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D1.2 · Distinguish zero, constant, and changing rates

Learn to distinguish zero, constant, and changing rates through clear examples and targeted practice.

Ontario Grade 12 Mathematics

Characteristics of Functions

Recognizing patterns in numerical and graphical rates of change

A rate of change describes how an output changes as an input changes. For example, distance can change as time passes. The distance might stay the same, change by equal amounts over equal time intervals, or change by different amounts. These patterns are called zero, constant, and changing rates. You can identify them by using numerical values and graphs.

What you will learn

1. Prerequisite bridge: compare changes

A quantity is something that can be measured or counted, such as time, distance, or temperature. An input is a value you choose or observe first. An output is a value that depends on the input. In a table of time and distance, time is often the input and distance is the output.
To find a change between two observations, subtract the earlier value from the later value. The change in output tells how much the dependent quantity moved. The change in input tells how much the input moved. Use the same order for both changes.
A rate compares these two changes. It answers this question: how much does the output change for each unit of input? Units matter. If distance is measured in kilometres and time in hours, the rate is measured in kilometres per hour.
The rate over an interval is the change in output divided by the change in input. An interval is the part of the input values between two selected points. The input values must be different so the division is possible.
If the input changes from x1x_1 to x2x_2 and the output changes from y1y_1 to y2y_2, the rate compares the output change with the input change. The subscripts identify the first and second points.
y2−y1x2−x1\frac{y_2-y_1}{x_2-x_1}

2. Three rate patterns in plain language

A zero rate means that the output does not change as the input changes over the interval being considered. For example, if a parked bicycle stays in the same position while time passes, its position has a zero rate during that time. On a graph with input on the horizontal axis and output on the vertical axis, this appears as a horizontal segment.
A constant rate means that the output changes by the same amount for each equal-sized change in input. A constant rate can be positive, so the output rises by equal amounts, or negative, so it falls by equal amounts. On a graph, a constant rate appears as a straight line. A horizontal straight line is the special case with rate zero.
A changing rate means that the rate is not the same across the intervals being compared. The output may keep increasing or keep decreasing. What matters is that the output change per input unit does not stay the same. A graph that bends rather than following one straight line shows a changing rate across that region.
To decide whether a rate is constant or changing, compare rates over intervals with the same input width. This makes the comparison fair and easy to read. If interval widths differ, compare output change per input unit rather than output changes alone.
y2−y1x2−x1\frac{y_2-y_1}{x_2-x_1}

3. Numerical, graphical, and symbolic representations

A table lets you compare rates by calculating changes between rows. When the input steps are equal, compare the output changes. Repeated equal output changes show a constant rate. No output change shows a zero rate. Unequal output changes show a changing rate.
If input steps are not equal, do not compare output changes alone. Divide each output change by its matching input change. For example, an output change of six over an input change of two has the same rate as an output change of nine over an input change of three. Each rate is three output units per input unit.
On a graph, look at the direction and shape. A horizontal segment has zero rate. A straight rising or falling segment has a constant positive or negative rate. A graph whose steepness changes indicates a changing rate. When accuracy matters, use plotted values or clearly marked intervals rather than guessing from appearance.
Symbols provide a compact way to express the same comparison. If the input is represented by xx and the output by yy, select two points and calculate the change in yy divided by the change in xx. Repeat this for other intervals. If the results match, the rates are constant over those intervals. If the results differ, the rate is changing. If the output does not change, the rate is zero.
A function is a rule that gives an output for each input. Make a short table of input values, calculate the corresponding outputs, and compare rates across equal input steps. This numerical approach ties the decision to the values and interval being studied.
y2−y1x2−x1\frac{y_2-y_1}{x_2-x_1}

4. Applying the distinction

When a situation is described in words, identify the input and output first. Then ask what happens to the output as the input advances. No change suggests a zero rate. Repeated equal changes for equal input steps suggest a constant rate. Different changes per equal input step suggest a changing rate.
In a real situation, measurements may be rounded or affected by small recording differences. Base your classification on the information given. If a problem says a quantity changes by the same amount each hour, treat that as a constant rate. If a table gives exact values, calculate the interval rates rather than guessing from the overall trend.
Do not treat increase or decrease as the classification. Increasing describes direction, not whether the rate is constant. A quantity can increase at a constant rate or at a changing rate. The key question is whether the amount of change per input unit stays the same.
A useful routine is to name the input and output, mark the intervals being compared, calculate or read the changes, and then classify the pattern. State the interval in your conclusion. This avoids making a claim about an entire situation when the evidence covers only part of it.

A quick comparison for equal input steps

Output changesRate patternWhat it means
0, 0, 0ZeroThe output stays the same.
4, 4, 4ConstantThe output changes by the same amount each step.
2, 3, 5ChangingThe output changes by different amounts.

Worked example

Classifying rates from a table

A small robot's distance from its starting point is recorded at equal one-second intervals. The distances are 2 m at 0 s, 5 m at 1 s, 8 m at 2 s, and 12 m at 3 s. Classify the rate over the full record and explain how you know.
  1. Identify the quantities
    Time is the input, and distance is the output. Each time step is one second, so the output changes can be compared directly.
    Δt=1 s\Delta t=1\text{ s}
  2. Compare consecutive changes
    Subtract each earlier distance from the next distance. The changes are three metres, three metres, and four metres. They are not all equal.
    5−2=3,8−5=3,12−8=45-2=3,\quad 8-5=3,\quad 12-8=4
  3. Classify the pattern
    Because the input steps are equal but the output changes are not all equal, the rate is changing over the full record. The distance increases throughout, but that does not make its rate constant.
    3, 3, 4 m/s3,\ 3,\ 4\text{ m/s}
Answer: The rate is changing over the recorded intervals.
Check: The rate is not zero because the distance changes. It is not constant because the distance changes by four metres in the last second, rather than three metres.

Common mistakes and how to avoid them

Calling every increasing output a constant rate.
Correction: Increasing describes direction only. Compare output changes over equal input steps to decide whether the rate is constant.
Comparing output changes when the input intervals have different widths.
Correction: Divide each output change by its matching input change. Then compare the rates.
Calling a horizontal graph a changing rate because its output has a value.
Correction: The output may be nonzero, but it does not change as the input changes. Its rate is zero.
Classifying a whole situation from only one interval.
Correction: A single interval gives one rate. Compare multiple intervals to decide whether the rate stays the same or changes.

Lesson summary

Check your understanding

Question 1

For equal input steps, the output changes by 6, 6, and 6 units. What is the rate pattern?
  1. Zero rate
  2. Constant rate
  3. Changing rate
  4. The output has no rate
Show answer and explanation
Constant rate
The output changes by the same amount for each equal input step, so the rate is constant.

Question 2

A graph is horizontal across an interval. What does this say about the output?
  1. It stays the same as the input changes.
  2. It rises by equal amounts.
  3. It changes by larger amounts each step.
  4. It falls by equal amounts.
Show answer and explanation
It stays the same as the input changes.
A horizontal graph has no vertical change while the input changes, so the rate is zero.

Question 3

Over equal input steps, an output rises by 2 units, then 5 units, then 3 units. What is the rate pattern?
  1. Zero rate
  2. Constant rate
  3. Changing rate
  4. No change in output
Show answer and explanation
Changing rate
The output changes by different amounts over equal input steps, so the rate is changing.

Key terms

Input
The value that is chosen or observed first in a relationship.
Output
The value that depends on the input.
Rate of change
A comparison of how much the output changes for a change in the input.
Interval
The part of the input values between two selected points.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D1.2. It is a study resource, not an official curriculum publication.

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