Question 743·Medium·Math

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Each dot in the scatterplot represents one day of data for a solar panel. The horizontal coordinate $x$ is the number of hours of direct sunlight that day, and the vertical coordinate $y$ is the amount of energy produced, in kilowatt-hours (kWh), that day.

The graph of which equation gives a line that most closely fits the data?

Dots scattered around a rising straight path call for a line of best fit, not a line that passes through every dot. Enter the approximate coordinates in a Desmos table and use a linear regression to estimate the slope and intercept. Compare both numbers with the choices. A low dot still counts, but it shouldn't determine the line by itself.

Hints

Hint 1
A data pair keeps one day's sunlight hours and energy together. Read its horizontal coordinate first and its vertical coordinate second. The leftmost dot is near $(1,2)$ ; how would you enter all the pairs in a Desmos table?
Hint 2
A linear regression finds a line that follows the data as a group. Type $y_1\sim mx_1+b$ using the table, and include the low dot even though it sits away from most of the others.
Hint 3
In $y=mx+b$ , the slope $m$ tells you the energy increase per extra hour of sunlight. The $y$ -intercept $b$ is the model's value at zero hours. Which equation is closest on both counts?

Step-by-step

Approach 1: Fit the plotted points in Desmos

Step 1
Enter each day's pair
Sunlight hours are the horizontal $x$ -values, and energy in kWh is the vertical $y$ -value, so put each same-day pair in a Desmos table. Read the dots approximately; Desmos plots each row. Include both dots at $x=5$ , even though one sits below the main group.
Step 2
Fit a line to all eight dots
Below the table, type $y_1\sim mx_1+b$ . The $\sim$ asks Desmos for a linear regression: it adjusts $m$ and $b$ to fit the whole group. Under PARAMETERS, Desmos shows $m\approx1.38$ and $b\approx0.57$ . A best-fit line follows the whole cloud, not one dot. It doesn't have to pass through the low dot.
Step 3
Match both parts of the equation
In $y=mx+b$ , the slope $m$ is the modeled kWh gained per extra sunlight hour; the $y$ -intercept $b$ is the model's value at zero hours. The fitted $1.38$ is closest to $1.5$ , and $0.57$ is closest to positive $0.5$ . So the closest model for daily energy is $y=1.5x+0.5$ . Choice A.

Approach 2: Estimate a line through the middle

Step 1
Pick points on the trend
Trace an imaginary line through the middle of the dots. It runs near $(7,11)$ and crosses the vertical axis near $(0,0.5)$ . These are points on your estimated best-fit line, not necessarily observed days. Don't bend the line toward the low dot near $(5,6)$ .
Step 2
Find the rise per hour
The slope is the change in energy divided by the change in sunlight hours. From the two points on your estimated line, type $\frac{11-0.5}{7-0}$ in Desmos. It displays $1.5$ , so the line rises about $1.5$ kWh per hour.
Step 3
Write the fitted equation
Put the slope $1.5$ and the estimated $y$ -intercept $0.5$ into $y=mx+b$ : $y=1.5x+0.5$ . This models daily energy in kWh from sunlight hours while following the overall trend. Choice A.

Question 743·Medium·Math

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