Question

there are three scatter plots (blue, red, black dots) with x and y axes from 0 to 5. on the right, three equations: $y = 0.5x + 2$, $y = 0.25x + 2$, $y = 0.25x + 3$. there are dashed boxes next to each scatter plot, likely for matching the best - fitting line.

Explanation:

Step1: Analyze the y-intercept

The y - intercept of a line in the form \(y = mx + b\) is \(b\). For the blue scatter plot, when \(x = 0\), the points are around \(y = 2\) (looking at the first graph, at \(x = 0\), the blue points are near \(y = 2\)). For the red scatter plot, at \(x = 0\), the red points are around \(y=3\). For the black scatter plot, at \(x = 0\), the black points are around \(y = 2\).

Step2: Analyze the slope

The slope \(m\) represents the rate of change. For the blue scatter plot, as \(x\) increases by \(4\) (from \(x = 0\) to \(x = 4\)), \(y\) increases by about \(2\) (from \(y\approx2\) to \(y\approx4\)), so the slope \(m=\frac{2}{4}=0.5\). For the red scatter plot, as \(x\) increases by \(4\) (from \(x = 0\) to \(x = 4\)), \(y\) increases by about \(1\) (from \(y\approx3\) to \(y\approx4\)), so the slope \(m = \frac{1}{4}=0.25\). For the black scatter plot, as \(x\) increases by \(4\) (from \(x = 0\) to \(x = 4\)), \(y\) increases by about \(1\) (from \(y\approx2\) to \(y\approx3\)), so the slope \(m=\frac{1}{4} = 0.25\).

Step3: Match the equations

• Blue scatter plot: slope \(m = 0.5\), y - intercept \(b = 2\), so the equation is \(y=0.5x + 2\).
• Red scatter plot: slope \(m=0.25\), y - intercept \(b = 3\), so the equation is \(y = 0.25x+3\).
• Black scatter plot: slope \(m = 0.25\), y - intercept \(b = 2\), so the equation is \(y=0.25x + 2\).

Answer:

• Blue scatter plot: \(y = 0.5x+2\)
• Red scatter plot: \(y=0.25x + 3\)
• Black scatter plot: \(y = 0.25x+2\)