Use a trend line/model to make predictions or estimate values. | Two-variable data: Models and scatterplots | SAT Math

核心知识

A trend line (line of best fit) lets you estimate the value of one variable given the other — either by reading the graph or plugging into the equation.

深入理解

A trend line is the straight line drawn through a scatterplot that best represents the overall pattern. Once you have it, you can predict values that weren't directly measured.

Two ways the SAT tests this:

From the graph — They give you a scatterplot with a line drawn through it and ask: "Based on the line of best fit, what is the predicted value of $𝑦$ when $𝑥 = 50$ ?" Go to $𝑥 = 50$ on the horizontal axis, move straight up to the line, then read across to the $𝑦$ -axis.
From the equation — They give you something like $𝑦 = 2.3 𝑥 + 15$ and ask for the predicted $𝑦$ when $𝑥 = 20$ . Plug in: $𝑦 = 2.3 (20) + 15 = 61$ .

Predictions within the range of the data (interpolation) are more reliable than predictions outside it (extrapolation). The SAT sometimes asks whether a prediction is reasonable — if the $𝑥$ -value is far beyond the data, the prediction is less trustworthy.

A predicted value almost never matches an actual data point exactly. The trend line shows the general pattern, not exact outcomes.

分步讲解

Identify whether you're working from a graph or an equation.
If from a graph: locate the x-value, go up (or down) to the trend line, then read the corresponding y-value.
If from an equation: substitute the given x-value and compute y.
Check whether the x-value falls within the range of the original data. If it's far outside, note that the prediction involves extrapolation.

常见误解

Reading values from data points instead of from the line of best fit. The question asks for the predicted value from the line, not the actual data point.
Assuming the predicted value must match an actual observation. Predictions are estimates based on the trend, not guarantees.
Treating extrapolated predictions as equally reliable as interpolated ones.

题目

示例解析

The line of best fit for a set of data relating the age of a used car (in years) to its sale price (in thousands of dollars) is $𝑝 = - 1.8 𝑎 + 28$ , where $𝑎$ is age and $𝑝$ is price. Based on this model, what is the predicted sale price, in thousands of dollars, of a car that is 6 years old?

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