Gradients of lines (Edexcel GCSE Maths): Revision Notes

Gradients of lines

What is a gradient?

The gradient of a straight line tells you how steep that line is. Think of it as measuring how quickly the line rises or falls as you move across the graph. A steep mountain path has a large gradient, while a gentle slope has a small gradient.

When working with graphs, gradients help you understand the relationship between two variables and how one changes relative to the other.

The gradient formula

To find the gradient of any straight line, you can use this essential formula:

$\text{Gradient} = \frac{\text{Distance up}}{\text{Distance across}}$

This means you need to measure:

• Distance up: How far the line rises (or falls) vertically
• Distance across: How far you move horizontally along the line

How to calculate gradients step by step

Drawing your triangle

To work out a gradient accurately, you need to draw a right-angled triangle using the line:

• Choose two clear points on the line that are reasonably far apart
• Draw a horizontal line from one point
• Draw a vertical line from the other point
• This creates your triangle for measuring

Measuring the distances

Once you have your triangle:

• Distance across: Count the horizontal distance between your two points
• Distance up: Count the vertical distance between your two points
• Always check the scales on both axes - they might be different!

Working through an example

Understanding positive and negative gradients

Positive gradients

When a line slopes upwards from left to right, it has a positive gradient. This means:

• As one value increases, the other value also increases
• Both variables move in the same direction
• The line climbs as you read from left to right

Negative gradients

When a line slopes downwards from left to right, it has a negative gradient. This means:

• As one value increases, the other value decreases
• The variables move in opposite directions
• The line falls as you read from left to right

Zero gradients

A perfectly horizontal line has a gradient of zero. This means:

• One variable stays constant regardless of changes in the other
• There is no relationship between the two variables

Top tips for accurate calculations

These practical tips will help you avoid common mistakes and improve your gradient calculations:

• Use large triangles: Bigger triangles give more accurate results and reduce the chance of calculation errors
• Check the scales: Don't just count grid squares - always read the numbers on the axes
• Choose clear points: Pick points where the line passes through grid intersections when possible
• Double-check your subtraction: Make sure you subtract the coordinates correctly when finding distances

Interpreting gradients in real contexts

The gradient tells you the rate of change between two variables. In the film example:

• A gradient of 0.1 means spending an extra £10 million on a film typically increases opening weekend earnings by about £1 million
• This helps production companies understand the relationship between budget and initial success