GCSE — mathsFoundation2 min read

Straight-Line Graphs

Plot and interpret straight-line graphs for GCSE Maths. Understand gradient, y-intercept, and the equation y = mx + c.

Tutor AI Team2026-02-25T01:45:43.857Z

Straight-line graphs represent linear relationships. The equation $y = mx + c$ describes every straight line on a coordinate grid, where $m$ is the gradient and $c$ is the y-intercept.

Core Concepts

The Equation $y = mx + c$

$m$ = gradient (slope) = $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$
$c$ = y-intercept (where the line crosses the y-axis)

Gradient

Positive gradient: line slopes upward (left to right).
Negative gradient: line slopes downward.
Zero gradient: horizontal line.
Undefined gradient: vertical line.

Parallel and Perpendicular Lines

Parallel lines have the same gradient: $m_1 = m_2$ .
Perpendicular lines have gradients that multiply to $-1$ : $m_1 \times m_2 = -1$ .

Finding the Equation of a Line

Given gradient $m$ and a point $(x_1, y_1)$ :

$y - y_1 = m(x - x_1)$

Midpoint and Distance

Midpoint: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Distance: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Worked Example: Example 1

Problem

Line through $(1, 3)$ and $(4, 9)$ . Gradient = $\frac{9-3}{4-1} = 2$ .

$y - 3 = 2(x - 1)$ → $y = 2x + 1$ .

Solution

Worked Example: Example 2

Problem

Line parallel to $y = 3x + 1$ through $(2, 5)$ : $m = 3$ .

$y - 5 = 3(x - 2)$ → $y = 3x - 1$ .

Solution

Worked Example: Example 3

Problem

Line perpendicular to $y = 2x + 3$ : $m = -\frac{1}{2}$ .

Solution

Practice Problems

1. Find the gradient between $(2, 5)$ and $(6, 13)$ .
1. Write the equation of a line with gradient $-3$ through $(1, 4)$ .
1. Are $y = 4x + 1$ and $y = -\frac{1}{4}x + 3$ perpendicular?

Want to check your answers and get step-by-step solutions?

Key Takeaways

✓
$y = mx + c$ : $m$ = gradient, $c$ = y-intercept.
✓
Parallel: same gradient. Perpendicular: $m_1 \times m_2 = -1$ .
✓
Gradient = $\frac{\Delta y}{\Delta x}$ .

Tags:

#gcse#maths#algebra#graphs#straight-lines#gradient

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Core Concepts

The Equation y=mx+cy = mx + cy=mx+c

Gradient

Parallel and Perpendicular Lines

Finding the Equation of a Line

Midpoint and Distance

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Practice Problems

Key Takeaways

Related Topics

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Biodiversity and Conservation

Sampling and Field Investigations

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The Equation $y = mx + c$