Choosing a model, naming its assumptions

Function families are distinguished by their rates of change, not by how their graphs look in one window. Linear functions change by equal amounts over equal input steps. Quadratic functions have first differences that themselves change by equal amounts: constant second differences. In general, a degree-$n$ polynomial has constant $n$th differences. Rational models show up when two quantities trade off multiplicatively, as when doubling one halves the other.

Shape is a weak witness. Over a short window, an exponential curve, a quadratic, and even a steep line can look nearly identical. The rate signature is the fingerprint; the silhouette is a costume.

Given a table, write the differences between consecutive outputs in the margin. Constant first differences: linear. First differences that grow or shrink by a constant amount: quadratic. Keep taking differences until they go constant, and the number of layers is the degree.

The rule that makes this legal: the inputs must be equally spaced. If the x-values step 0, 1, 2, 4, the raw output differences are not comparable, because the last one covers two units of input. Divide each output change by its input change first (that is, compute average rates of change), and only then compare. A table can look nonlinear while its rates are perfectly constant.

1. Check the input spacing. Unequal? Work with $\frac{\Delta y}{\Delta x}$ instead of raw differences.
2. Constant first rate: linear. Constant second differences: quadratic. Constant $n$th: degree $n$.
3. Growing gaps alone do not mean exponential; quadratics grow by more each step too.

Graph features put a floor under the degree. A polynomial with three distinct real zeros needs degree at least 3. Two turning points likewise demand degree at least 3, since a degree-$n$ polynomial has at most $n - 1$ turns. These are minimums, not identifications: a degree-5 polynomial can also cross three times, so data alone fixes the floor, and the context or the differences pick the ceiling.

Inverse relationships pull toward rational models. If a fixed job is shared by $w$ workers, or a fixed distance driven at speed $s$, the product of the two quantities is constant, and the model is $k/x$ shaped: large at small inputs, leveling toward zero, never linear.

Choosing a family is an assumption: this rate behavior continues. The CED asks you to articulate it. A linear population model assumes the same absolute growth every year, forever. That may be defensible across the observed three years and indefensible fifty years out. A quadratic height model assumes gravity is the only force and ends the moment the ball lands.

Context also writes the domain. A model can be algebraically happy at inputs the situation forbids: negative times, populations below zero, prices beyond what anyone would pay. State the restriction with the model, because the formula will not volunteer it.