Meaning of Model Components

Two-Variable Data: Meaning of Model Components

These questions show a scatterplot with a line of best fit and ask you to pick the equation that matches the line, or to interpret what the slope and y-intercept mean in context. The key skill is identifying the sign of the slope and the sign of the y-intercept from the graph.

The Two Things to Check

For any linear equation $y = mx + b$:

1. Slope ($m$): Does the line go up (positive) or down (negative) from left to right?
2. Y-intercept ($b$): Where does the line cross the vertical axis — above zero (positive) or below zero (negative)?

Once you know the signs of $m$ and $b$, there's usually only one answer choice that matches.

Worked Example 1 — Positive Slope, Positive Intercept

A scatterplot shows the relationship between study hours ($h$) and test score ($S$), along with a line of best fit. Which equation best represents the line?

A) $S = 55 + 4h$
B) $S = 55 - 4h$
C) $S = -55 + 4h$
D) $S = -55 - 4h$

SOLUTION

Step 1 — Slope: The line goes up from left to right → slope is positive ($+4$). This eliminates B (slope $-4$) and D (slope $-4$).
Step 2 — Y-intercept: The line crosses the y-axis above zero (around 55) → intercept is positive ($+55$). This eliminates C (intercept $-55$).
Only A has both positive slope and positive y-intercept.
Answer: A) $S = 55 + 4h$

Worked Example 2 — Positive Slope, Positive Intercept (Different Context)

An ice cream shop tracked daily sales ($s$, in dollars) vs. temperature ($t$, in °C). The scatterplot and line of best fit are shown. Which equation best describes the relationship?

A) $s = 100 - 15t$
B) $s = -100 - 15t$
C) $s = 100 + 15t$
D) $s = -100 + 15t$

SOLUTION

Slope: Higher temperature → higher sales → line goes up → slope is positive ($+15$). Eliminates A and B.
Y-intercept: The line crosses the y-axis at a positive value (around $100). Eliminates D.
Answer: C) $s = 100 + 15t$

Interpreting Slope and Intercept in Context

Some questions don't ask for the equation — they ask what the slope or intercept means:

• Slope $=$ rate of change per unit of $x$.
  Example: "For each additional hour of study, the predicted score increases by 4 points."

• Y-intercept $=$ predicted value when $x = 0$.
  Example: "A student who studies 0 hours is predicted to score 55."

Common Gotchas

• Mixing up slope and intercept. The slope is the coefficient of the variable ($m$ in $mx + b$). The intercept is the constant ($b$). Don't swap them.
• Getting the slope sign wrong. A line that goes down from left to right has a negative slope. Read the direction carefully — don't just glance.
• Interpreting slope as a one-time change. The slope is a per-unit rate: "for each additional hour," "per degree increase," etc. It is not a total or a one-time adjustment.
• Taking the y-intercept too literally. "When $x = 0$, the predicted $y$ is $b$" may not make real-world sense (e.g., a car at age 0 being worth exactly $25,000). The SAT still expects you to match the graph, even if the intercept is just a mathematical artifact.

What to Do on Test Day

• Check the slope sign first (up = positive, down = negative). This eliminates half the choices immediately.
• Then check the y-intercept sign (above zero = positive, below zero = negative). This narrows to one choice.
• You don't need to compute exact values. Just identify the signs and match.
• If asked about meaning: slope = "for each additional unit of $x$, the predicted $y$ changes by [slope amount]." Y-intercept = "when $x$ is 0, the predicted $y$ is [intercept value]."
• These take 15–20 seconds once you know the sign-checking technique.