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3.11 · Paper 2C
Reading and criticising graphs, linear models, sketching quadratic and cubic graphs, and solving equations graphically.
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On a graph, steeper means a faster rate of change, flat means no change, and a jump (discontinuity) means a sudden change such as a new price band. A linear model $y = mx + c$ has gradient $m$ (the rate of change, with units) and $y$-intercept $c$ (the starting or fixed value). Always say where the model stops being sensible.
When sketching, mark the intercepts with the axes, the turning points and the overall shape. A quadratic is a $\cup$ or $\cap$ shape; its turning point is halfway between its roots. A cubic has an S-shape. Quadratic models describe things like the height of a thrown ball or the area of a rectangle.
The solutions of $f(x) = g(x)$ are the $x$-coordinates where the graphs of $y = f(x)$ and $y = g(x)$ intersect. The roots of $f(x) = 0$ are where the curve crosses the $x$-axis. To solve a new equation, rearrange it so that one side is the equation of a curve you already have, then draw the line for the other side.
A taxi charges £3 plus £2 per kilometre. Write down the cost $C$ for $d$ km, and find the cost of a 7.5 km journey.
The fixed charge is £3 (the $y$-intercept) and the rate is £2 per km (the gradient), so $C = 3 + 2d$.
For $d = 7.5$: $C = 3 + 2 \times 7.5 = 3 + 15 = $ £18.
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