Modelling with quadratic, cubic and power functions

The shapes and their key features

• Quadratic ($y = ax^2 + bx + c$): a parabola with exactly one turning point, the vertex. It opens upward when $a > 0$ (vertex is a minimum) and downward when $a < 0$ (vertex is a maximum). It is symmetric about the axis of symmetry $x = -\frac{b}{2a}$, meets the y-axis at $(0, c)$, and has up to two roots where it crosses the x-axis.

• Cubic ($y = ax^3 + bx^2 + cx + d$): typically an 'S' shape with up to two turning points (a local maximum and a local minimum) and up to three roots. It meets the y-axis at $(0, d)$. Good for growth that speeds up, levels off, then changes again.

• Power ($y = ax^n$): a single-term curve whose shape is set by $n$. For $n > 0$ it rises from the origin with no turning point; for $n < 0$ it decays like a reciprocal curve and is undefined at $x = 0$. It is monotonic — it cannot rise then fall — which is the key clue for when NOT to use it.

Fitting a model with your GDC

• Step 1 — Enter. In the statistics menu, put the independent variable (e.g. time) in one list and the dependent variable (e.g. height) in a second list.

• Step 2 — Choose. In the regression sub-menu pick the model: QuadReg for quadratic, CubicReg for cubic, PwrReg for power.

• Step 3 — Record. Run it, checking the correct lists are used. Write the full equation, quoting each parameter to at least 3 significant figures. If shown, note the $R^2$ value as a measure of fit.

• Step 4 — Store. Save the equation into a function variable (e.g. Y1). Now you can graph it and use max/min, zero and value tools without retyping long decimals — and without introducing rounding error.

Worked example 1 — a quadratic model

Interpretation and prediction

Common mistakes examiners penalise

• Confusing the vertex with a coefficient — the maximum/minimum of $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$ (or read from the GDC), NOT at $x = c$ or at the largest coefficient. The vertex is a point $(x, y)$; quote both coordinates with units.

• Keeping a non-physical root — a quadratic or cubic model of a real quantity often has two or three roots, but negative times, negative lengths and values outside the modelled window must be rejected with a brief reason ('time cannot be negative'). Take the meaningful root only.

• Forcing a power model onto humped data — $y = ax^n$ is monotonic and passes through the origin (for $n > 0$); it cannot rise then fall. Data with a peak or trough needs a quadratic or cubic, not a power model.

• Reading a feature off the wrong place — get the vertex from the GDC's 'maximum'/'minimum' feature and the roots from 'zero'; do not read the largest data value as the maximum or the value of $c$ as the vertex.

• Extrapolating without comment — predicting outside the data range is allowed, but you must flag it as extrapolation and question its reliability. Treating a far extrapolation as a firm prediction loses the reasoning mark.

• Over-rounding the parameters — quoting $a$, $b$, $c$ to 3 s.f. and then computing with the rounded values corrupts later answers. Store the exact regression equation and evaluate from it; round only the final answer.

• Writing the parameters but not the model — 'find the model' means write the full equation $h(t) = -5t^2 + 30t + 10$, not a bare list $a = -5, b = 30, c = 10$.

• Dropping units and context — a height is '$55$ m', a value is '£110 000'. A bare number left unlabelled, or a parameter never interpreted in context, throws away the interpretation marks these questions are designed to test.

Model answer — marked the way our engine marks it

Where this leads