Rates of Change of Graphs

‘Real life graphs’ won’t just relate to distance, speed, and time. They can show the relationship between any two variables, although they most commonly show how something changes with time.

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 at a point?

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 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)
If the graph is a curve, you can draw a tangent at a point on the graph and find its gradient

This will be an estimate of the rate of change at that point
This is also known as the instantaneous rate of change

The rate of change is larger when the graph is steeper (a higher gradient)

In the below image, tangents drawn at points A and B show that the graph is steeper at B
Therefore the rate of change at B is greater

rates-of-change-of-graphs-tangents

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

To find the average rate of change between two points, find the gradient of the chord between those two points.
- In the image below, the average rate of change between A and B is the gradient of the chord between A and B.

ocr-rn-rates-of-change-of-graphs-image-2

Exam Tip

When considering the rate of change of a curved graph, sketch tangents on the graph at different points to help you
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

(a)

Each of the graphs below show the depth of water, d cm, in a container that is being filled with water at a constant rate.

Match each of the graphs 1, 2, 3, 4 with the containers A, B, C, D

Considering graph 1, the gradient is constant, so the rate of change is constant. So the depth increases at the same rate throughout.
This matches container D which has vertical sides, so the rate of filling does not change.

Graph 1 is Container D

Considering graph 2, the gradient starts shallow and becomes steeper, meaning that the depth increases faster and faster.
This matches container A, gets narrower towards the top, meaning the rate of filling increases.

Graph 2 is Container A

Considering graph 3, the gradient starts steep and becomes shallower, meaning that the depth increases at a slower and slower rate.
This matches container B, which gets wider towards the top, meaning the rate of filling decreases.

Graph 3 is Container B

Considering graph 4, the gradient starts steep, then becomes shallow, then becomes steep again. This means that the depth increases quickly, then slowly, then quickly again.
This matches container C, which is narrow at the bottom (fast filling), gets wider in the middle (slow filling) and narrow again at the top (faster filling)

Graph 4 is Container C

(b)

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.

(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.

Worked example

The graph shows the distance in metres travelled by a car over 6 seconds.

(a) Use the graph to estimate the speed after 3 seconds, giving your answer correct to 3 significant figures.

Speed is the rate of change of distance with respect to time.

Draw a tangent at t = 3. The tangent should touch but not cross the curve.

Use $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ or $\frac{rise}{run}$ to find the gradient of your tangent. Because the tangent line is drawn by eye, there will be some margin of error permitted in the answer. But according to our tangent line above,

speed = gradient = $\frac{40}{4.5} = 8 . \dot{8}$

Round to 3 significant figures as instructed.

Speed = 8.89 m/s

(b) Use the graph to find the average speed over the first 3 seconds, giving your answer correct to 3 significant figures.

Draw a chord between t = 0 and t = 3.