Lines of Best Fit and Predictions

Objective: Evaluate a model at a specified input.

Prerequisite skills: slope, intercepts, and scatterplots. A fitted line summarizes a two-variable trend. Use its equation for a prediction at the requested input, while recognizing that individual observations need not lie exactly on the line. Common error: treating a model prediction as a guaranteed observed value. Restrictions and interpretation: Predictions are estimates; extrapolation beyond observed inputs can be unreliable.

Objective: Interpret a per-unit predicted change without assuming causation.

The slope of a fitted line is the change in predicted y associated with one additional x-unit. It is not an assertion that manipulating x will cause that change. Units identify what the coefficient represents. Common error: adding the intercept to a predicted change. Restrictions and interpretation: Predictions are estimates; extrapolation beyond observed inputs can be unreliable.

Objective: Classify a prediction relative to the observed input range.

Interpolation predicts inside the range of measured inputs; extrapolation goes outside it. Even a strong fit can fail outside the observed range because the relationship may change. Neither type guarantees an individual outcome. Common error: calling any future-looking calculation an interpolation. Restrictions and interpretation: Predictions are estimates; extrapolation beyond observed inputs can be unreliable. 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.