Gradient of a Line | IGCSE Maths Revision

Syllabus coverage

Find the gradient of a straight line, reading it from a grid
Extended only: calculate the gradient of a straight line from the coordinates of two points on it
Find the gradient of a graph from its equation, for example the gradient of $y = 6 x + 3$
Recognise a positive, a negative, a zero and an undefined gradient
Interpret the gradient as a rate of change when the graph is in a practical context

Gradient as rise over run on y = 2x + 1, with a shallower line of gradient 0.5 drawn on the same axes for comparison.

What the gradient measures

The gradient of a straight line measures its steepness: how far the line rises or falls for each unit you travel across. Write it as a fraction, rise over run, and simplify it. A line that climbs 6 units while travelling 4 units across has gradient $\frac{6}{4} = \frac{3}{2}$ , which is also 1.5.

The sign is part of the answer. A line that goes up from left to right has a positive gradient, and a line that comes down has a negative gradient. Always read the line from left to right, because the same line read the other way would appear to have the opposite sign.

The gradient does not depend on which part of the line you measure. Any two points on a straight line give the same rise over run, and that is exactly what makes the line straight.

Gradient = rise divided by run = the vertical change divided by the horizontal change
Read the line from left to right
Up to the right is positive, down to the right is negative
Give the answer as a simplified fraction or an exact decimal

A gradient of

\frac{3}{2}

and a gradient of 1.5 are the same answer, and both score. A gradient of 2 is steeper than a gradient of

\frac{1}{2}

, because each unit across climbs four times as far.

Finding the gradient from a grid

Choose two points on the line that sit exactly on grid intersections, as far apart as possible. Draw a right-angled triangle between them, with one horizontal side and one vertical side meeting at a right angle. The vertical side is the rise and the horizontal side is the run.

Count squares only when one square is one unit on both axes. Otherwise read the values off the axis labels. If the $x$ -axis is numbered every 5 units and the run is 3 squares, the run is 15, not 3.

Working from left to right makes the sign automatic: if the second point is higher than the first, the rise is positive, and if it is lower, the rise is negative. The run is always positive when you work left to right.

The triangle must be right-angled, with both points on the line
The vertical side is the rise and the horizontal side is the run
Read the axis labels rather than the squares when one square is not one unit
A line that falls has a negative rise, so its gradient is negative

Pick points far apart. On a line of gradient 2, two points one square apart can be misread by a square and give a wildly wrong answer; points six squares apart give 12 over 6 and any error is obvious.

Finding the gradient from two points

When the coordinates of two points are given, the gradient is $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ : the difference in the $y$ -coordinates divided by the difference in the $x$ -coordinates. For $(2, 3)$ and $(8, 15)$ , $m = \frac{15 - 3}{8 - 2} = \frac{12}{6} = 2$ .

Subtract in the same order both times. Taking the second point minus the first for the $y$ values and the first minus the second for the $x$ values flips the sign of the answer. Either order works, as long as both differences run the same way.

A negative answer is not a mistake: it means the line falls from left to right. For $(1, 9)$ and $(4, - 3)$ , $m = \frac{- 3 - 9}{4 - 1} = \frac{- 12}{3} = - 4$ .

Points	Change in $y$	Change in $x$	Gradient
$(2, 3)$ and $(8, 15)$	$15 - 3 = 12$	$8 - 2 = 6$	2
$(1, 9)$ and $(4, - 3)$	$- 3 - 9 = - 12$	$4 - 1 = 3$	-4
$(0, 5)$ and $(4, 1)$	$1 - 5 = - 4$	$4 - 0 = 4$	-1
$(- 2, - 1)$ and $(2, 7)$	$7 - (- 1) = 8$	$2 - (- 2) = 4$	2

Reading the gradient from the equation

In $y = m x + c$ the gradient is the number multiplying $x$ . So $y = 6 x + 3$ has gradient 6, and $y = 7 - 4 x$ has gradient $- 4$ , because the $x$ -term is $- 4 x$ .

If $x$ has no number in front of it the gradient is 1, as in $y = x + 5$ . If the terms are the other way round the gradient is unchanged, because $y = 2 + 3 x$ and $y = 3 x + 2$ are the same line.

A horizontal line $y = c$ has gradient 0, because the rise is 0 however far you travel across. A vertical line $x = k$ has no gradient at all: the run is 0 and division by zero is undefined.

The gradient is the coefficient of $x$ in $y = m x + c$
$y = 6 x + 3$ has gradient 6 and intercept 3
$y = 7 - 4 x$ has gradient $- 4$
$y = c$ has gradient 0 and $x = k$ has an undefined gradient

Extended: if

y

is not on its own, rearrange first.

3 x + 2 y = 5

becomes

2 y = 5 - 3 x

, so

y = \frac{5 - 3 x}{2}

and the gradient is

- \frac{3}{2}

Positive, negative, zero and undefined gradients

Four cases cover every straight line you will meet. A positive gradient climbs from left to right. A negative gradient falls from left to right. A gradient of 0 belongs to a horizontal line. An undefined gradient belongs to a vertical line.

The size of the gradient is the steepness, and the sign is the direction. A gradient of 5 is much steeper than a gradient of 0.5, but the second may still be the steeper-looking line on the page if the two axes use different scales, so trust the numbers rather than the picture.

Two lines with the same gradient are parallel, whatever their intercepts. That is why the gradient is the number worth finding first: once you have it, the direction and the steepness of the line are both settled.

Gradient	The line	Example
positive	rises from left to right	$y = 2 x + 1$
negative	falls from left to right	$y = - 3 x + 4$
0	horizontal	$y = 5$
undefined	vertical	$x = - 2$

A vertical line is the only line whose gradient is undefined, and the only kind of line that cannot be written in the form

y = m x + c

What a gradient means in context

When the axes carry real quantities, the gradient is a rate: how much the vertical quantity changes for each unit of the horizontal one. On a distance-time graph with distance in kilometres and time in hours, the gradient is a speed in km/h. On a graph of cost against quantity it is the price of one item.

The units come from the axes, so write them down. A gradient of 0.4 on a graph of kilometres against minutes means 0.4 km per minute, which is 24 km/h. The same number on a different pair of axes would mean something else entirely.

A gradient can also describe steepness directly. A ramp that rises 0.75 m over a horizontal run of 3 m has gradient $\frac{0.75}{3} = 0.25$ , so it rises 1 m for every 4 m travelled along. In context a negative gradient means the quantity is falling, as with the depth of water in a draining tank.

Reading a gradient off a real graph works exactly like reading one off a plain grid, but the answer needs units. On a graph of a car's value against its age, a gradient of $- 1500$ dollars per year means the value drops by 1500 dollars each year: the negative sign says the value is falling and the 1500 says how quickly.

Distance-time graph: the gradient is the speed
Cost against quantity: the gradient is the price of one item
A negative gradient means the vertical quantity is falling
Attach the units the axes give you to every gradient in context

The line

y = 2 x - 3

. Stepping one unit to the right from

(0, - 3)

and two units up lands back on the line, so the gradient is 2.

Key terms you must be able to define

Gradient: How steep a straight line is: the vertical change divided by the horizontal change, rise over run.

Rise: The vertical change between two points on a line, positive if the line climbs and negative if it falls.

Run: The horizontal change between two points on a line, counted from left to right so that it is positive.

Positive gradient: A line that rises from left to right, with a gradient greater than zero.

Negative gradient: A line that falls from left to right, with a gradient less than zero.

Zero gradient: The gradient of a horizontal line: the rise is 0, so the gradient is 0.

Undefined gradient: The gradient of a vertical line: the run is 0, and division by zero is not defined.

y-intercept: The value of $y$ where a line crosses the $y$ -axis, written $c$ in $y = m x + c$ .

y = mx + c: The form of the equation of a straight line, where $m$ is the gradient and $c$ is the $y$ -intercept.

Rate of change: A gradient read as a rate, with the units taken from the two axes of the graph.

Parallel lines: Lines with the same gradient, so they point the same way and never meet.

Line segment: A straight piece of a line between two given points, usually named by its endpoints.

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

From a graph

On the graph of $y = 2 x - 3$ , take the points $(1, - 1)$ and $(3, 3)$ . The run is $3 - 1 = 2$ and the rise is $3 - (- 1) = 4$ , so the gradient is $\frac{4}{2} = 2$ .

From two points

The gradient of the line through $(2, 1)$ and $(6, 9)$ is $\frac{9 - 1}{6 - 2} = \frac{8}{4} = 2$ . Subtracting the other way round gives $\frac{1 - 9}{2 - 6} = \frac{- 8}{- 4} = 2$ , the same answer, because both differences changed sign together.

A negative gradient

The line through $(0, 7)$ and $(4, - 1)$ has gradient $\frac{- 1 - 7}{4 - 0} = \frac{- 8}{4} = - 2$ , so it falls 2 units for every 1 unit across.

From the equation

$y = 6 x + 3$ has gradient 6 and intercept 3. Extended: $3 x + 2 y = 5$ rearranges to $2 y = 5 - 3 x$ , so $y = 2.5 - 1.5 x$ and the gradient is $- 1.5$ , which is $- \frac{3}{2}$ .

In context

A taxi firm's graph of fare against distance is a straight line through $(0, 3)$ and $(5, 13)$ . Its gradient is $\frac{13 - 3}{5 - 0} = 2$ , so the fare rises by 2 dollars for every kilometre travelled, and the intercept 3 is the fixed charge.

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1How do you work out the gradient of a straight line?

Divide the rise by the run: the vertical change divided by the horizontal change.

Q2What is the gradient of a horizontal line?

0. The rise is zero however far across you go, so the line is flat.

Q3What is the gradient of a vertical line?

Undefined. The run is 0, and you cannot divide by 0.

Q4What is the gradient of the line through

(2, 3)

and

(8, 15)

2, because

\frac{15 - 3}{8 - 2} = \frac{12}{6} = 2

Q5What is the gradient of

y = 7 - 4 x

- 4

. The

x

-term is

- 4 x

, so the sign travels with the number.

Q6What is the gradient of

y = 6 x + 3

6. The gradient is the coefficient of

x

; the 3 is the

y

-intercept.

Q7What does a negative gradient look like?

A line that falls from left to right.

Q8Which is steeper, a gradient of 3 or a gradient of

\frac{1}{3}

A gradient of 3: it climbs 3 units for every 1 across, against 1 unit for every 3 across.

Q9What does the gradient represent on a distance-time graph?

The speed: the change in distance divided by the change in time, in km/h or m/s.

Q10How do you find the gradient of a line from its equation?

Rearrange it into the form

y = m x + c

if necessary, then read off

m

, the number multiplying

x

Q11What is the gradient of the line

3 x + 2 y = 5

- \frac{3}{2}

. Rearrange to

y = \frac{5 - 3 x}{2}

, so

m = - \frac{3}{2}

Q12What does it mean if two lines have the same gradient?

They are parallel: the same steepness and the same direction, so they never meet.

Q13A ramp rises 0.6 m over a horizontal run of 4 m. What is its gradient?

0.15, because

0.6 \div 4 = 0.15

. The ramp rises 1 m for roughly every 6.7 m along.

0 of 13 marked as known

Exam-style questions

7 questions, 16 marks. Write your answer, then reveal the mark scheme and tick each point you actually made. Answering honestly here is what makes the rest of the site useful.

Write downAO11 mark

Write down the gradient of the straight line with equation

y = 6 x + 3

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

B16, the coefficient of $x$ in $y = m x + c$ not checked yet

Examiner tip: In

y = m x + c

the gradient is the number in front of

x

, and the intercept is the number standing on its own. Here the gradient is 6 and the 3 is the intercept.

Work outAO12 marks

The line

L

passes through the points

(2, 3)

and

(8, 15)

. Work out the gradient of

L

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

M1the change in $y$ is $15 - 3 = 12$ and the change in $x$ is $8 - 2 = 6$
A1 $\frac{12}{6} = 2$ not checked yet

Examiner tip: Write the two differences down before dividing. Both must be worked out in the same order, here second point minus first point.

Work outAO13 marks

The straight line

P Q

passes through

P (1, 9)

and

Q (4, - 3)

. Work out the gradient of

P Q

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

M1attempts the change in $y$ divided by the change in $x$
M1 $- 3 - 9 = - 12$ and $4 - 1 = 3$ , subtracting in the same order for both
A1 $\frac{- 12}{3} = - 4$ not checked yet

Examiner tip: A negative answer is expected here: the line falls from left to right, because the second

y

-coordinate is smaller than the first.

Work outAO22 marks

A straight ramp rises 1.2 m over a horizontal distance of 4.8 m. Work out the gradient of the ramp.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

M1uses rise divided by run: $1.2 \div 4.8$
A10.25not checked yet

Examiner tip: Both measurements are in metres, so the units cancel and the gradient is a plain number. A gradient of 0.25 means the ramp rises 1 m for every 4 m travelled along.

Write downAO12 marks

The points

(3, 5)

and

(7, 5)

lie on a straight line. Write down the gradient of the line.

Your answer

Write a short answer in words.

Mark scheme — tick each point you actually made

B1the two points have the same $y$ -coordinate, so the rise is 0
B1gradient 0, so the line is horizontalnot checked yet

Examiner tip: Equal

y

-coordinates mean a horizontal line, so the gradient is 0. Equal

x

-coordinates mean a vertical line, and that gradient is undefined instead.

DetermineAO12 marks

A straight line passes through the points

(4, - 1)

and

(4, 6)

. Determine the gradient of the line.

Your answer

Write a short answer in words.

Mark scheme — tick each point you actually made

M1the change in $x$ is $4 - 4 = 0$
B1undefined, because dividing by 0 is not definednot checked yet

Examiner tip: Both points have

x = 4

, so the line is vertical. Any attempt to work out the gradient ends in division by zero, so the answer is 'undefined' rather than 0.

Work outAO24 marks

A boat hire firm charges a fixed amount plus a charge for each hour. The graph of the total cost, in dollars, against the number of hours

h

is a straight line through the points

(0, 20)

and

(4, 60)

. Work out the gradient of the line, and state what the gradient represents in this context.

Your answer

Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

B1the change in cost is $60 - 20 = 40$ dollars
M1the change in time is $4 - 0 = 4$ hours, and $40 \div 4 = 10$
B1the gradient is the cost of each extra hour: 10 dollars per hour
A110not checked yet

Examiner tip: The gradient is in dollars per hour, because the axes are dollars and hours. Read the two points off the graph first, then write the change in cost over the change in time.

Exam technique for this topic

Draw the right-angled triangle on the graph and write the rise and the run beside it before dividing. The two differences are worth a method mark, and they show where the sign of the answer came from.
Use points that sit exactly on grid intersections and are as far apart as possible. Two points close together turn a small counting error into a large error in the gradient.
Check the scale on each axis before counting squares. If one square is 2 units across and 5 units up, counting squares gives a gradient that is wrong by a factor of 2.5.
Subtract in the same order for the $y$ values and for the $x$ values. Mixing the orders flips the sign, and a sign error loses the accuracy mark even when the method is right.
Leave the gradient as a simplified fraction when it does not come out exactly, and give the units whenever the axes carry them - a gradient of 0.4 on a distance-time graph is a speed, not just a number.

Why students lose marks here

A student wroteThe gradient is the run divided by the rise.

Why it loses marksThat is the reciprocal: the horizontal change over the vertical change. On a line that rises 4 while travelling 2 across it gives

\frac{1}{2}

instead of 2.

The answer that scoresGradient = rise divided by run. The line through

(1, 3)

and

(3, 7)

has run 2 and rise 4, so the gradient is

\frac{4}{2} = 2

A student wroteThe gradient of the line through

(1, 4)

and

(5, 2)

is 0.5.

Why it loses marksThe line falls from left to right, so the gradient must be negative. The student has worked out

\frac{4 - 2}{5 - 1}

, subtracting the

y

values in the opposite order to the

x

values.

The answer that scores

\frac{2 - 4}{5 - 1} = \frac{- 2}{4} = - 0.5

. Subtracting in the same order on both rows keeps the sign right.

A student wroteThe gradient of the vertical line

x = 3

is 0.

Why it loses marksOn

x = 3

the run between any two points is 0, and dividing by 0 is undefined. It is a horizontal line that has gradient 0.

The answer that scores

x = 3

has an undefined gradient, because the change in

x

is 0. The line

y = 3

is the one with gradient 0.

A student wroteThe gradient of

y = 6 x + 3

is 3.

Why it loses marksThe 3 is the

y

-intercept, the value of

y

when

x = 0

. The gradient is the number multiplying

x

The answer that scores

y = 6 x + 3

has gradient 6 and

y

-intercept 3: the line crosses the

y

-axis at

(0, 3)

and rises 6 for every 1 across.