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]
