# 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.

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.

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).