Transformation to Achieve Linearity vs Linear Model

When a scatterplot curves, you can re-express one or both variables and fit the line to the new pairs instead. Growth that multiplies is the classic case: the points $(1, 6)$, $(2, 18)$, and $(3, 54)$ triple at every step, and taking $\log_{10} y$ turns them into $(1, 0.778)$, $(2, 1.255)$, and $(3, 1.732)$, which climb by a constant $0.477$ per step and therefore lie on a straight line. You then fit $\log \hat{y} = a + bx$ (read y-hat for $\hat{y}$) and back-transform to predict in the original units, checking the residual plot of the transformed data to confirm the curve is gone. The Fall 2026 course still asks you to spot departures from linearity in a residual plot, but it has no topic on re-expressing data to straighten a curve, so transformation is college and textbook content rather than AP content.

A linear model summarizes a scatterplot with one straight line, so any $x$ you choose turns into a prediction. It is written $\hat{y} = a + bx$, where $\hat{y}$ (read y-hat) is the predicted response, $a$ is the y-intercept, and $b$ is the slope. For example, if $\hat{y} = 12 + 3x$ predicts a plant's height in centimeters from weeks of growth, then at $x = 5$ weeks the model gives $12 + 3(5) = 27$ centimeters. Fit a linear model only when the scatterplot and the residual plot show a linear form, since the equation will happily produce numbers for curved data too.