Compare linear and non-linear models of real-world data and select the most appropriate model

What this dot point is asking

NESA wants you to look at a table or graph of data and decide whether the underlying relationship is linear, quadratic, exponential or reciprocal. The diagnostic rules are simple and worth memorising.

The answer

Four standard model shapes

• Linear $y = m x + c$. First differences (consecutive $y$ values subtracted) are constant.
• Quadratic $y = a x^2 + b x + c$. First differences are not constant but second differences are.
• Exponential $y = a b^x$. Ratios of consecutive $y$ values are constant (equal to $b$).
• Reciprocal $y = \frac{k}{x}$. Products $x y$ are constant.

Diagnostic checklist

Given a table of $(x, y)$ pairs with $x$ equally spaced:

1. Compute consecutive differences $y_{i+1} - y_i$. If constant, linear.
2. Compute consecutive ratios $\frac{y_{i+1}}{y_i}$. If constant, exponential.
3. Compute products $x y$. If constant, reciprocal.
4. If none of the above, try second differences for a quadratic, or try other forms.

Shape clues from a graph

• Straight line. Linear.
• Smooth curve through the origin, increasing faster and faster. Quadratic or exponential. Use the diagnostic to decide.
• Curve rising sharply at first then flattening towards an asymptote on the right. Exponential decay if downward, or possibly a logarithm.
• Two branches with both axes as asymptotes. Reciprocal.
• Parabolic shape with one turning point. Quadratic.

Practical clues from context

• Money charged per unit plus a flat fee: linear.
• Compound growth or decay (interest, depreciation, population): exponential.
• A fixed total split among a variable number of parts (fuel cost for a fixed trip, time for a fixed job): reciprocal.
• Projectile or revenue-maximising problem: quadratic.

When two models could work

Sometimes a small dataset fits multiple models reasonably well. The question will normally guide you (for example, "the growth percentage is constant" tells you exponential). If asked to compare fit, plot residuals: the better model has smaller residuals across the dataset.