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:
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 x-axis, read off from x=2 to find the corresponding y-coordinate on the graph. Going up from x=2 until you meet the line, then going across until you meet the y-axis, you will find that this corresponding y-coordinate is 100. Hence, in the first two hours, 100 km has been covered.
Gradients
The gradient of any line segment represents the rate at which the y-variable changes with respect to the x-variable. This rate of change essentially means that as the x-variable increases by one, the y-variable will increase by the value of the gradient. Note: if the gradient is negative, then the y-variable decreases as the x-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
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 2 and when time is 3:
speed=change in xchange in y
=2−34−2
=−12
=−2
Thus the speed is −2 kilometres per hour. Strictly speaking, this is velocity, since the negative indicates a direction. What this means is that the speed is 2 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 40 cm. The bath starts empty and the tap is run for 15 minutes such that the water runs at a constant rate. Once the depth reaches 25 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 10 minutes until the depth of the water is at 35 cm. The tap is then turned off for 5 minutes. For the following 5 minutes, the tap is still off, but also water is let out of the bath until the depth is 30 cm. Finally, the tap is turned on for another 10 minutes until it is full (at a depth of 40 cm). Draw a graph that reflects this scenario.
This graph will have time in minutes on the x-axis and water depth in cm on the y-axis. It will be made up of line segments (since each stage has a consistent flow of water) that join the points (0,0) to (15,25) to (25,35) to (30,35) to (35,30) to (45,40):