Domain: Problem-Solving and Data Analysis | Skill: Two-variable data: models and scatterplots | Difficulty: Medium

Mastering Scatterplots and Linear Models on the SAT

Welcome to your guide for tackling one of the most common question types in the SAT Math section: two-variable data, specifically focusing on scatterplots and lines of best fit. These questions are a staple of the Problem-Solving and Data Analysis domain. At the Medium difficulty level, you’re expected to move beyond simply reading a point on a graph. You need to interpret the model—usually a line of best fit—that represents the trend in the data. Mastering this skill is about understanding the story the data tells and using the language of algebra to describe it.

Decoding Scatterplot Questions

The SAT uses these questions to test your ability to connect a real-world scenario to a graphical representation and its algebraic model. Let’s break down the common formats.

Typical Format	What It Tests	Quick Strategy
A scatterplot is shown with a line of best fit. The question asks for the equation of the line.	Calculating the slope (rate of change) and identifying the y-intercept (starting value) from the graph or given points.	Use two points on the line (not just any data points) to calculate the slope with \( m = \dfrac{y_2 – y_1}{x_2 – x_1} \).
A linear equation is given to model a real-world situation (e.g., “A bakery sells loaves…”). The question asks to interpret the slope or y-intercept.	Understanding the real-world meaning of the components of a linear equation (\(y = mx + b\)).	The slope (\(m\)) is the “rate” or change per one unit of \(x\). The y-intercept (\(b\)) is the “initial” or “starting” value when \(x=0\).
A scatterplot shows data points. The question asks which of the given equations is the most appropriate line of best fit.	Estimating the slope and y-intercept visually and using the process of elimination.	First, estimate the y-intercept to eliminate choices. Then, determine if the slope should be positive or negative and estimate its value.

Real SAT-Style Example

The scatterplot above shows the number of students \( s \) enrolled in an online course \( t \) months after its launch, where \( 0 \leq t \leq 12 \). A line of best fit is also shown. The line passes through the points \( (0,500) \) and \( (8,1400) \). Which of the following could be an equation of the line of best fit shown?

A) \( s = 500 + 112.5 t \) ✅

B) \( s = 500 + 900 t \)

C) \( s = 112.5 + 500 t \)

D) \( s = 900 + 500 t \)

Step-by-Step Solution

An equation for a line is typically written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this problem, the variables are \( s \) and \( t \), so our equation will be in the form \( s = mt + b \).

1. Find the y-intercept (\(b\)): The y-intercept is the value of \( s \) when \( t = 0 \). The problem states the line passes through the point \( (0,500) \). Therefore, the y-intercept \( b \) is 500.
2. Calculate the slope (\(m\)): The slope represents the rate of change. We can use the slope formula \( m = \dfrac{\text{change in } s}{\text{change in } t} = \dfrac{s_2 – s_1}{t_2 – t_1} \) with the two given points, \( (t_1, s_1) = (0,500) \) and \( (t_2, s_2) = (8,1400) \).

\[ m = \dfrac{1400 – 500}{8 – 0} = \dfrac{900}{8} = 112.5 \]

3. Construct the equation: Now we combine our slope \( m = 112.5 \) and our y-intercept \( b = 500 \) into the equation \( s = mt + b \).

\[ s = 112.5t + 500 \]

4. Match with the options: This equation is the same as option A, \( s = 500 + 112.5 t \).

Your 4-Step Strategy for Scatterplot Models

1. Identify Variables and Context: Before you calculate anything, understand what the axes represent. What does \(x\) mean? What does \(y\) mean? What are the units? In our example, \(t\) is months and \(s\) is the number of students.
2. Pinpoint Key Features of the Model: Look for the two key parts of the linear model: the y-intercept and the slope. The question will always give you enough information to find them, either with explicit points or through the graph itself.
3. Calculate the Slope and Intercept: Use the given points on the line of best fit (not just any data points from the scatterplot) to calculate the slope using \( m = \dfrac{y_2 – y_1}{x_2 – x_1} \). The y-intercept is the point where the line crosses the vertical axis (where \(x=0\)).
4. Build the Equation and Eliminate: Assemble your calculated slope and y-intercept into the \(y = mx + b\) format. Compare your result with the answer choices. Even if you only find the intercept first, you can often eliminate two or three incorrect options immediately.

Applying the Strategy to Our Example

Let’s walk through the example problem again, this time using our formal strategy step-by-step.

Step 1 Applied: Identify Variables and Context

The problem states that \(s\) is the number of students enrolled and \(t\) is the number of months after launch. The model will predict the number of students based on the number of months that have passed.

Step 2 Applied: Pinpoint Key Features of the Model

We need to find the equation for the line of best fit. This requires two features: the slope (\(m\)) and the y-intercept (\(b\)). The problem gives us two explicit points that lie on this line: \((0, 500)\) and \((8, 1400)\).

Step 3 Applied: Calculate the Slope and Intercept

First, the y-intercept. This is the value of \(s\) when \(t=0\). The point \((0, 500)\) tells us directly that the y-intercept, \(b\), is 500.

Next, the slope. Using our points \((0, 500)\) and \((8, 1400)\) and the slope formula: \[ m = \dfrac{s_2 – s_1}{t_2 – t_1} = \dfrac{1400 – 500}{8 – 0} = \dfrac{900}{8} = 112.5 \] So, the slope \(m\) is 112.5.

Step 4 Applied: Build the Equation and Eliminate

Using the form \(s = mt + b\), we substitute our values: \(s = 112.5t + 500\). This can also be written as \(s = 500 + 112.5t\). Looking at the options, this perfectly matches choice A. We could have also used the y-intercept of 500 to immediately eliminate choices C and D, which have incorrect intercepts.

Ready to Try It on Real Questions?

Now that you understand the strategy, it’s time to practice with authentic SAT questions! Head to mytestprep.ai and follow these steps:

1. Login using your account or signup on mytestprep.ai
2. Click on Practice Sessions once you are on the dashboard. You will see the link on the left side navigation menu of the dashboard
3. Click on Create New Session
4. Start with Co-Pilot Mode on with hints and explanations—it’s like having a personal coach who explains exactly why each answer is right or wrong
5. Select Math as your subject
6. Select Problem-Solving and Data Analysis, Two-variable data: models and scatterplots and Medium Difficulty
7. Start practicing. Happy Practicing!

Key Takeaways

• Context is King: Always start by understanding what the variables in the problem represent.
• Slope is the Rate: The slope (\(m\)) of a line of best fit tells you how much the \(y\)-variable is predicted to change for every one-unit increase in the \(x\)-variable.
• Intercept is the Start: The y-intercept (\(b\)) is the predicted value of the \(y\)-variable when the \(x\)-variable is zero.
• Use the Line: When calculating slope, use points that are explicitly stated to be on the line of best fit, not just any two data points from the scatterplot itself.