Fit a linear function to bivariate numerical data on a scatter plot, interpret the slope and intercept in context, and use the model to make predictions (MA.912.DP.2.4, MA.912.DP.2.5).

What this topic is asking

MA.912.DP.2 asks you to work with bivariate (two-variable) numerical data on a scatter plot: describe the association, fit a line of best fit, interpret its slope and intercept, and predict. The B.E.S.T. Algebra 1 EOC ties this directly to linear functions, so the slope and intercept skills from the algebra strand reappear here.

Describing the association

Read three things from the scatter plot:

• Direction. Upward (as $x$ increases, $y$ increases) is positive; downward is negative.
• Strength. Points hugging a line are a strong association; a loose cloud is weak.
• Form. A straight trend is linear; a curve is nonlinear.

The line of best fit

A line of best fit is a line that passes through the middle of the trend, minimizing the overall distance to the points. Its equation $\hat{y} = mx + b$ is interpreted just like any linear function:

• Slope $m$: the predicted change in $y$ for each one-unit increase in $x$.
• $y$-intercept $b$: the predicted $y$ when $x = 0$.

Predicting: interpolation versus extrapolation

• Interpolation: predicting within the range of the data. Generally reliable, because the trend is observed there.
• Extrapolation: predicting far beyond the data. Risky, because the trend may not continue (the -\200$ intercept above is an extrapolation artifact).

How the B.E.S.T. EOC examines this topic

• Multiple choice. Describe the association, or interpret a slope or intercept.
• Equation editor and number entry. Use a line of best fit to predict a value.
• GRID. Identify or place a reasonable trend line on a scatter plot.

A clarifying idea: a line of best fit is a linear model of a trend, so its slope and intercept carry the same meaning as in any linear function, only now they describe a predicted, average relationship rather than an exact rule. That is why interpretations use words like "predicted" or "associated."

Why extrapolation is risky

Predicting well outside the data range is unreliable because the line of best fit is only known to describe the trend where data exists. Inside that range, the points confirm the linear pattern, so interpolating is trustworthy. Far beyond the data, nothing guarantees the relationship stays linear: ice-cream sales cannot keep rising forever as temperature climbs, a plant's growth levels off, and a negative intercept (like -\200$in sales) is meaningless because the real relationship bends or stops before reaching$x = 0$. The model captured a local straight-line trend, not a universal law, so pushing it into unobserved territory can give absurd answers. This is why the EOC flags interpretations at$x = 0$ when zero lies far from the data, and why prediction questions stay near the observed range.

Connecting to the algebra strand

Because the line of best fit is just $y = mx + b$, every skill from writing and graphing linear functions transfers: finding the slope between two points on the line, reading the intercept, and substituting to predict. The difference is interpretive, the data only suggest a trend, so the line summarizes a relationship rather than defining one exactly. Recognizing the trend line as an ordinary linear function lets you reuse the algebra you already know on the statistics items.

Try this

Q1. A line of best fit is $\hat{y} = -3x + 40$. What does the slope mean? [1 point]

• Cue. Each one-unit increase in $x$ predicts a decrease of about 3 in $y$.

Q2. Using $\hat{y} = 2x + 5$, predict $y$ when $x = 12$. [1 point]

• Cue. $\hat{y} = 2(12) + 5 = 29$.