Real-life graphs

Real-life graphs

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Real-life graphs

​​In a nutshell

Graphs can be used to represent a number of real-life scenarios and relationships. Straight line graphs show a relationship with a constant change, and the gradient of the line or line segment represents the rate of change.

Real-life graphs

Example 1

Consider the following graph:

Maths; Graphs; KS4 Year 10; Real-life graphs

The graph tracks distance travelled against time. Read off the graph to find the distance travelled from the start in two hours.

Since time is on the xx-axis, read off from x=2x=2 to find the corresponding yy-coordinate on the graph. Going up from x=2x=2 until you meet the line, then going across until you meet the yy-axis, you will find that this corresponding yy-coordinate is 100100. Hence, in the first two hours, 100 km\underline{100\text{ km}} has been covered.


The gradient of any line segment represents the rate at which the yy-variable changes with respect to the xx-variable. This rate of change essentially means that as the xx-variable increases by one, the yy-variable will increase by the value of the gradient. Note: if the gradient is negative, then the yy-variable decreases as the xx-variable increases.

Some rates of change will be familiar to you: the rate of change of distance with respect to time is known as speed. The rate of change of speed with respect to time is known as acceleration. 

Example 2
Maths; Graphs; KS4 Year 10; Real-life graphs

According to this graph, find the speed of the motion between hour two and hour three.

In a distance-time graph, the gradient gives the speed. The gradient between hour two and hour three is found by using the coordinates on the graph when time is 22 and when time is 33:

speed=change in ychange in x\text{speed}=\frac{\text{change in }y}{\text{change in }x}




Thus the speed is 2\underline{-2} kilometres per hour. Strictly speaking, this is velocity, since the negative indicates a direction. What this means is that the speed is 22 kilometres per hour but back in the direction toward the start position.

Horizontal lines indicate no change since the gradient is zero.

Plotting a real-life graph

You can plot a real-life graph if you are given information about the scenario.

Example 3

A bathtub is full once the depth is at 4040 cm. The bath starts empty and the tap is run for 1515 minutes such that the water runs at a constant rate. Once the depth reaches 2525 cm, the tap is turned down such that the water is running at a lower, but still constant, rate. The tap runs at this state for 1010 minutes until the depth of the water is at 3535 cm. The tap is then turned off for 55 minutes. For the following 55 minutes, the tap is still off, but also water is let out of the bath until the depth is 3030 cm. Finally, the tap is turned on for another 1010 minutes until it is full (at a depth of 4040 cm). Draw a graph that reflects this scenario.

This graph will have time in minutes on the xx-axis and water depth in cm on the yy-axis. It will be made up of line segments (since each stage has a consistent flow of water) that join the points (0,0)(0,0) to (15,25)(15,25) to (25,35)(25,35) to (30,35)(30,35) to (35,30)(35,30) to (45,40)(45,40):

Maths; Graphs; KS4 Year 10; Real-life graphs

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FAQs - Frequently Asked Questions

What are some examples of rates of change?

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