Home / Graphical Methods / Graphs and linear models
3.11a · Graphical Methods · Sub-skill
Read a graph as a story, use $y = mx + c$, and build and criticise linear models in context.
See it first
Scroll to the graphic and it plays by itself — press Replay to watch it again.
Build it up, step by step
Click each step below to reveal it — work through them in order the first time round.
Read the axes and units first. A line that goes up means the quantity is increasing, flat means it is not changing, and down means decreasing. The steeper the line, the faster the rate of change. A jump (discontinuity) means a sudden change, like a price that switches at a threshold.
A straight line has equation $y = mx + c$. The gradient $m = \dfrac{\text{change in } y}{\text{change in } x}$ is the rate of change; $c$ is the $y$-intercept, where the line crosses the $y$-axis (the value when $x = 0$).
Many real costs are fixed charge + rate × amount, giving $C = c + mx$. The intercept is the fixed charge and the gradient is the rate. Use the equation to find values, or rearrange it to find $x$ when you know $C$.
A model is only valid for sensible values: a distance or time cannot be negative, and a real price may change beyond a certain amount. Comment on whether the straight-line assumption is reasonable and what the gradient and intercept mean.
A taxi charges £3 plus £2 per kilometre. Write the cost $C$ for $d$ km and find the cost of 7.5 km.
$C = 3 + 2d$ (fixed charge £3, gradient £2 per km). For $d = 7.5$: $C = 3 + 15 = $ £18.
Test yourself
A plumber charges a call-out fee of £40 plus £30 for each hour of work.
(a) Write down a formula for the cost $C$ of a job that takes $h$ hours.
(b) Find the cost of a job lasting 3.5 hours.
(c) Explain what the numbers 40 and 30 mean in the formula.
(d) A job costs £220. How long did it take? [5 marks]
(a) $C = 40 + 30h$. (1 mark)
(b) $C = 40 + 30 \times 3.5 = 40 + 105 = $ £145. (1 mark)
(c) £40 is the fixed call-out fee (the intercept); £30 is the charge per hour (the gradient). (1 mark)
(d) $220 = 40 + 30h$, so $30h = 180$ and $h = $ 6 hours. (2 marks)
Practice
Five short questions. Work through them, then reveal the mark scheme to check.