Which model deserves the data?

Two points determine a line. They also determine an exponential, and infinitely many quadratics pass through them too. So "my model goes through (1, 10) and (2, 20)" eliminates nothing: several families claim those same points and then disagree everywhere else.

The same warning applies to checking one prediction. If a model predicts 14 and the data says 14, you have one agreement, which every rival model through that point also enjoys. A model is validated by its behavior across the whole data set, never by a hit at a single input.

Each family changes in its own signature way, and equal-spaced data lets you read the signature directly:

1. Linear: first differences constant. Equal steps in, equal amounts out.
2. Quadratic: first differences change, second differences constant.
3. Exponential: ratios of consecutive outputs constant. Equal steps in, equal factors out.

Run the tests, do not eyeball them. Outputs 4, 7, 12, 19 have growing gaps, which "looks exponential," but the ratios drift (1.75, then 1.71, then 1.58) while the second differences sit at exactly 2: quadratic. Growing gaps are shared by every accelerating family; only the tests distinguish them.

A residual is actual minus predicted, one per data point. Plotted in order, residuals tell you whether the model's shape is right:

Random scatter around zero, no trend, means the model family fits and the leftovers are noise. A systematic pattern, a bow, an arch, a run of negatives then positives then negatives, means the model is bending the wrong way or not bending enough: the family is wrong even if every residual is small.

Aggregate summaries can hide the pattern. Residuals of −8, −2, +3, +5, +3, −2, −9 nearly cancel in total, yet their arch shape says the data curves relative to the model. Look at the pattern, not the sum.

Competing models that agree inside the data window can diverge wildly outside it; that is where the family choice matters most and where validation evidence is thinnest. A perfect five-year fit says little about year thirty, and the context's assumptions (room to grow, resources, saturation) usually break before the algebra does.

The AP habit: justify the family with a rate argument or residual argument, state the assumption your model leans on, and flag any prediction made outside the window as an extrapolation resting on that assumption.