Family Overview
In Plain Language
Topic 2.6 is about choosing and validating among competing models—usually linear, quadratic, and exponential—fit to the same data set.
- Build multiple candidates: slightly changing rates can sometimes be fit by more than one model type (2.6.A.1).
- Compare with context: unbounded steady change, leveling growth, or symmetric rise/fall point toward different families (2.6.A.2).
- Validate with residuals: residual = actual $-$ predicted; a residual plot with no clear pattern supports the model (2.6.B.1).
- Error in context: depending on the scenario, an overestimate or underestimate may be safer or more useful (2.6.B.2).
Why It Matters
Real data rarely announce their model type. AP questions ask students to fit competing models, read residual plots, and defend a choice with context—not just pick the highest $R^2$.
How Families Can Help
- Make three fits side by side: linear, quadratic, exponential.
- Sketch residual plots: random scatter vs curve vs fan shape.
- Ask: "What would go wrong if we used the wrong family?"
- Discuss safety contexts: medicine (avoid overdose), inventory (avoid stockout).
What Not to Do
- Do not choose a model from the calculator alone without residuals or context.
- Do not ignore a curved residual plot.
- Do not assume exponential fits every increasing data set.
- Do not confuse residual with percent error without defining terms.