Gradient as a Rate of Change

Grade 5

Gradient tells you the rate

Gradient isn't just a number — it has a meaning depending on what the axes represent. On a distance-time graph, the gradient of the line is the speed. A steeper line means a faster object.

On a speed-time graph, the gradient is acceleration — how quickly speed is changing. A flat line means constant speed (zero acceleration). A rising line means speeding up.

How to read gradient in context

The units of the gradient come directly from the axes. If the y-axis is distance in metres and the x-axis is time in seconds, the gradient has units of m/s — metres per second. That's speed.

Rate of change = (change in y-axis quantity) ÷ (change in x-axis quantity)

Always write the units of your gradient answer. "The gradient is 4" is incomplete. "The gradient is 4 m/s" tells the reader what it actually means.

Worked example — distance-time graph

EXAMPLE

On a distance-time graph, a line passes through (0, 0) and (30, 240), where time is in seconds and distance is in metres. What is the speed?

Gradient = rise ÷ run = (240 − 0) ÷ (30 − 0) = 240 ÷ 30 =8 m/s.

What a negative gradient means in context

On a distance-time graph, a negative gradient means the object is moving back towards the starting point — its distance from the origin is decreasing. On a speed-time graph, a negative gradient means the object is decelerating (slowing down).

Try this one

The diagram shows a distance-time graph for two runners, A and B, in a race.

(a) Find the speed of runner A. [2]

(b) Find the speed of runner B. [1]

Nova's hint:

This question uses the gradient of a distance-time graph to find speed — speed = distance ÷ time.

Ask yourself: 'What are the start and end coordinates for each line, and how do I use these to find gradient?'

Then subtract the two distances at t = 40 to find the difference.

Try to Solve it with Nova?