Mathfolis

Two-Variable Data: Models & Scatterplots

Unit 3 · Problem-Solving & Data Analysis

What the SAT asks here

Two-variable data uses scatterplots and best-fit models. The SAT focuses on interpreting the slope and y-intercept of a fitted line in context, choosing the right model family for a given pattern, and computing or interpreting residuals. About one problem per test.

Model families

Roughly linear pattern
y=mx+by = mx + b
Rapid growth or decay
y=a⋅bxy = a \cdot b^x
U-shaped pattern
y=ax2+bx+cy = ax^2 + bx + c

Residual

Residual = actual y − predicted y
SAT Tip: Slope of a fitted line = predicted change in y per +1 unit in x. State the units explicitly when answering an interpretation question.
Caution: Extrapolating far outside the observed x-range is unreliable. The SAT often penalises this kind of overreach.
Type 1

Interpret slope or intercept in context

Translate m and b into a sentence about the real-world units of x and y.

Example 1
A scatterplot of (hours studied, test score) is modeled by ŷ = 6x + 50. What does the slope 6 represent?

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Type 2

Choose the best model

Match the visible shape of the data to a function family: linear for steady, exponential for constant percent change, quadratic for symmetric U-curves.

Example 2
A population doubles every 5 years. Which model best fits the data — linear, quadratic, or exponential?

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Type 3

Compute or interpret a residual

Residual = actual y − predicted y. Positive means the data point sits above the model line.

Example 3
A model predicts ŷ = 2x + 1. For the data point (3, 9), find the residual.

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