Gradient of a Line

What you'll learn

What the gradient of a straight line means.
How to find gradient from a graph using rise and run.
How to calculate gradient from two coordinate points.
How to use a given gradient to find a missing value such as $k$ .

Coordinates: the starting point

A point on a graph is written as an ordered pair, like $(3, 5)$ .

The first number is the $x$ -coordinate: how far left or right.
The second number is the $y$ -coordinate: how far up or down.

So $(3, 5)$ means 3 across and 5 up.
A straight line has many points on it, but you only need two points to find its gradient.

Example

Choosing two points from a line

A straight line passes through the clear grid points $(- 1, - 4)$ , $(0, - 1)$ and $(1, 2)$ . Find two points you could use to find its gradient.

Look for points where the line crosses exact grid intersections.
Choose any two clear points on the line, for example $(- 1, - 4)$ and $(1, 2)$ .
It is usually best to choose points that are not too close together, because the change is easier to see.

Tip

Use clear grid points

When finding gradient from a graph, do not guess points between squares. Pick two points where the line passes exactly through grid intersections.

What gradient means

Definition

Gradient

The gradient of a line is a measure of how steep it is. It tells you how much the $y$ -value changes for each increase of 1 in the $x$ -value.

You may hear gradient described as:

gradient=riserun\text{gradient}=\frac{\text{rise}}{\text{run}}

The rise is the vertical change.
The run is the horizontal change.

The diagram shows a line where the run is +3 and the rise is +3, so the gradient is 1.

A coordinate grid showing a straight line through A(1,2) and B(4,5), with run +3 and rise +3 marked.

Key Idea

Rise over run

To find the gradient, move from one point on the line to another: count the vertical change, then divide by the horizontal change.

Example

Finding gradient from a graph

A line passes through the points $(- 2, - 3)$ and $(0, 5)$ . Find its gradient.

Choose two clear points on the line: $(- 2, - 3)$ and $(0, 5)$ .
Find the change in $y$ : from -3 to 5 is an increase of 8, so the rise is 8.
Find the change in $x$ : from -2 to 0 is an increase of 2, so the run is 2.
Divide rise by run: $gradient=82=4\text{gradient}=\frac{8}{2}=4$ .

Positive, negative and fractional gradients

A line with a positive gradient slopes upwards from left to right.

A line with a negative gradient slopes downwards from left to right.

A gradient can also be a fraction. For example, if the line goes up 1 square for every 3 squares across, the gradient is $13\frac{1}{3}$ .

The diagram shows a negative gradient: as you move right, the line goes down.

A coordinate grid showing a downward sloping line through C(1,4) and D(4,-2), with run +3 and rise -6 marked.

Example

A negative gradient from a graph

A line passes through the points $(- 3, 4)$ and $(3, 2)$ . Find its gradient.

Choose the two points $(- 3, 4)$ and $(3, 2)$ .
Find the change in $y$ : from 4 to 2 is a decrease of 2, so the rise is -2.
Find the change in $x$ : from -3 to 3 is an increase of 6, so the run is 6.
Divide rise by run: $gradient=−26=−13\text{gradient}=\frac{-2}{6}=-\frac{1}{3}$ .

Common Mistake

Forgetting the negative sign

If the line slopes down from left to right, the gradient must be negative. A common error is to count the vertical change as positive even though the line goes down.

Calculating gradient from two coordinates

When you are not given a graph, use the coordinate formula.

If the two points are $x_1, y_1)$ and $x_2, y_2)$ , then:

gradient=y2−y1x2−x1\text{gradient}=\frac{y_2-y_1}{x_2-x_1}

This means:

subtract the $y$ -coordinates to find the vertical change;
subtract the $x$ -coordinates to find the horizontal change;
divide the vertical change by the horizontal change.

Example

Gradient from two points

Find the gradient of the line passing through $(2, 3)$ and $(6, 15)$ .

Label the coordinates: $x_1=2$ , $y_1=3$ , $x_2=6$ , $y_2=15$ .
Subtract the $y$ -coordinates: $15 - 3 = 12$ .
Subtract the $x$ -coordinates: $6 - 2 = 4$ .
Divide: $gradient=124=3\text{gradient}=\frac{12}{4}=3$ .

Example

Gradient with negative coordinates

Find the gradient of the line passing through $(- 4, 7)$ and $(2, - 5)$ .

Label the coordinates carefully: $x_1=-4$ , $y_1=7$ , $x_2=2$ , $y_2=-5$ .
Find the change in $y$ : $- 5 - 7 = - 12$ .
Find the change in $x$ : $2 - (- 4) = 6$ .
Divide: $gradient=−126=−2\text{gradient}=\frac{-12}{6}=-2$ .

Common Mistake

Mixing up the order

If you start with the second point on the top, you must also start with the second point on the bottom. For example, $y2−y1x2−x1\frac{y_2-y_1}{x_2-x_1}$ is fine, and $y1−y2x1−x2\frac{y_1-y_2}{x_1-x_2}$ is also fine, but do not mix them.

Horizontal and vertical lines

A horizontal line is flat. Its $y$ -value does not change, so its gradient is 0.

A vertical line goes straight up and down. Its $x$ -value does not change, so the formula would involve dividing by 0, which is not allowed.

Example

A horizontal line

Find the gradient of the line passing through $(- 2, 4)$ and $(5, 4)$ .

Notice that both points have the same $y$ -coordinate, 4.
Find the change in $y$ : $4 - 4 = 0$ .
Find the change in $x$ : $5 - (- 2) = 7$ .
Divide: $gradient=07=0\text{gradient}=\frac{0}{7}=0$ .

Common Mistake

Vertical lines

A vertical line has no gradient because the change in $x$ is 0, and you cannot divide by 0. Do not write its gradient as 0; only horizontal lines have gradient 0.

Finding a missing value using gradient

Sometimes you are given the gradient and one coordinate contains an unknown, such as $k$ .

Use the same formula, substitute what you know, then solve the equation.

Example

Finding a missing y-coordinate

The line through $A (3, - 2)$ and $B (7, k)$ has gradient 4. Find $k$ .

Write the gradient formula using the two points: $gradient=k−(−2)7−3\text{gradient}=\frac{k-(-2)}{7-3}$ .
Replace the gradient with 4: $4=k+244=\frac{k+2}{4}$ .
Multiply both sides by 4: $16 = k + 2$ .
Subtract 2 from both sides: $k = 14$ .

Example

Finding a missing x-coordinate

The line through $(- 1, 6)$ and $(k, 18)$ has gradient 3. Find $k$ .

Put the known values into the formula: $3=18−6k−(−1)3=\frac{18-6}{k-(-1)}$ .
Simplify the top and the bottom: $3=12k+13=\frac{12}{k+1}$ .
Multiply both sides by $k + 1$ : $3 (k + 1) = 12$ .
Divide by 3: $k + 1 = 4$ .
Subtract 1: $k = 3$ .

Tip

Check your answer makes sense

After finding $k$ , imagine moving from the first point to the second. If the given gradient is positive, the overall movement should be up as you move right. If the given gradient is negative, it should go down as you move right.

Exam technique

In the exam

Pick two clear points if you are using a graph; avoid estimating from unclear parts of the line.
Use $x\text{gradient}=\frac{\text{change in }y}{\text{change in }x}$ and keep the subtraction order consistent.
Check the sign: upward left-to-right means positive, downward left-to-right means negative.

Self review

Check yourself

If a line goes up 6 squares while going right 2 squares, what is its gradient?
How can you tell from the graph whether a gradient should be negative?
What formula would you use to find the gradient between $x_1, y_1)$ and $x_2, y_2)$ ?

Gradient of a Line

What you'll learn

What the gradient of a straight line means.
How to find gradient from a graph using rise and run.
How to calculate gradient from two coordinate points.
How to use a given gradient to find a missing value such as $k$ .

Coordinates: the starting point

A point on a graph is written as an ordered pair, like $(3, 5)$ .

The first number is the $x$ -coordinate: how far left or right.
The second number is the $y$ -coordinate: how far up or down.

So $(3, 5)$ means 3 across and 5 up.
A straight line has many points on it, but you only need two points to find its gradient.

Example

Choosing two points from a line

A straight line passes through the clear grid points $(- 1, - 4)$ , $(0, - 1)$ and $(1, 2)$ . Find two points you could use to find its gradient.

Look for points where the line crosses exact grid intersections.
Choose any two clear points on the line, for example $(- 1, - 4)$ and $(1, 2)$ .
It is usually best to choose points that are not too close together, because the change is easier to see.

Tip

Use clear grid points

When finding gradient from a graph, do not guess points between squares. Pick two points where the line passes exactly through grid intersections.

What gradient means

Definition

Gradient

The gradient of a line is a measure of how steep it is. It tells you how much the $y$ -value changes for each increase of 1 in the $x$ -value.

You may hear gradient described as:

gradient=riserun\text{gradient}=\frac{\text{rise}}{\text{run}}

The rise is the vertical change.
The run is the horizontal change.

The diagram shows a line where the run is +3 and the rise is +3, so the gradient is 1.

A coordinate grid showing a straight line through A(1,2) and B(4,5), with run +3 and rise +3 marked.

Key Idea

Rise over run

To find the gradient, move from one point on the line to another: count the vertical change, then divide by the horizontal change.

Example

Finding gradient from a graph

A line passes through the points $(- 2, - 3)$ and $(0, 5)$ . Find its gradient.

Choose two clear points on the line: $(- 2, - 3)$ and $(0, 5)$ .
Find the change in $y$ : from -3 to 5 is an increase of 8, so the rise is 8.
Find the change in $x$ : from -2 to 0 is an increase of 2, so the run is 2.
Divide rise by run: $gradient=82=4\text{gradient}=\frac{8}{2}=4$ .

Positive, negative and fractional gradients

A line with a positive gradient slopes upwards from left to right.

A line with a negative gradient slopes downwards from left to right.

A gradient can also be a fraction. For example, if the line goes up 1 square for every 3 squares across, the gradient is $13\frac{1}{3}$ .

The diagram shows a negative gradient: as you move right, the line goes down.

A coordinate grid showing a downward sloping line through C(1,4) and D(4,-2), with run +3 and rise -6 marked.

Example

A negative gradient from a graph

A line passes through the points $(- 3, 4)$ and $(3, 2)$ . Find its gradient.

Choose the two points $(- 3, 4)$ and $(3, 2)$ .
Find the change in $y$ : from 4 to 2 is a decrease of 2, so the rise is -2.
Find the change in $x$ : from -3 to 3 is an increase of 6, so the run is 6.
Divide rise by run: $gradient=−26=−13\text{gradient}=\frac{-2}{6}=-\frac{1}{3}$ .

Common Mistake

Forgetting the negative sign

If the line slopes down from left to right, the gradient must be negative. A common error is to count the vertical change as positive even though the line goes down.

Calculating gradient from two coordinates

When you are not given a graph, use the coordinate formula.

If the two points are $x_1, y_1)$ and $x_2, y_2)$ , then:

gradient=y2−y1x2−x1\text{gradient}=\frac{y_2-y_1}{x_2-x_1}

This means:

subtract the $y$ -coordinates to find the vertical change;
subtract the $x$ -coordinates to find the horizontal change;
divide the vertical change by the horizontal change.

Example

Gradient from two points

Find the gradient of the line passing through $(2, 3)$ and $(6, 15)$ .

Label the coordinates: $x_1=2$ , $y_1=3$ , $x_2=6$ , $y_2=15$ .
Subtract the $y$ -coordinates: $15 - 3 = 12$ .
Subtract the $x$ -coordinates: $6 - 2 = 4$ .
Divide: $gradient=124=3\text{gradient}=\frac{12}{4}=3$ .

Example

Gradient with negative coordinates

Find the gradient of the line passing through $(- 4, 7)$ and $(2, - 5)$ .

Label the coordinates carefully: $x_1=-4$ , $y_1=7$ , $x_2=2$ , $y_2=-5$ .
Find the change in $y$ : $- 5 - 7 = - 12$ .
Find the change in $x$ : $2 - (- 4) = 6$ .
Divide: $gradient=−126=−2\text{gradient}=\frac{-12}{6}=-2$ .

Common Mistake

Mixing up the order

Horizontal and vertical lines

A horizontal line is flat. Its $y$ -value does not change, so its gradient is 0.

A vertical line goes straight up and down. Its $x$ -value does not change, so the formula would involve dividing by 0, which is not allowed.

Example

A horizontal line

Find the gradient of the line passing through $(- 2, 4)$ and $(5, 4)$ .

Notice that both points have the same $y$ -coordinate, 4.
Find the change in $y$ : $4 - 4 = 0$ .
Find the change in $x$ : $5 - (- 2) = 7$ .
Divide: $gradient=07=0\text{gradient}=\frac{0}{7}=0$ .

Common Mistake

Vertical lines

A vertical line has no gradient because the change in $x$ is 0, and you cannot divide by 0. Do not write its gradient as 0; only horizontal lines have gradient 0.

Finding a missing value using gradient

Sometimes you are given the gradient and one coordinate contains an unknown, such as $k$ .

Use the same formula, substitute what you know, then solve the equation.

Example

Finding a missing y-coordinate

The line through $A (3, - 2)$ and $B (7, k)$ has gradient 4. Find $k$ .

Write the gradient formula using the two points: $gradient=k−(−2)7−3\text{gradient}=\frac{k-(-2)}{7-3}$ .
Replace the gradient with 4: $4=k+244=\frac{k+2}{4}$ .
Multiply both sides by 4: $16 = k + 2$ .
Subtract 2 from both sides: $k = 14$ .

Example

Finding a missing x-coordinate

The line through $(- 1, 6)$ and $(k, 18)$ has gradient 3. Find $k$ .

Put the known values into the formula: $3=18−6k−(−1)3=\frac{18-6}{k-(-1)}$ .
Simplify the top and the bottom: $3=12k+13=\frac{12}{k+1}$ .
Multiply both sides by $k + 1$ : $3 (k + 1) = 12$ .
Divide by 3: $k + 1 = 4$ .
Subtract 1: $k = 3$ .

Tip

Check your answer makes sense

Exam technique

In the exam

Pick two clear points if you are using a graph; avoid estimating from unclear parts of the line.
Use $x\text{gradient}=\frac{\text{change in }y}{\text{change in }x}$ and keep the subtraction order consistent.
Check the sign: upward left-to-right means positive, downward left-to-right means negative.

Self review

Check yourself

If a line goes up 6 squares while going right 2 squares, what is its gradient?
How can you tell from the graph whether a gradient should be negative?
What formula would you use to find the gradient between $x_1, y_1)$ and $x_2, y_2)$ ?

Gradient of a Line

What you'll learn

Coordinates: the starting point

What gradient means

Positive, negative and fractional gradients

Calculating gradient from two coordinates

Horizontal and vertical lines

Finding a missing value using gradient

Recap questions

You've reached the end

Gradient of a Line

What you'll learn

Coordinates: the starting point

What gradient means

Positive, negative and fractional gradients

Calculating gradient from two coordinates

Horizontal and vertical lines

Finding a missing value using gradient

Recap questions

You've reached the end

Gradient of a Line

What you'll learn

Coordinates: the starting point

What gradient means

Positive, negative and fractional gradients

Calculating gradient from two coordinates

Horizontal and vertical lines

Finding a missing value using gradient

Recap questions

You've reached the end

Gradient of a Line Revision Guide

Gradient of a Line

What you'll learn

Coordinates: the starting point

What gradient means

Positive, negative and fractional gradients

Calculating gradient from two coordinates

Horizontal and vertical lines

Finding a missing value using gradient

Recap questions

You've reached the end

Gradient of a Line Revision Guide