Rukasha Maths

GCSE maths Beta

Straight-line graphs · Foundation notes

Foundation · Algebra · AQA references A8, A9, A10

Every point on a straight-line graph satisfies its equation. In y = mx + c, the value c is where the line crosses the y-axis because x is zero there. The gradient m measures how much y changes for each increase of 1 in x. Use change in y divided by change in x between two points; keep the subtraction order the same. A positive gradient rises to the right and a negative gradient falls. To plot a line, calculate several coordinate pairs, mark them accurately and join them with a straight line.

Learning objectives

  • Identify a gradient and vertical intercept.
  • Find the equation of a line from two points.

Key ideas

The gradient is the coefficient of the horizontal variable; the constant is the vertical intercept.

y=mx+cy=mx+c

Gradient is vertical change divided by horizontal change.

m=y2−y1x2−x1(x2≠x1)m=\frac{y_2-y_1}{x_2-x_1}\quad(x_2\ne x_1)

Find the vertical intercept by substituting a point into the line equation.

Distinct parallel non-vertical lines have equal gradients. A vertical line has undefined gradient.

Coordinates give the horizontal position first.

(x,y)(x,y)

Worked examples

Example 1

Find the line through the two given points.

(1,5), (3,9):m=9−53−1=2(1,5),\ (3,9):\qquad m=\frac{9-5}{3-1}=2

Substitute the first point.

5=2(1)+c⟹c=35=2(1)+c\quad\Longrightarrow\quad c=3

Check the second point.

2(3)+3=92(3)+3=9

Answer

y=2x+3y=2x+3

Example 2

Substitute a negative horizontal coordinate.

y=4x+1,x=−2:y=4(−2)+1=−7y=4x+1,\quad x=-2:\qquad y=4(-2)+1=-7

Answer

y=−7y=-7

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Plot a line or find its gradient, intercept or equation.

AO2 · Reasoning

Explain what a gradient or intercept represents, or whether lines are parallel.

AO3 · Problem solving

Use a linear model to compare costs and identify a break-even point.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

The constant is the vertical intercept, not the gradient. Keep brackets around negative values when substituting.

Try it yourself

Attempt each question before opening its answer.

Check 1: What are the gradient and y-intercept of y = -3x + 4?

Answer: Gradient -3; y-intercept 4

The coefficient of x is m = -3 and the constant is c = 4.

Check 2: Find y when x = -2 on y = 4x + 1.

Answer: y = -7

Substitute carefully: y = 4 × (-2) + 1 = -8 + 1 = -7.

Explain it in your own words

  • Make a table of values and plot y = 2x - 1.
  • Explain what a gradient of -2 means when x increases by 1.
  • Write two different parallel lines and identify their shared gradient.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes

Specification source

AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.