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G7 · Explain the meanings of slope and intercept on a graph

Learn to explain the meanings of slope and intercept on a graph through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Modelling Linear Relations

Read what a straight-line graph says about change and where it begins

A graph can show how one quantity changes as another quantity changes. For example, it might show distance over time or cost against the number of items. Two useful features of a straight-line graph are its slope and its intercept. Slope describes how the graph changes as you move along it. An intercept marks where the graph crosses an axis. To explain either feature, first check what each axis measures and what units it uses.

What you will learn

Explain what slope tells you about how a graph changes.
Identify an intercept on a graph and explain its meaning.
Use axis labels and units to describe slope and intercept in context.
Distinguish the meaning of slope from the meaning of an intercept.

1. Grade 9 bridge: start with the axes

A coordinate graph has two number lines called axes. The horizontal axis runs from left to right. The vertical axis runs up and down. Their crossing point is called the origin. It is written as

(0,0)

A point on a graph is written as an ordered pair, such as

(2,6)

. The first number tells the position along the horizontal axis. The second number tells the position along the vertical axis. If the horizontal axis shows time in minutes and the vertical axis shows distance in kilometres, the point

(2,6)

means 2 minutes and 6 kilometres.

Before interpreting a graph, read its labels and units. Also check which direction shows increasing values. A line that rises as you look from left to right does not always mean that something is physically rising. Its meaning depends on what the vertical axis measures.

Read the labels and units on both axes.
In an ordered pair, the first value is horizontal and the second is vertical.
A graph feature gets its meaning from the quantities shown on the axes.

2. Slope: how the graph changes

Slope describes how much the vertical quantity changes compared with the horizontal quantity. In plain language, it tells how quickly one quantity changes as the other changes. The phrase rate of change means the amount of change in one quantity for a change in another.

To understand slope on a straight line, compare two points on the line. Find how much the vertical value changes. Then compare that with how much the horizontal value changes. For example, if distance increases by 6 kilometres while time increases by 2 minutes, the distance changes at 3 kilometres per minute. The units matter: kilometres per minute tells which quantity is changing and which quantity is being compared.

The direction of the line gives a quick clue. A line that rises from left to right has positive slope: the vertical values increase as the horizontal values increase. A line that falls from left to right has negative slope: the vertical values decrease as the horizontal values increase. A horizontal line has zero slope because its vertical value does not change as the horizontal value changes.

Slope is not the same as how steep a line looks on a page. The spacing and scale on the axes can affect the line’s appearance. Use the values on the axes and the units, rather than the visual angle alone, to describe the change.

\frac{\text{vertical change}}{\text{horizontal change}}

Slope compares vertical change with horizontal change.
A rising line has positive slope; a falling line has negative slope.
A horizontal line has no change in its vertical value.
Describe slope with units when the graph provides them.

3. Intercept: where the graph crosses an axis

An intercept is a point where a graph meets one of the axes. The vertical intercept is where the graph crosses the vertical axis. At that point, the horizontal value is zero. Its vertical value can often be read directly from the graph.

A vertical intercept can describe a starting or initial value in a situation. For example, on a graph of distance against time, it tells the distance shown when time is zero. Do not assume this value must be zero. The graph might show that an object is already some distance from a chosen starting point at time zero.

A graph can also cross the horizontal axis. This is called the horizontal intercept. At this point, the vertical value is zero. Its meaning depends on the labels. On a graph showing water remaining over time, a horizontal intercept could mark the time when no water remains.

Slope and intercept answer different questions. Slope describes change along the line. An intercept gives the value at the point where the line crosses an axis. An intercept is a location on the graph, not a rate of change.

At the vertical intercept, the horizontal value is zero.
At the horizontal intercept, the vertical value is zero.
Use the graph’s labels and units to explain what an intercept means.
An intercept does not have to be zero.

4. Guided example and independent practice

When you explain a graph, connect its features to the quantities shown. First identify the axes. Then describe the change between points to explain slope. Finally, read where the line crosses an axis to explain the intercept. The worked example uses points that are supplied on a straight-line graph.

For independent practice, explain each feature in words using the graph’s context. A graph of cost against the number of identical items rises from left to right. What does its slope describe? A graph of temperature against time is horizontal. What does that tell you about temperature during the time shown? A line crosses the vertical axis at 4 on a graph where the vertical axis shows height in metres. What height is shown when the horizontal value is zero?

In each response, name the quantity and include its units when they are given. Avoid describing a graph only as “up,” “down,” or “at 4.” A clear explanation tells what changes, what stays the same, or what value is shown at an intercept.

Use two points on a straight line to compare vertical and horizontal changes.
Read the vertical intercept where the horizontal value is zero.
Explain what a graph feature means in the situation, not only what it is called.

Reading common graph features

Feature	What to look for	Meaning
Slope	Compare vertical and horizontal changes between points	How much the vertical quantity changes as the horizontal quantity changes
Vertical intercept	Where the line crosses the vertical axis	The vertical value when the horizontal value is zero
Horizontal intercept	Where the line crosses the horizontal axis	The horizontal value when the vertical value is zero
Horizontal line	The line stays at one vertical value	The vertical quantity does not change

Worked example

Interpreting a cycling graph

A straight-line graph shows a cyclist’s distance from a starting point, in kilometres, against time, in minutes. The line passes through

(0,2)

and

(3,11)

. Explain the slope and the vertical intercept.

Identify the quantities
Time is on the horizontal axis, and distance is on the vertical axis. The two points show the cyclist’s distance at two different times.
$(0,2),\ (3,11)$
Compare the changes
From the first point to the second, time increases by 3 minutes and distance increases by 9 kilometres. Compare the distance change with the time change to describe the slope.
$\frac{11-2}{3-0}=\frac{9}{3}=3$
Explain the slope
The slope is 3 kilometres per minute. In this graph, the cyclist’s distance from the starting point increases by 3 kilometres for each minute of time shown.
$3\text{ km/min}$
Read the vertical intercept
The point $(0,2)$ is on the vertical axis because its horizontal value is zero. It shows a distance of 2 kilometres at time zero.
$2\text{ km}$

Answer: The slope is 3 kilometres per minute. The vertical intercept is 2 kilometres, meaning the graph shows the cyclist 2 kilometres from the starting point at time zero.

Check: The distance increases from 2 to 11 kilometres, a change of 9 kilometres, while time increases by 3 minutes. The change is therefore 3 kilometres per minute. The point with time zero gives the vertical intercept.

Common mistakes and how to avoid them

Calling the vertical intercept the slope.

Correction: The intercept is where the line crosses an axis. Slope describes change along the line.

Comparing horizontal change with vertical change when describing slope.

Correction: Compare vertical change with horizontal change. Then use the axis units to state the rate clearly.

Assuming the vertical intercept must be zero.

Correction: Read the value where the line actually crosses the vertical axis. It may be a non-zero value.

Giving a slope without saying what it measures.

Correction: Use the graph’s labels and units. For example, say kilometres per minute rather than only giving a number.

Lesson summary

Slope describes how the vertical quantity changes compared with the horizontal quantity.
A line rising from left to right has positive slope; a falling line has negative slope; a horizontal line has zero slope.
The vertical intercept is where the graph crosses the vertical axis, where the horizontal value is zero.
The horizontal intercept is where the graph crosses the horizontal axis, where the vertical value is zero.
Use the axis labels and units to explain what slope and intercept mean in context.

Check your understanding

Question 1

A graph shows water remaining, in litres, against time, in minutes. The line falls from left to right. What does this tell you?

The amount of water decreases as time increases.
The amount of water increases as time increases.
The amount of water stays constant as time increases.
The graph’s vertical intercept must be zero.

Show answer and explanation

The amount of water decreases as time increases.

As you move right, time increases. The falling line shows that the vertical value, the amount of water, decreases.

Question 2

A line crosses the vertical axis at 7 on a graph of distance in metres against time in seconds. What does the vertical intercept mean?

The distance changes by 7 metres each second.
At time zero, the distance is 7 metres.
At distance zero, the time is 7 seconds.
The distance stays at 7 metres for all times.

Show answer and explanation

At time zero, the distance is 7 metres.

At the vertical intercept, the horizontal value is zero. Since time is on the horizontal axis, the graph shows a distance of 7 metres at time zero.

Question 3

On a straight-line graph, the vertical value increases by 12 units while the horizontal value increases by 4 units. What is the slope?

3 vertical units per horizontal unit
One-third of a vertical unit per horizontal unit
8 vertical units per horizontal unit
16 vertical units per horizontal unit

Show answer and explanation

3 vertical units per horizontal unit

Compare the vertical change with the horizontal change: 12 divided by 4 is 3. The slope is 3 vertical units per horizontal unit.

Key terms

Axis: One of the number lines on a coordinate graph.
Slope: A measure of how much the vertical value changes compared with the horizontal value.
Rate of change: How much one quantity changes as another quantity changes.
Intercept: A point where a graph meets an axis.
Vertical intercept: Where a graph crosses the vertical axis; the horizontal value there is zero.
Horizontal intercept: Where a graph crosses the horizontal axis; the vertical value there is zero.
Origin: The point where the two axes meet, written as $(0,0)$ .

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