Choosing among algebraic model families

Choosing among algebraic model families

Choosing among algebraic model families Choose and critique linear, quadratic, power and rational models using both data and context. This precalculus practice asks what a proposed formula explains, what restrictions it must satisfy and what further evidence would distinguish it from competing formulas. Questions include unequal input spacing, constant-area constraints, a square-root inverse relation, bounded growth and two models that agree at existing observations. Residual and validation tasks compare numerical fit with the structure of the errors. A final algebraic counterexample shows why finitely many exact observations do not establish one universal model. Prerequisites include linear and quadratic models, power functions, rational behavior and the meaning of a residual. All datasets are synthetic and supplied in the questions. Single-select and multiselect items progress from direct model selection to designing a useful comparison and evaluating the limits of the evidence. Check formulas against observations, but also check units, allowed inputs, expected long-run behavior and the reason a relationship should hold. Detailed explanations separate a best choice among named candidates from a proof about every possible function. Use independent validation evidence and inspect error patterns when supplied; a small training error or an attractive curve alone does not guarantee reliable extrapolation. The set supports reasoned model selection without requiring advanced regression theory.