How to Plot Straight Line Graphs — KS3 Maths

To plot a straight line graph, substitute a set of x values into the equation to find the corresponding y values, plot those coordinate pairs on a grid, and draw a single straight line through them. Every straight line can be written in the form y = mx + c.

Understanding y = mx + c

Every straight line graph in KS3 maths has an equation of the form:

y = mx + c

Where:

m is the gradient (steepness of the line)

c is the y-intercept (where the line crosses the y-axis)

Part of y = mx + c | What it tells you
m (gradient) | How steep the line is; positive = slopes up left to right, negative = slopes down
c (y-intercept) | The y-coordinate where the line crosses the y-axis

Example: In y = 3x + 2, the gradient is 3 and the y-intercept is 2.

How to plot a straight line graph — step by step

Step 1 — build a table of values

Choose at least three x values (usually the question specifies a range). Substitute each x value into the equation to find y.

Step 2 — write the coordinate pairs

Write each (x, y) pair. These are the points you will plot.

Step 3 — plot the points on a grid

Mark each point accurately on your coordinate grid.

Step 4 — draw the line

Use a ruler to draw a straight line through all the points. Extend the line to the edges of the grid unless the question specifies otherwise.

Worked example 1 — plot y = 2x + 1 for x from −2 to 3

Table of values:

x | 2x + 1 | y
−2 | 2(−2) + 1 | −3
−1 | 2(−1) + 1 | −1
0 | 2(0) + 1 | 1
1 | 2(1) + 1 | 3
2 | 2(2) + 1 | 5
3 | 2(3) + 1 | 7

Coordinate pairs: (−2, −3), (−1, −1), (0, 1), (1, 3), (2, 5), (3, 7)

The gradient is 2 (rises 2 units for every 1 unit across) and the y-intercept is 1. Plot each point and draw a straight line through them.

Worked example 2 — plot y = −x + 4 for x from 0 to 5

Table of values:

x | −x + 4 | y
0 | −0 + 4 | 4
1 | −1 + 4 | 3
2 | −2 + 4 | 2
3 | −3 + 4 | 1
4 | −4 + 4 | 0
5 | −5 + 4 | −1

The gradient is −1 (line slopes downward) and the y-intercept is 4. The line crosses the x-axis at x = 4 (where y = 0).

Worked example 3 — plot y = ½x − 3 for x from −2 to 6

Table of values:

x | ½x − 3 | y
−2 | ½(−2) − 3 | −4
0 | ½(0) − 3 | −3
2 | ½(2) − 3 | −2
4 | ½(4) − 3 | −1
6 | ½(6) − 3 | 0

Choosing even x values when the gradient is ½ avoids fractions in the y column. The y-intercept is −3.

Reading the gradient and y-intercept from a graph

If you are given a graph (rather than an equation), you can find m and c by inspection.

Finding c: read off the y-value where the line crosses the y-axis.

Finding m: choose two points on the line (preferably where the line crosses grid lines). Then use:

gradient = rise ÷ run = (change in y) ÷ (change in x)

Worked example 4 — find the equation from a graph

A line passes through (0, −2) and (4, 6).

c = −2 (y-intercept read directly)

Gradient = (6 − (−2)) ÷ (4 − 0) = 8 ÷ 4 = 2

Equation: y = 2x − 2

Special cases — horizontal and vertical lines

Line type | Equation form | Example
Horizontal line | y = k (a constant) | y = 3 (all points have y-coordinate 3)
Vertical line | x = k (a constant) | x = −2 (all points have x-coordinate −2)

Vertical lines cannot be written in y = mx + c form — their gradient is undefined.

Straight lines in the national curriculum

The DfE's KS3 mathematics programme of study requires pupils to use and interpret graphs in the form y = mx + c, and to identify and interpret the gradient and y-intercept of linear functions graphically and algebraically. The statutory national curriculum for Key Stage 3 and 4 confirms that straight line graphs are among the most regularly examined algebra topics at both KS3 and GCSE.

Common mistakes when plotting straight line graphs

Mistake | How it shows up | How to avoid it
Arithmetic error in the table of values | Points do not lie on a straight line | Check at least one substitution twice; if the points do not align, recheck your arithmetic
Plotting (y, x) instead of (x, y) | Graph looks wrong | Always write x first in the coordinate pair
Ignoring negative y values | Leaving points off the grid | Extend the y-axis downward if needed
Drawing a curve through the points | Loses marks even if points are correct | Straight line graphs MUST be drawn with a ruler

Frequently asked questions

How do I find the gradient of a straight line graph?

Choose two points on the line where it crosses grid lines clearly. Gradient = (difference in y-values) ÷ (difference in x-values). If the line goes up from left to right, the gradient is positive. If it goes down, the gradient is negative. For example, through (1, 2) and (4, 8): gradient = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2.

What does the y-intercept mean on a straight line graph?

The y-intercept is the point where the line crosses the y-axis — that is, where x = 0. In the equation y = mx + c, the value c is the y-intercept. You can read it directly off a graph, or substitute x = 0 into the equation to calculate it.

How many points do I need to plot a straight line?

Mathematically, two points define a straight line. However, always plot at least three points. If the third point does not lie on the line through the other two, you have made an arithmetic error somewhere and can spot it before drawing. Using three points is standard practice in KS3 and GCSE examinations.

What is the equation of a horizontal line?

A horizontal line has gradient zero. Its equation is y = c, where c is the constant y-value. For example, the line y = 5 passes through (0, 5), (3, 5), (−2, 5), and all other points with y-coordinate 5. Vertical lines have equations of the form x = k and cannot be written as y = mx + c.

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