Interpreting Linear and Nonlinear Models

Interpreting Linear Models

For $\hat{y} = mx + b$:

• Slope $m$: for each 1-unit increase in $x$, $y$ changes by $m$ units.
• Y-intercept $b$: the predicted value of $y$ when $x = 0$.

Interpreting Nonlinear Models

Quadratic: $y = ax^2 + bx + c$. The vertex gives the optimal value. $a$ determines whether there's a maximum or minimum.

Exponential: $y = a \cdot b^x$. $a$ is the initial value. $b > 1$ means growth; $0 < b < 1$ means decay.

Choosing the Right Model

• Linear: constant rate of change. Scatterplot shows a straight-line trend.
• Quadratic: data rises then falls (or vice versa). Scatterplot shows a curved U or ∩ shape.
• Exponential: data increases increasingly rapidly (or decays). Scatterplot curves upward.

Residual Plots

A residual plot shows residuals vs. $x$-values. If the model is appropriate:

• Residuals are randomly scattered around 0.

If the model is inappropriate:

• Residuals show a pattern (e.g., curved).

Tip 1: Focus on What the Question Asks

"What does the slope represent?" or "What does the y-intercept represent?" — connect to context.

Tip 2: Check if the Model Is Appropriate

If a linear model is applied to curved data, the residual plot will show a pattern.

Tip 3: Predictions Are Only Valid Within the Data Range

Models may not hold outside the range of observed data.