Straight lines and parabolas are only the start. Extended IGCSE asks you to recognise three more curve families on sight: the ever-steepening exponential y = aˣ, the S-shaped cubic y = x³, and the two-branch reciprocal y = a/x. Knowing their shapes lets you sketch, solve and model real growth and decay.
Why this matters: exponential curves model population, compound interest, radioactive decay and viral spread; reciprocals model inverse proportion; cubics appear in volume problems. Cambridge routinely gives a table of values to plot, then asks you to estimate a gradient or read a solution — recognising the shape stops silly errors.
1. The three shapes
| Curve | Shape | Key features |
|---|---|---|
| y = aˣ (a > 1) | Rises ever faster | Passes (0, 1); asymptote y = 0; never negative |
| y = x³ | S-shape through origin | Passes (0, 0); goes from bottom-left to top-right |
| y = a/x | Two curved branches | Asymptotes at both axes; never touches them |
2. Asymptotes and intercepts
- An asymptote is a line the curve approaches but never reaches. For y = aˣ the x-axis (y = 0); for y = a/x both axes.
- y = aˣ always cuts the y-axis at (0, 1) because a⁰ = 1.
- For decay use 0 < a < 1 (e.g. y = 0.5ˣ), which falls towards the x-axis.
3. Estimating a gradient and a growth problem
- To estimate the gradient at a point on a curve, draw a tangent there and find rise ÷ run.
- To estimate an area under a curve, count squares or use the trapezium rule.
Growth worked: A colony of 200 bacteria doubles every hour, so N = 200 × 2ᵗ. After 3 hours N = 200 × 2³ = 200 × 8 = 1600. After 5 hours N = 200 × 2⁵ = 200 × 32 = 6400.
Reading a value: On y = x³, at x = 4, y = 4³ = 64; at x = −2, y = (−2)³ = −8. The negative cube stays negative — a giveaway of the S-shape.
Common exam mistake: drawing y = aˣ crossing the x-axis. It never does — the x-axis is an asymptote, so the curve only gets closer. Also, (−2)³ = −8, not +8: an odd power keeps the sign.
Exam technique: when asked to estimate the gradient, show the tangent triangle and write rise ÷ run — the working earns the marks even if your read-off is slightly out. For Sketch, mark the key point ((0,1) for exponential, origin for cubic) and the asymptotes.