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 has a particular meaning
Example:
This graph shows the cost of petrol.
It shows that 20 litres will cost 拢23 or 拢15 will buy 13 litres.
\(gradient = \frac{change~up}{change~right}\) or \(\frac{change~in~y}{change~in~x}\)
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 \(\frac{change~in~y}{change~in~x} = \frac{change~in~cost}{change~in~litres} = \frac{change~in~拢}{change~in~l} = 拢/l.\)
The gradient shows the cost per litre. Petrol costs 拢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 拢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 \(\frac{change~in~y}{change~in~x} = \frac{change~in~cost}{change~in~days} = \frac{change~in~拢}{change~in~days} = 拢/day.\)
The gradient shows the cost per day. It costs 拢3 per day to hire the ladder.
The graph crosses the vertical axis at (0, 7).
There is an additional cost of 拢7 on top of the 拢3 per day (this might be a delivery charge for example).