Compare rates of change in linear and non-linear graphs

Ontario Grade 9 Mathematics

Algebra

MTH1W — Expectation C3.1: Recognizing Constant and Changing Rates of Change

Imagine two delivery cyclists leave the same warehouse at the same time. Cyclist A covers exactly 15 km every hour, no matter what. Cyclist B starts out slow, then speeds up a lot. After three hours both have travelled the same total distance — but their journeys looked very different on a graph. The key difference is in their rates of change. This lesson teaches you how to spot, calculate, and compare rates of change for linear (straight-line) and non-linear (curved) graphs. These are core skills you will use throughout the course whenever you analyze a relationship between two quantities.

What you will learn

Explain what rate of change means and calculate it between two points on a graph.
Identify a linear graph by its constant rate of change.
Identify a non-linear graph by its changing rate of change.
Compare the rates of change of a linear and a non-linear graph over specified intervals.
Use a table of values to decide whether a relationship is linear or non-linear.

Prerequisite Review: What Is a Rate of Change?

A rate of change tells you how much one quantity changes compared to another quantity. You already worked with this idea in Grade 8 when you compared unit rates — for example, 'CAD 3 per litre' or '12 metres per second'. Both of those are rates of change.

On a graph, the two quantities are the horizontal axis (usually called

x

) and the vertical axis (usually called

y

). The rate of change between two points tells you: for every 1 unit

x

increases, how much does

y

increase (or decrease)?

To calculate a rate of change between two points, use this plain-language rule: divide the change in

y

by the change in

x

. The change in a quantity just means the ending value minus the starting value. So if you move from point A to point B, the rate of change is the rise (vertical change) over the run (horizontal change).

rate of change =

\frac{change in y}{change in x}

=

\frac{y_2 - y_1}{x_2 - x_1}

Rate of change = (change in $y$ ) ÷ (change in $x$ ).
Change in $y$ means: $y$ -value at the end minus $y$ -value at the start.
Change in $x$ means: $x$ -value at the end minus $x$ -value at the start.
A positive rate of change means $y$ is increasing as $x$ increases.
A negative rate of change means $y$ is decreasing as $x$ increases.

Linear Graphs: A Constant Rate of Change

A linear relationship produces a straight-line graph. The most important feature of a linear graph is that its rate of change is the same everywhere on the line — it never speeds up or slows down. This is called a constant rate of change.

Think of Cyclist A from the introduction: she always travels 15 km per hour. Whether you look at hours 0–1, hours 1–2, or hours 2–3, the rate of change of distance with respect to time is always 15. On a distance-time graph, her journey is a straight line.

You can confirm a constant rate of change using a table of values. Calculate the rate of change for every consecutive pair of rows. If every answer is identical, the relationship is linear. If even one interval gives a different answer, the relationship is not linear.

On a straight-line graph, the constant rate of change is also the steepness of the line. A larger constant rate of change means the line rises (or falls) more steeply.

\frac{y_2 - y_1}{x_2 - x_1}

= same value for every pair of points on a linear graph

A linear graph is a straight line.
Its rate of change is constant — the same value at every interval.
Equal increases in $x$ always produce equal increases (or decreases) in $y$ .
A steeper line has a larger constant rate of change.

Non-Linear Graphs: A Changing Rate of Change

A non-linear relationship produces a curved graph. The key feature is that its rate of change is not the same at every interval — it changes as

x

increases. This is called a changing (or varying) rate of change.

Think of Cyclist B: he starts slowly and then speeds up dramatically. In the first hour he might travel only 5 km, but in the third hour he travels 25 km. Each hour the rate of change of distance is different, so his graph curves upward.

You calculate the rate of change on a non-linear graph the same way as on a linear graph — (change in

y

) ÷ (change in

x

) — but you must state which interval you are talking about, because the answer will be different for each interval.

The rate of change can increase over time (the curve gets steeper), decrease over time (the curve flattens), or do both at different sections. The shape of the curve shows you the pattern: a curve that bends upward is getting steeper, meaning the rate of change is increasing.

rate of change over interval =

\frac{y_2 - y_1}{x_2 - x_1}

(value changes with each interval)

A non-linear graph is a curve, not a straight line.
Its rate of change is different at different intervals.
You must name the interval when you state a rate of change for a non-linear graph.
A curve bending upward means the rate of change is increasing; a curve flattening means it is decreasing.

Comparing Rates of Change: Linear vs. Non-Linear

When you compare a linear and a non-linear graph, the central question is: how does the rate of change behave in each graph, and how do those behaviours differ?

For a linear graph, you only need to calculate the rate of change once — it is the same everywhere. For a non-linear graph, you calculate the rate of change separately for each interval you want to examine. Then you compare those values to describe how the two graphs differ.

A useful strategy is to build a table of values for both relationships, calculate the rate of change for each interval, and then write a sentence comparing them. For example: 'Between

x = 0

and

x = 2

, the linear graph has a rate of change of 4, while the non-linear graph has a rate of change of 1. Between

x = 4

and

x = 6

, both still show a rate of change of 4 for the linear graph, but the non-linear graph now shows a rate of change of 9.' This tells a clear story: the linear graph stays steady while the non-linear graph starts slowly and then grows much faster.

You can also compare graphs visually. A straight line and a curve that start at the same point will eventually separate. The curve may be below the line first and then overtake it, or it may start above and then fall below. Tracking where one rate of change becomes larger than the other is a key part of comparing them.

Comparison:

\frac{\Delta y_{linear}}{\Delta x}

= constant,

\frac{\Delta y_{non-linear}}{\Delta x}

varies by interval

To compare, calculate the rate of change over the same intervals for each graph.
State the interval clearly whenever you report a rate of change from a non-linear graph.
A table of values makes it easy to see where rates of change differ.
A linear graph has one rate of change; a non-linear graph has a different rate of change at each interval.

Reading the Story in the Graph

Every graph tells a story about how two quantities are related. When you compare rates of change, you are reading that story carefully.

Ask three questions about any graph: (1) Is the rate of change increasing, decreasing, or staying the same? (2) Is the change happening quickly or slowly? (3) How does this graph's behaviour compare to the other graph at the same interval? Answering these three questions in plain language is exactly what 'comparing rates of change' means in this course.

Remember that a larger rate of change means a steeper section of the graph. So if a curve is getting steeper, its rate of change is increasing. If a line is straight, its rate of change never changes. When a curve overtakes a line on a graph — crossing from below to above — that is the moment the non-linear rate of change has grown larger than the constant linear rate of change.

\text{steeper section} \Rightarrow \text{larger } \frac{\Delta y}{\Delta x} \quad ; \quad \text{flatter section} \Rightarrow \text{smaller } \frac{\Delta y}{\Delta x}

A steeper section of a graph = a larger rate of change at that interval.
A flat section of a graph = a small (close to zero) rate of change.
When a curve crosses a line, the two rates of change at that point are equal.
Describe rates of change in plain language: 'increasing', 'decreasing', 'constant', 'faster', 'slower'.

Rate of Change Comparison: Plant L (Linear) vs. Plant N (Non-Linear)

Interval (Days)	Plant L Height Change (cm)	Plant L Rate (cm/day)	Plant N Height Change (cm)	Plant N Rate (cm/day)
0 to 2	4	2	1	0.5
2 to 4	4	2	3	1.5
4 to 6	4	2	5	2.5
6 to 8	4	2	7	3.5
8 to 10	4	2	9	4.5

Worked example

Comparing a Linear and a Non-Linear Relationship Using a Table

Two plants are growing in a classroom. Plant L (linear) and Plant N (non-linear) were both 0 cm tall on Day 0. The table below shows their heights at the end of each 2-day interval. Calculate the rate of change of height for each plant over each interval. Then write two sentences comparing their rates of change.

Day 0: Plant L = 0 cm, Plant N = 0 cm
Day 2: Plant L = 4 cm, Plant N = 1 cm
Day 4: Plant L = 8 cm, Plant N = 4 cm
Day 6: Plant L = 12 cm, Plant N = 9 cm
Day 8: Plant L = 16 cm, Plant N = 16 cm
Day 10: Plant L = 20 cm, Plant N = 25 cm

Step 1 — Set up the rate of change formula
The rate of change formula is (change in height) ÷ (change in days). Each interval is 2 days wide, so the change in $x$ is always 2 for both plants.
rate of change = $\frac{height at end of interval - height at start of interval}{2}$
Step 2 — Calculate Plant L's rate of change for each interval
Days 0–2: change in height = $4 - 0 = 4$ , so rate = $4 ÷ 2 = 2$ cm/day. Days 2–4: change = $8 - 4 = 4$ , rate = $2$ cm/day. Days 4–6: change = $12 - 8 = 4$ , rate = $2$ cm/day. Days 6–8: change = $16 - 12 = 4$ , rate = $2$ cm/day. Days 8–10: change = $20 - 16 = 4$ , rate = $2$ cm/day. Every interval gives the same answer.
$\frac{4-0}{2} = \frac{8-4}{2} = \frac{12-8}{2} = \frac{16-12}{2} = \frac{20-16}{2} = 2 \text{ cm/day}$
Step 3 — Calculate Plant N's rate of change for each interval
Days 0–2: change = $1 - 0 = 1$ , rate = $1 ÷ 2 = 0.5$ cm/day. Days 2–4: change = $4 - 1 = 3$ , rate = $3 ÷ 2 = 1.5$ cm/day. Days 4–6: change = $9 - 4 = 5$ , rate = $5 ÷ 2 = 2.5$ cm/day. Days 6–8: change = $16 - 9 = 7$ , rate = $7 ÷ 2 = 3.5$ cm/day. Days 8–10: change = $25 - 16 = 9$ , rate = $9 ÷ 2 = 4.5$ cm/day. Each interval gives a different answer — and each answer is larger than the last.
$\frac{1-0}{2}=0.5,\quad \frac{4-1}{2}=1.5,\quad \frac{9-4}{2}=2.5,\quad \frac{16-9}{2}=3.5,\quad \frac{25-16}{2}=4.5 \text{ cm/day}$
Step 4 — Compare the two plants
Plant L's rate of change is constant at 2 cm/day throughout, which confirms it is linear. Plant N's rate of change starts at 0.5 cm/day and increases to 4.5 cm/day, which confirms it is non-linear. Notice that Plant N starts growing slower than Plant L but ends up growing faster. The two plants reach the same height (16 cm) at Day 8, which is where their graphs would cross.
Plant L: rate = 2 cm/day (constant) ; Plant N: rate grows from 0.5 to 4.5 cm/day

Answer: Plant L has a constant rate of change of 2 cm/day, confirming a linear relationship. Plant N has a rate of change that increases from 0.5 cm/day to 4.5 cm/day, confirming a non-linear relationship. Plant N grows more slowly than Plant L at first, but overtakes Plant L after Day 8.

Check: Verify Plant L: every height difference over 2 days equals 4, and

4 ÷ 2 = 2

. ✓ Verify Plant N at Day 8–10:

25 - 16 = 9

and

9 ÷ 2 = 4.5

, which is greater than Plant L's rate of 2. ✓ Both plants are at 16 cm on Day 8:

0.5 \times 4 = 2

intervals of growth check confirms Plant N is indeed non-linear because consecutive differences (1, 3, 5, 7, 9) are not equal. ✓

Common mistakes and how to avoid them

Calculating the rate of change for a non-linear graph once and treating it as constant for the whole graph.

Correction: For a non-linear graph, always calculate the rate of change separately for each interval. The whole point of a non-linear graph is that its rate of change is different at different intervals. Always name the interval when you report the rate of change.

Subtracting

x

from

y

instead of dividing: writing

y_2 - y_1 - (x_2 - x_1)

as the rate of change.

Correction: Rate of change is a division, not a subtraction. The formula is

\frac{y_2 - y_1}{x_2 - x_1}

. Think of it as 'how many units does

y

change per one unit of

x

'.

Concluding a graph is linear just because

y

is increasing throughout.

Correction: A graph can increase and still be non-linear. The test is whether the rate of change is the same at every interval — not simply whether

y

goes up. Check the rate of change for multiple intervals before deciding.

Reversing the order of subtraction in the numerator but not the denominator (or vice versa), giving the wrong sign.

Correction: Always subtract in the same order in both the numerator and denominator. If you use end minus start for

y

, you must also use end minus start for

x

:

\frac{y_2 - y_1}{x_2 - x_1}

. Mixing the order flips the sign of your answer.

Describing a non-linear graph's overall rate of change with a single number without specifying an interval.

Correction: A single number cannot describe the rate of change of a non-linear graph overall. You must always say 'between

x = a

and

x = b

, the rate of change is …'. Without naming the interval, the statement is incomplete and likely wrong.

Lesson summary

A rate of change measures how much $y$ changes for each unit increase in $x$ , and is calculated as $\frac{y_2 - y_1}{x_2 - x_1}$ .
A linear graph is a straight line with a constant rate of change — the same value at every interval.
A non-linear graph is a curve with a changing rate of change — a different value at each interval.
To compare the two types, calculate the rate of change over matching intervals and describe how they differ: one stays constant, the other increases or decreases.
A steeper section of any graph means a larger rate of change; a flatter section means a smaller rate of change.
Always name the interval when reporting the rate of change for a non-linear graph, because that value only applies to that specific interval.

Check your understanding

Question 1

A table of values shows that every time

x

increases by 1,

y

increases by 5. What does this tell you about the graph?

The graph is non-linear because $y$ is always increasing.
The graph is linear because the rate of change is constant at 5.
The graph is linear because $y$ and $x$ are both positive.
The graph is non-linear because the rate of change equals 5.

Show answer and explanation

The graph is linear because the rate of change is constant at 5.

A constant rate of change — here, always 5 — means the graph is a straight line, which is linear. The fact that

y

is increasing or that the numbers are positive does not determine whether a graph is linear. Only a constant rate of change confirms linearity.

Question 2

A non-linear graph passes through the points

(2, 3)

,

(4, 11)

, and

(6, 27)

. What is the rate of change between

x = 4

and

x = 6

?

$\frac{11-3}{4-2} = 4$
$\frac{27-3}{6-2} = 6$
$\frac{27-11}{6-4} = 8$
$\frac{27-11}{6-4} = 16$

Show answer and explanation

\frac{27-11}{6-4} = 8

For the interval from

x = 4

to

x = 6

: change in

y = 27 - 11 = 16

and change in

x = 6 - 4 = 2

, so the rate of change

= 16 ÷ 2 = 8

. Option A uses the wrong interval (0–2). Option B averages over two intervals at once. Option D gives the change in

y

without dividing by the change in

x

.

Question 3

Plant A grows 3 cm each day (linear). Plant B grew 1 cm on Day 1, 3 cm on Day 2, 5 cm on Day 3, and 7 cm on Day 4 (non-linear). On which day does Plant B's daily rate of change first exceed Plant A's constant rate of change?

Day 1
Day 2
Day 3
Day 4

Show answer and explanation

Day 3

Plant A's constant rate of change is 3 cm/day. Plant B's daily rates are: Day 1 = 1 cm/day, Day 2 = 3 cm/day, Day 3 = 5 cm/day, Day 4 = 7 cm/day. On Day 2, Plant B matches Plant A exactly (both 3 cm/day). On Day 3, Plant B's rate of 5 cm/day first exceeds Plant A's rate of 3 cm/day.

Question 4

Which statement correctly compares a linear and a non-linear graph?

Both graphs can have a constant rate of change over different sections.
A linear graph has a constant rate of change; a non-linear graph has a rate of change that varies by interval.
A non-linear graph always has a larger rate of change than a linear graph.
A linear graph curves upward because its rate of change is constant.

Show answer and explanation

A linear graph has a constant rate of change; a non-linear graph has a rate of change that varies by interval.

The defining difference is that a linear graph has exactly one rate of change that stays the same everywhere, while a non-linear graph has a different rate of change at each interval. A non-linear graph is not always faster than a linear graph — it depends on the interval. A linear graph is a straight line, not a curve.

Key terms

Rate of change: A measure of how much one quantity ( $y$ ) changes for each unit increase in another quantity ( $x$ ). Calculated as (change in $y$ ) ÷ (change in $x$ ).
Constant rate of change: A rate of change that stays the same value at every interval. This is the feature that makes a graph linear (a straight line).
Changing rate of change: A rate of change that is different at different intervals. This is the feature that makes a graph non-linear (a curve).
Linear graph: A graph that forms a straight line, produced when the rate of change between any two points is always the same.
Non-linear graph: A graph that forms a curve, produced when the rate of change is not the same at every interval.
Interval: A specific section of a graph or table, defined by a starting $x$ -value and an ending $x$ -value — for example, 'from $x = 2$ to $x = 4$ '.
Change in $y$ (written $\Delta y$ ): The difference between two $y$ -values: ending $y$ -value minus starting $y$ -value. The symbol $\Delta$ (delta) simply means 'change in'.
Change in $x$ (written $\Delta x$ ): The difference between two $x$ -values: ending $x$ -value minus starting $x$ -value.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C3.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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C3.1 · Compare rates of change in linear and non-linear graphs

Ontario Grade 9 Mathematics

MTH1W — Expectation C3.1: Recognizing Constant and Changing Rates of Change

What you will learn

Prerequisite Review: What Is a Rate of Change?

Linear Graphs: A Constant Rate of Change

Non-Linear Graphs: A Changing Rate of Change

Comparing Rates of Change: Linear vs. Non-Linear

Reading the Story in the Graph

Rate of Change Comparison: Plant L (Linear) vs. Plant N (Non-Linear)

Comparing a Linear and a Non-Linear Relationship Using a Table

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review