Finding the Equation From a Graph

Read where a drawn line crosses the y-axis and how steep it is, then write its equation as y = mx + c.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 11 real past-paper questions below

Two Things to Read

Every straight line (except an upright one) has an equation of this form:

Formula

y = m x + c

To find it from a graph you read just two numbers:

Letter	What it is	Where to look
$m$	the gradient: how steep	up ÷ across between two points
$c$	the y-intercept	where the line crosses the y-axis

Key idea

For more on what m and c mean, see The Equation y = mx + c.

Read c, Then m

The 2–3 mark question: a line is drawn on a grid, find its equation. Pick two points where the line crosses grid corners exactly.

Worked example

1
$crosses the y-axis at 2$ — Find where the line cuts the upright axis. That height is c, so c = 2.
2
$(0, 2) and (2, - 4)$ — Pick two points where the line passes exactly through a grid corner. Exact points give an exact gradient.
3
$up = - 4 - 2 = - 6$ — y goes from 2 down to −4. Going down means up is negative: −6.
4
$across = 2 - 0 = 2$ — x goes from 0 to 2, so the line moves 2 to the right.
5
$m = \frac{- 6}{2} = - 3$ — Gradient = up ÷ across. Negative, which matches a line going down to the right.
6
$y = - 3 x + 2$ — Put m in front of x and c on its own, as in $y = m x + c$ .
7
$x = 2 : - 3 \times 2 + 2 = - 4$ — Check with the second point. It gives y = −4, so the equation fits.

Common mistake

Writing $y = 3 x + 2$ . The line goes down to the right, so the gradient must be negative.

Key idea

Show up ÷ across in your working, and write the answer as $y = \dots$ , not $L = \dots$ . A correct gradient or a correct c still earns a mark, even if the other one is wrong.

When the Gradient Is a Fraction

Some lines rise slowly. Count across to the next exact grid corner: the gradient is then a fraction.

Worked example

1
$c = - 1$ — The line cuts the y-axis below 0, at −1. So c = −1, minus sign included.
2
$(0, - 1) and (3, 1)$ — Move right from (0, −1) until the line passes exactly through a grid corner. The first one is (3, 1).
3
$up = 1 - (- 1) = 2$ — y goes from −1 up to 1. Taking away a negative adds, so 1 + 1 = 2.
4
$across = 3 - 0 = 3$ — x goes from 0 to 3, so the line moves 3 to the right.
5
$m = \frac{2}{3}$ — Up ÷ across. 2 does not divide exactly by 3, so leave the gradient as a fraction.
6
$y = \frac{2}{3} x - 1$ — Put m and c into $y = m x + c$ . The −1 keeps its minus.

Common mistake

Doing across ÷ up and getting $\frac{3}{2}$ . Up always goes on top.

Key idea

Check with a third point. At $(6, 3)$ : $\frac{2}{3} \times 6 - 1 = 3$ . It fits.

When One Square Is Not 1

Exams often use axes where one square is not 1. Read the numbers on the axes, never just count squares.

Worked example

1
$x : 1 square = 2, y : 1 square = 0.5$ — Read the scales first. Here 5 squares reach 10 on the x-axis, and 2 squares reach 1 on the y-axis.
2
$c = 3$ — The line crosses the y-axis at the number 3. Read the label, not the square count.
3
$(0, 3) and (8, 5)$ — Pick a second point where the line passes exactly through a grid corner, and read both numbers off the axes.
4
$up = 5 - 3 = 2$ — Use the axis numbers. It is 4 squares, but each square is 0.5, so up = 2.
5
$across = 8 - 0 = 8$ — Also 4 squares, but each square is 2, so across = 8.
6
$m = \frac{2}{8} = 0.25$ — Up ÷ across, then simplify. 2 ÷ 8 = 0.25, which is the same as $\frac{1}{4}$ .
7
$y = 0.25 x + 3$ — Put m and c into $y = m x + c$ .
8
$x = 4 : 0.25 \times 4 + 3 = 4$ — Check with a point on the line, (4, 4). It fits.

Common mistake

Counting squares: 4 up and 4 across gives $m = 1$ . That is only right when both scales are the same.

Which Method? Read the Question

Look at the line and what the question asks.

You see	You do
A sloping line drawn on a grid	read c, then up ÷ across
A flat line	$y =$ the height it sits at
An upright line	$x =$ where it crosses the x-axis
A sloping line through $(0, 0)$	c = 0, so $y = m x$
A line that does not reach the y-axis	find m, then put a point in for c: Equation of a Line Through Two Points
Only two points, no graph	Equation of a Line Through Two Points
“Draw the line”	Drawing Straight-Line Graphs

Key idea

Check the axis scales before counting squares. One square is not always 1.

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