Real-life graphs

The concepts of gradient and rate of change are explored

All real-life graphs can be used to estimate or read-off values.The actual meaning of the values will depend on the labels and units shown on each axis. Sometimes:

  • the gradient of the line or curve has a particular meaning
  • the y-intercept (where the graph crosses the vertical axis) has a particular meaning
  • the area under the graph has a particular meaning

Example:

This graph shows the cost of petrol.

It shows that 20 litres will cost £23 or £15 will buy 13 litres.

A graph showing the cost of petrol. The y axis shows cost in pounds from zero to 30 and the X axis shows litres from zero to 24. The graph shows that 20 litres will cost £23 or £15 will buy 13 litres.

Gradient = \frac{\text{change up}}{\text{change right}} or \frac{\text{change in y}}{\text{change in x}}

Using the points (0, 0) and (20, 23), the gradient = \frac{23}{20} = 1.15.

The units of the axes help give the gradient a meaning.

The calculation was: \frac{change~in~y}{change~in~x} = \frac{change~in~cost}{change~in~litres} = \frac{change~in~£}{change~in~l} = £/l.

The gradient shows the cost per litre. Petrol costs £1.15 per litre.

The graph crosses the vertical axis at (0, 0). This is known as the intercept.

It shows that if you buy 0 litres, it will cost £0.

Example:

This graph shows the cost of hiring a ladder for various numbers of days.

A graph showing the cost of ladder hire. The y axis is cost in pounds from zero to 40 and the X axis shows time in days from zero to 10. The gradient shows that it costs £3 per day to hire the ladder.

Using the points (1, 10) and (9, 34), the gradient = \frac{change~up}{change~right} or \frac{change~in~y}{change~in~x} = \frac{34-10}{9-1} = \frac{24}{8} = 3.

The units of the axes help give the gradient a meaning.

The calculation was: \frac{\text{change in y}}{\text{change in x}} = \frac{\text{change in cost}}{\text{change in days}} = \frac{\text{change in £}}{\text{change in days}}= £/day

The gradient shows the cost per day. It costs £3 per day to hire the ladder.

The graph crosses the vertical axis at (0, 7).

There is an additional cost of £7 on top of the £3 per day (this might be a delivery charge for example).