Rates of Change of Graphs

What is a rate of change?

A rate of change describes how a variable changes with time (or another variable)
All of the following are examples of rates of change

Speed (change in distance divided by time)
Acceleration (change in speed divided by time)
The depth of water in a container as it is filled with water (change in depth divided by time)
The volume of air inside an inflating balloon as the radius of the balloon increases (change in volume divided by change in radius)
The number of ice-creams sold as the weather gets warmer (change in ice-creams sold divided by change in temperature)

How can I use a graph to find the rate of change?

We can use the same methods that were used with distance-time and speed-time graphs
To find the rate of change of a unit on the y-axis per unit change in the x-axis (often time) we can find the gradient of the graph
- The steeper the gradient, the higher the rate of change
The units of the rate of change will be the units of the y-axis, divided by the units of the x-axis
- If the graph showed volume in cm³ on the y-axis and time in seconds on the x-axis, the rate of change would be measured in cm³/s or cm³s^-1
If the graph is a straight line the rate of change is constant
If the graph is horizontal, the rate of change is zero (y is not changing as x changes)

Examiner Tip

The units of the gradient can help you understand what is happening in the context of the question.
- For example, if the y-axis is in dollars and the x-axis is in hours, the gradient represents dollars per hour.

Worked example

The graph below shows a model of the volume, v litres, of diesel in the tank of George’s truck after it has travelled a distance of d kilometres.

BxxbOqQK_rates-of-change-of-graphs-we-b-graph

(i)

Find the gradient of the graph, stating its units.

$Gradient = \frac{change in y}{change in x}$

$Gradient = \frac{- 54 litres}{600 kilometres}$

-0.09 litres per kilometre

(ii)

Interpret what the gradient of the graph represents.

Consider the units of the gradient; litres per kilometre

The gradient represents the amount of diesel used to travel each kilometre
Travelling 1km requires 0.09 litres of diesel

(iii)

Give one reason why this model may not be realistic.

The consumption of fuel may not be linear (a straight line); it is more likely to be curved
A reason for this could be that as fuel is used up the truck becomes lighter, so becomes more fuel efficient

Rates of Change of Graphs (Edexcel GCSE Maths: Foundation)

Revision Note