Quick questions on Choosing between linear and non-linear models for HSC Maths Standard 2

Each model has a signature shape and a signature diagnostic. The panels below put all four on identical axes so the differences are visible side by side. The data points sit exactly on each curve; the caption names the test that confirms it.

If you are given a graph rather than a table, the shape narrows it down quickly:

A straight line, $y = mx + c$. Equal steps in $x$ produce equal steps in $y$, so the first differences are constant. This is the only model with no curvature at all.

A single smooth bend (a parabola), $y = ax^2 + bx + c$. The first differences are not constant, but the differences of those differences, the second differences, are constant. One turning point is the visual tell.

A curve that climbs slowly then ever more steeply (or, for decay, falls fast then flattens), $y = ab^x$. Equal steps in $x$ multiply $y$ by the same factor, so the ratios of consecutive $y$ values are constant (and that constant ratio is the base $b$).

A curve that falls steeply then flattens towards the axes, $y = \dfrac{k}{x}$. As $x$ rises, $y$ shrinks towards zero, and the product $xy$ is constant (it always equals $k$). This is the shape of a fixed total shared among more and more parts.