Finding the Best-Fitting Equation for Scatterplot Data
To find the equation that best fits the data on a scatterplot, we look for the line that closely follows the trend of the data points.
From what I can see in the image:
1. The slope of the line should be positive, as the points seem to increase as we move from left to right.
2. The y-intercept is the point where the line crosses the y-axis. At x = 0, the line should cross the y-axis around the value of 9.
This information helps us eliminate some of the options:
- Option A (\(y = \frac{1}{2}x + 9\)) has a positive slope and the correct y-intercept.
- Option B (\(y = 2x + 9\)) also has a positive slope and the correct y-intercept.
- Option C (\(y = \frac{1}{2}x - 9\)) has the correct slope direction but the incorrect y-intercept.
- Option D (\(y = 2x + 9\)) is the same as option B, which appears to be an error in the image as it's duplicated and does not provide another distinct option.
Now, to distinguish between the remaining options A and B (or D, since it's the same), we need to consider the steepness of the line in the scatterplot. Option A has a gentler slope (\(\frac{1}{2}\)) than Option B (2).
By inspecting the graph, it looks like the data points rise slower than a slope of 2 for every increase in x (since a line with slope 2 would rise too steeply for these data points), which suggests that Option A with a slope of \(\frac{1}{2}\) and a y-intercept at 9 would be the best fit for the data.
Thus, Option A, \(y = \frac{1}{2}x + 9\), is most likely the equation that best fits the data on the scatterplot.
