 12 Mathematics

Characteristics of Functions

Use tables, graphs, and numerical changes to describe how quantities change.

A rate of change describes how one quantity changes as another quantity changes. For example, a cyclist’s distance changes as time passes. A rate can be positive, negative, or zero. In this lesson, you will calculate average rates between two inputs and estimate rates near one input. You will use tables and graphs, then explain what each result means. Always identify what is changing, what it is changing with respect to, and what the answer’s units tell you.

What you will learn

1. Prerequisite bridge: change and graph slope

A change is a final value minus an initial value. If a tank’s water level rises from 1212 cm to 2020 cm, its change is 88 cm. If it falls from 2020 cm to 1212 cm, its change is −8-8 cm. A positive change means an increase. A negative change means a decrease.
Slope describes how much a graph’s output changes compared with how much its input changes. The horizontal axis usually shows the input, such as time. The vertical axis shows the output, such as distance. A graph rising from left to right has positive slope. A graph falling from left to right has negative slope. A horizontal segment has slope zero.
A rate of change is a slope interpreted in a situation. It compares output change with input change. If distance is measured in kilometres and time in hours, the rate is measured in kilometres per hour. In general, the units are output units divided by input units.
To find a rate from two points, divide the output change by the input change. Use the same order for both subtractions. This keeps the signs consistent and makes the result match the graph’s direction.
In words, the rate compares the change in the output with the change in the input.

2. Average rate of change over an interval

An average rate of change describes the overall change between two input values. It does not say that the output changed at this rate at every point between them. The output could rise quickly and then slowly while having the same average across the full interval.
Choose the two endpoints of the interval. Find the output change and input change using the same endpoint order. Divide the output change by the input change. For a function with output f(x)f(x), the average rate from input aa to input bb is shown in the formula.
On a graph, use the points at the interval’s endpoints. The slope of the straight line through those two points gives the average rate over the interval. A table gives the same information when it lists the output values at the chosen inputs.
Interpret the answer in context. A rate of −3-3 metres per second means the measured height decreases by an average of 33 metres for each second over the interval. The negative sign shows direction; it is not a mistake.
f(b)−f(a)b−a\frac{f(b)-f(a)}{b-a}

3. Estimating a rate near one input

A question may ask how fast a quantity is changing near one particular input. A table or graph might not give an exact rate there. You can estimate it from nearby values. An estimate is based on the information available, so use words such as “about” or “approximately.”
For a numerical estimate, select two nearby inputs around the target, if the data allow it. Find the average rate between those inputs. For example, to estimate the rate near 55 minutes, you might use values at 44 and 66 minutes. Their average rate gives information about how the quantity is changing near 55 minutes.
A short interval near the target often gives a useful estimate, but the data must be reliable. Rounded measurements or uneven changes can affect the result. Report only as much precision as the information supports.
For a graph-based estimate, read the graph near the target input. Compare nearby points on the curve and estimate their output change divided by their input change. The graph’s scale matters: check the intervals marked on both axes before calculating. If the curve is rising near the target, the estimate should be positive; if falling, it should be negative. If the curve is nearly flat, the estimate should be near zero.

4. Apply and communicate the result

Rate-of-change applications can describe motion, temperature, height, cost, or another changing quantity. First identify the input and output. Note the interval or target input, and identify the units of both quantities. The input is the quantity shown on the horizontal axis or the quantity that is varied in the situation. The output is the quantity being measured.
For a question about an interval, calculate an average rate. For a question about one input, estimate a rate from nearby numerical values or graph points. An average rate summarizes a span. A local estimate describes change near one input. Match your method to the question.
A complete answer includes the numerical rate, its units, and a sentence in context. For a negative rate, wording such as “decreases by” makes the direction clear. For an estimate, say “about” or “approximately.”
If a table or graph uses rounded values, the calculated rate is also approximate. Do not report more precision than the data support. A rate averaged over a large interval can differ from the rate near either endpoint, especially when the graph changes direction or steepness. Check that the sign, size, and units fit the situation.
output unitsinput units\frac{\text{output units}}{\text{input units}}

Tea temperature data

Time (min)Temperature (°C)
278
470
566
662
854

Worked example

Average and nearby cooling rates

A cup of tea cools according to the table. Find its average rate of cooling from 22 to 88 minutes, and estimate its rate near 55 minutes.
  1. Identify the quantities
    Time is the input, measured in minutes. Temperature is the output, measured in degrees Celsius. A negative rate will mean that the temperature is decreasing.
  2. Calculate the average rate
    From 22 to 88 minutes, the temperature changes from 7878 degrees Celsius to 5454 degrees Celsius. Divide the output change by the input change. The units are degrees Celsius per minute.
    54−788−2=−4 °C/min\frac{54-78}{8-2}=-4\ \mathrm{°C/min}
  3. Estimate near five minutes
    Use the nearby values at 44 and 66 minutes. Their average rate gives an estimate near 55 minutes because the interval is short and surrounds the target.
    62−706−4=−4 °C/min\frac{62-70}{6-4}=-4\ \mathrm{°C/min}
  4. Interpret both rates
    The first rate describes the overall cooling across six minutes. The second estimates the cooling rate near five minutes. Both are negative because the temperature falls as time increases. The data support an estimate, not a claim that the tea cools at exactly this rate throughout.
Answer: The average rate from 22 to 88 minutes is −4-4 degrees Celsius per minute. The estimated rate near 55 minutes is about −4-4 degrees Celsius per minute.
Check: The temperature drops by 2424 degrees Celsius over 66 minutes, giving an average of 44 degrees Celsius per minute. The nearby values drop by 88 degrees Celsius over 22 minutes, also giving 44 degrees Celsius per minute. The negative signs indicate cooling.

Common mistakes and how to avoid them

Subtracting input values in one order and output values in the opposite order.
Correction: Use the same endpoint order in both differences. If the output change is final minus initial, the input change must also be final minus initial.
Leaving out units or reporting only a number.
Correction: Divide output units by input units, then state what the rate means in the situation.
Treating an average rate over an interval as the rate at every point in that interval.
Correction: An average summarizes the whole interval. Use nearby table values or graph points to estimate the rate near one input.
Changing a negative rate to positive without explaining the direction.
Correction: Keep the negative sign in the calculation. In words, explain that the quantity is decreasing.

Lesson summary

Check your understanding

Question 1

A plant grows from 1414 cm to 2626 cm over 66 days. What is its average rate of growth?
  1. 22 cm per day
  2. 66 cm per day
  3. 1212 cm per day
  4. −2-2 cm per day
Show answer and explanation
22 cm per day
The height increases by 1212 cm over 66 days, so the average rate is 22 cm per day.

Question 2

A graph of distance against time is falling from left to right. What does this indicate about the rate of change of distance?
  1. It is negative over that interval.
  2. It must be zero over that interval.
  3. It is positive over that interval.
  4. Its units are time per distance.
Show answer and explanation
It is negative over that interval.
A falling graph has negative slope, so distance decreases as time increases.

Question 3

A temperature table gives 3131 degrees Celsius at 99 minutes and 2525 degrees Celsius at 1111 minutes. What is the average rate from 99 to 1111 minutes?
  1. −3-3 degrees Celsius per minute
  2. −6-6 degrees Celsius per minute
  3. 33 degrees Celsius per minute
  4. 66 degrees Celsius per minute
Show answer and explanation
−3-3 degrees Celsius per minute
The temperature change is 25−31=−625-31=-6 degrees Celsius and the time change is 22 minutes. The rate is −6/2=−3-6/2=-3 degrees Celsius per minute.

Key terms

Rate of change
A comparison of how much an output changes for a change in its input.
Average rate of change
The rate calculated between two input values; it describes overall change across that interval.
Slope
A measure of how much a graph’s output changes compared with its input change.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D1.9. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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