Texas HS Precalculus Question of the Day

Take $\ln y$ versus $t$ and fit a line; declare an exponential model adequate based on a high $R^2$, without checking for saturation or residual patterns.

Choose a logistic function and adjust parameters by eye until the curve looks close; accept the fit without diagnostics or predictive checks.

Compute successive average growth rates; if they decrease, declare the model logistic and stop without parameter estimation or validation.

Inspect the data for saturation and propose $y(t)=\tfrac{L}{1+Ae^{-kt}}$; estimate $L$ from the observed plateau; linearize via $\ln!\left(\tfrac{L}{y}-1\right)=\ln A-kt$ and regress to estimate $k$ and $A$; back-transform to obtain the model; validate with residual plots (no curvature), compare errors to an exponential alternative, and test a withheld time point; verify the inflection occurs near when $y\approx L/2$.