How to Find Gradient in a Graph?

The gradient of a line on a graph measures its steepness, which can be determined by calculating the ratio of the change in the y-axis (vertical change) to the change in the x-axis (horizontal change) between two points on the line. This concept is explained in the video ["GCSE Maths - How to Find the Gradient of a Straight Line #65"]().

Calculating the Gradient

Here's a step-by-step guide on how to find the gradient of a straight line on a graph:

1. Identify Two Points:

   • Select two clear points on the line whose coordinates you can easily determine. Let's call these (x₁, y₁) and (x₂, y₂).

2. Calculate the Change in Y (Vertical Change):

   • Subtract the y-coordinate of the first point from the y-coordinate of the second point: Δy = y₂ - y₁

3. Calculate the Change in X (Horizontal Change):

   • Subtract the x-coordinate of the first point from the x-coordinate of the second point: Δx = x₂ - x₁

4. Calculate the Gradient:

   • Divide the change in y by the change in x. The formula for gradient (often denoted as 'm') is:

       m = Δy / Δx = (y₂ - y₁) / (x₂ - x₁)

Example Using Reference Data

According to the reference, the presenter calculates a gradient where:

• The change in y (vertical change) is -6.
• The change in x (horizontal change) is 3.

Therefore, using the formula, the gradient is:

m = -6 / 3 = -2

Practical Considerations

• Positive Gradient: A line that slopes upwards from left to right has a positive gradient.
• Negative Gradient: A line that slopes downwards from left to right has a negative gradient.
• Zero Gradient: A horizontal line has a gradient of zero (no vertical change).
• Undefined Gradient: A vertical line has an undefined gradient (infinite vertical change for no horizontal change).
• Consistency: The gradient of a straight line is constant - you'll get the same value no matter which two points you choose on the line.
• Units: The gradient does not have units unless the axes have units, in which case, the units would be units of y per unit of x.

Summary

To calculate the gradient of a straight line on a graph:

Step	Description	Formula/Calculation
1. Choose Two Points	Select two points on the line	(x₁, y₁) and (x₂, y₂)
2. Calculate Change in Y	Subtract the y-coordinate of the first point from the y-coordinate of the second point.	Δy = y₂ - y₁
3. Calculate Change in X	Subtract the x-coordinate of the first point from the x-coordinate of the second point.	Δx = x₂ - x₁
4. Calculate the Gradient	Divide the change in y by the change in x.	m = Δy / Δx = (y₂ - y₁) / (x₂ - x₁)