SAT Two-Variable Data Question 24 | Free SAT Prep | Aniko

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Question 24·Hard·Two-Variable Data: Models and Scatterplots

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The scatterplot above shows the number of baskets of produce harvested each week during a 10-week growing season.

Which choice gives a quadratic equation that best models the data in the scatterplot?

For a scatterplot that looks quadratic, use a few high-value checkpoints instead of trying to match every point: (1) decide whether the parabola opens up or down, (2) check the ends of the data range to eliminate models with the wrong vertical shift, and (3) use the approximate peak (both its location and height) to choose the curvature that best matches the middle of the data.

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Use the intercepts first

The point at week $0$ is near $23$ , and the point at week $10$ is also near $23$ . Check which equations give values near $23$ at $x=0$ and $x=10$ .

Then use the peak

After eliminating any equations that miss the ends, compare the remaining equations at the peak near $x=5$ (about $45.5$ baskets).

Don’t rely on just one checkpoint too early

If more than one equation matches one key point, use another key point (like the other end of the season or the maximum) to break the tie.

Desmos Guide

Enter the candidate equations

Type each option as a separate equation (for example, enter them as $y_1$ , $y_2$ , $y_3$ , and $y_4$ ).

Plot the scatterplot points in a table

Create a table with columns $x_1$ and $y_1$ , then enter the points from the scatterplot (weeks and baskets) so Desmos shows the same set of dots.

Compare overall closeness

Turn on all four graphs and look for the equation whose curve stays closest to the dots throughout the season (especially near week $0$ , week $10$ , and the peak around week $5$ ).

Step-by-step Explanation

Use key features from the scatterplot

From the scatterplot:

The harvest increases and then decreases, so a quadratic model should open downward (a negative $x^2$ coefficient).
The harvest is about $23$ baskets at week $0$ and also about $23$ baskets at week $10$ .
The maximum occurs near week $5$ and is about $45.5$ baskets.

Check the value at week 0

At $x=0$ , the model value is the constant term.

$y=-0.9x^2+9x+23 \Rightarrow y(0)=23$
$y=-0.85x^2+8.75x+23 \Rightarrow y(0)=23$
$y=-0.85x^2+8.5x+23 \Rightarrow y(0)=23$
$y=-0.85x^2+8.25x+25.5 \Rightarrow y(0)=25.5$

The last equation starts too high compared with the point near $(0,23)$ , so eliminate it.

Check the value at week 10

The scatterplot is also near $23$ baskets at week $10$ .

Evaluate the remaining equations at $x=10$ :

For $y=-0.9x^2+9x+23$ :

$y(10)=-0.9(100)+9(10)+23=-90+90+23=23$
For $y=-0.85x^2+8.75x+23$ :

$y(10)=-0.85(100)+8.75(10)+23=-85+87.5+23=25.5$
For $y=-0.85x^2+8.5x+23$ :

$y(10)=-0.85(100)+8.5(10)+23=-85+85+23=23$

The second equation gives $25.5$ at week $10$ , which is too high, so eliminate it.

Use the peak near week 5 to choose between the finalists

Now compare the two remaining equations at $x=5$ , where the scatterplot peaks near $45.5$ baskets:

For $y=-0.9x^2+9x+23$ :

$y(5)=-0.9(25)+9(5)+23=-22.5+45+23=45.5$
For $y=-0.85x^2+8.5x+23$ :

$y(5)=-0.85(25)+8.5(5)+23=-21.25+42.5+23=44.25$

The first equation matches the peak much more closely, so the best model is:

$y=-0.9x^2+9x+23$

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation