Question

1. which equation represents the best line of fit for the scatter plot? (lesson 4 - 3)

a ( y = 0.6x+1 )
b ( y = 0.5x + 2 )
c ( y=x - 2 )
d ( y = 0.75x )

Explanation:

Step1: Analyze the y - intercept

For a line of the form \(y = mx + b\), the y - intercept \(b\) is the value of \(y\) when \(x = 0\). Looking at the scatter - plot, when \(x\) is near \(0\), \(y\) is near \(1\) or \(2\). Option C (\(y=x - 2\)) has a y - intercept of \(- 2\), which is not in line with the scatter - plot data. Option D (\(y = 0.75x\)) has a y - intercept of \(0\), which is also not a good fit.

Step2: Calculate the slope

We can use two points \((x_1,y_1)\) and \((x_2,y_2)\) to approximate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's assume two points \((2,2)\) and \((8,6)\). Then \(m=\frac{6 - 2}{8 - 2}=\frac{4}{6}\approx0.67\).
For option A: \(y = 0.6x+1\), when \(x = 2\), \(y=0.6\times2 + 1=1.2 + 1=2.2\); when \(x = 8\), \(y=0.6\times8+1=4.8 + 1=5.8\).
For option B: \(y = 0.5x + 2\), when \(x = 2\), \(y=0.5\times2+2=1 + 2=3\); when \(x = 8\), \(y=0.5\times8+2=4 + 2=6\).

Comparing the values of \(y\) calculated from the equations with the approximate \(y\) - values of the scatter - plot points, the line \(y = 0.6x + 1\) gives values that are closer to the scatter - plot points on average.

Answer:

A. \(y = 0.6x+1\)