Question

look at this scatter plot. which equation could represent the line of best fit for this data?
options:
y = x + 2
y = \frac{1}{4}x + 4
y = \frac{1}{3}x + 2
y = 3x + 3
y = \frac{3}{2}x + 1
y = 4x + 3

Explanation:

Step1: Analyze the trend of the scatter plot

The scatter plot shows a positive linear trend, meaning as \( x \) increases, \( y \) generally increases. We can estimate the slope and y - intercept by looking at the general direction of the points.

Step2: Analyze the slope and y - intercept of each equation

• For \( y=x + 2\): The slope \( m = 1\), y - intercept \( b=2\).
• For \( y=\frac{1}{4}x + 4\): Slope \( m=\frac{1}{4}\), y - intercept \( b = 4\).
• For \( y=\frac{1}{3}x+2\): Slope \( m=\frac{1}{3}\), y - intercept \( b = 2\).
• For \( y = 3x+3\): Slope \( m = 3\), y - intercept \( b=3\).
• For \( y=\frac{3}{2}x + 1\): Slope \( m=\frac{3}{2}\), y - intercept \( b = 1\).
• For \( y=4x + 3\): Slope \( m = 4\), y - intercept \( b=3\).

Looking at the scatter plot, the slope should be a positive value less than 1 (since the rate of increase of \( y \) with respect to \( x \) is moderate) and the y - intercept should be around 2. Among the given equations, \( y=\frac{1}{3}x + 2\) has a slope of \(\frac{1}{3}\) (a moderate positive slope) and a y - intercept of 2, which seems to fit the general trend of the scatter plot. The other equations either have a slope that is too steep (like \( y = 3x+3\), \( y=4x + 3\), \( y=\frac{3}{2}x+1\)) or a y - intercept that does not match the general position of the points (like \( y=\frac{1}{4}x + 4\) has a relatively high y - intercept, \( y=x + 2\) has a slope that is steeper than what the scatter plot suggests).

Answer:

\( y=\frac{1}{3}x + 2 \)