Raquel throws darts at a coordinate grid centered at the origin. Her goal is to create a line of darts. Her darts actually hit the coordinate grid at [tex]\((-5, 0)\)[/tex], [tex]\((1, -3)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((-8, -6)\)[/tex], [tex]\((0, 2)\)[/tex], and [tex]\((9, 6)\)[/tex].

Which equation best approximates the line of best fit of the darts?

A. [tex]y = 0.6x + 0.6[/tex]
B. [tex]y = 0.1x + 0.8[/tex]
C. [tex]y = 0.8x + 0.1[/tex]
D. [tex]y = 0.5x + 0.6[/tex]



Answer :

To determine which equation best approximates the line of best fit for Raquel's dart throws, we need to perform a detailed step-by-step analysis. First, let's analyze the coordinates of the darts:

The coordinates where the darts hit are:
[tex]$ (-5, 0), \quad (1, -3), \quad (1, 5), \quad (-8, -6), \quad (0, 2), \quad (9, 6) $[/tex]

Here’s how we can determine the equation of the line of best fit:

1. Organize the Data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & 0 \\ 1 & -3 \\ 1 & 5 \\ -8 & -6 \\ 0 & 2 \\ 9 & 6 \\ \hline \end{array} \][/tex]

2. Calculate the Slope (m) and Intercept (c):

To find the line of best fit, you typically use the least squares method to perform a linear regression. After performing this regression, we find:
[tex]\[ \text{slope} = 0.6147859922178989, \quad \text{intercept} = 0.8715953307392992 \][/tex]

3. Compare with the Given Options:

Let's compare the calculated slope and intercept with the given equations to determine which is the closest match:

- [tex]\( y = 0.6x + 0.6 \)[/tex]
- [tex]\( y = 0.1x + 0.8 \)[/tex]
- [tex]\( y = 0.8x + 0.1 \)[/tex]
- [tex]\( y = 0.5x + 0.6 \)[/tex]

We see that the first equation, [tex]\( y = 0.6x + 0.6 \)[/tex], has a slope of [tex]\(0.6\)[/tex] and an intercept of [tex]\(0.6\)[/tex], which is closest to our calculated values.

4. Determine the Best Match:

By computing the absolute differences between the calculated slope and intercept with those provided in the options, the smallest differences align with the equation:
[tex]\[ y = 0.6x + 0.6 \][/tex]

Therefore, the equation that best approximates the line of best fit of the darts is:
[tex]$ y = 0.6x + 0.6 $[/tex]