Researchers measured the daily growth, , of a seedling for different numbers of hours of sunlight, . | Sunlight (hours) | Growth (millimeters) | |---|---| |2|1.2| |4|2.3| |6|3.0| |8|4.1| |10|5.2| Which equation best approximates the line of best fit for the data, where is predicted growth and is hours of sunlight?

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SAT Two-Variable Data Question 11 | Free SAT Prep | Aniko

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Question 11·Medium·Two-Variable Data: Models and Scatterplots

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Researchers measured the daily growth, $y$ , of a seedling for different numbers of hours of sunlight, $x$ .

Sunlight (hours) $x$	Growth (millimeters) $y$
2	1.2
4	2.3
6	3.0
8	4.1
10	5.2

Which equation best approximates the line of best fit for the data, where $y$ is predicted growth and $x$ is hours of sunlight?

For line-of-best-fit questions with a small table of nearly linear data, quickly estimate the slope using two convenient points that are far apart (often the first and last) so random fluctuations average out. Match that slope to the answer choices to eliminate options with clearly wrong slopes, then use one data point and $y = mx + b$ to find the intercept, or simply test the remaining equations by plugging in a couple of $x$ ‑values from the table to see which one’s predictions stay closest to all the data.

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Think about the type of relationship

The data show $y$ increasing as $x$ increases in a fairly steady way. What simple type of equation models a steady rate of change between two variables?

Estimate the rate of change (slope)

Use two points that are far apart in the table, such as $(2, 1.2)$ and $(10, 5.2)$ . Compute the change in $y$ divided by the change in $x$ to estimate the slope.

Find the intercept from one point

Once you have an approximate slope, plug that slope and any one of the data points into $y = mx + b$ to solve for $b$ . Then look for the answer choice whose $m$ and $b$ match your estimates.

Desmos Guide

Enter the data as a table

In Desmos, add a table and enter the $x$ ‑values 2, 4, 6, 8, 10 in the first column and the corresponding $y$ ‑values 1.2, 2.3, 3.0, 4.1, 5.2 in the second column. You should see the five data points plotted.

Graph each answer choice as a line

In separate expression lines, type each option exactly: y = 0.4x + 0.4, y = 0.5x + 0.2, y = 0.6x - 0.3, and y = 0.8x - 0.5. Desmos will draw all four lines on the same graph as the points.

Compare which line best matches the points

Look at how close each line is to all five data points. Focus especially on whether the line goes through or very near the first and last points and stays close to the middle ones. The equation whose line stays closest overall to the plotted points is the best approximation for the line of best fit.

Step-by-step Explanation

Write the general form of a linear model

Because the data appear roughly linear, we model the relationship between sunlight $x$ and growth $y$ with a line of the form

y = mx + b

where $m$ is the slope (growth per hour of sunlight) and $b$ is the y‑intercept (predicted growth when $x=0$ ). Our job is to estimate $m$ and $b$ from the table.

Estimate the slope from the data

Use the first and last points to estimate the slope. These are $(2, 1.2)$ and $(10, 5.2)$ .

The slope is

m = \frac{\Delta y}{\Delta x} = \frac{5.2 - 1.2}{10 - 2} = \frac{4.0}{8} = 0.5.

So the line of best fit should have slope about $0.5$ millimeters of growth per hour of sunlight.

Use a data point to find the y-intercept

Now plug the estimated slope $m = 0.5$ and one of the data points into $y = mx + b$ to solve for $b$ .

Using the point $(2, 1.2)$ :

\begin{aligned} 1.2 &= 0.5(2) + b \\ 1.2 &= 1 + b \\ b &= 0.2 \end{aligned}

So the y‑intercept of the best‑fit line should be about $0.2$ millimeters.

Write and check the equation

A line with slope $0.5$ and y‑intercept $0.2$ has equation

y = 0.5x + 0.2.

Check a couple of points:

For $x = 4$ : predicted $y = 0.5(4) + 0.2 = 2.2$ (actual is $2.3$ ).
For $x = 8$ : predicted $y = 0.5(8) + 0.2 = 4.2$ (actual is $4.1$ ).

The predictions are very close to the actual values, and the line passes exactly through the first and last points. Therefore, the best equation of the line of best fit is $y = 0.5x + 0.2$ .

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Step-by-step Explanation

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