Comparing Linear, Quadratic, and Exponential Models

Objective: Identify a linear pattern in an equally spaced table.

Prerequisite skills: linear and exponential change and quadratic functions. Constant first differences for equal input steps are the signature of a linear table. Constant positive ratios instead suggest an exponential model. Do not compare raw differences across unequal input intervals. Common error: checking ratios when the relevant pattern is additive. Restrictions and interpretation: Fit describes observed patterns; several models can fit a small sample and none guarantees causality or distant forecasts.

Objective: Identify multiplicative change in an equally spaced table.

A positive constant ratio between consecutive outputs at equal input steps is consistent with an exponential relationship. Its differences typically grow or shrink rather than remain fixed. Zero or negative outputs need separate care. Common error: using increasing first differences alone as proof of a quadratic. Restrictions and interpretation: Fit describes observed patterns; several models can fit a small sample and none guarantees causality or distant forecasts.

Objective: Recognize a quadratic pattern with equally spaced inputs.

For a quadratic y=ax²+bx+c at unit-spaced inputs, the second differences are constant and equal to 2a. A constant nonzero second difference distinguishes a quadratic table from a linear one. Common error: using half a first difference as the quadratic coefficient. Restrictions and interpretation: Fit describes observed patterns; several models can fit a small sample and none guarantees causality or distant forecasts. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.