To determine the slope of the linear function from the given table, we use the formula for the slope of a line passing through two points, which is:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} \][/tex]
Where [tex]\(\Delta y\)[/tex] is the change in the [tex]\(y\)[/tex]-coordinates (outputs) and [tex]\(\Delta x\)[/tex] is the change in the [tex]\(x\)[/tex]-coordinates (inputs).
Here is the table for reference:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 8 \\
\hline
-1 & 2 \\
\hline
0 & -4 \\
\hline
1 & -10 \\
\hline
2 & -16 \\
\hline
\end{array}
\][/tex]
Let's consider the first two points in the table to find the changes in [tex]\(y\)[/tex] and [tex]\(x\)[/tex]:
- Point 1: [tex]\((-2, 8)\)[/tex]
- Point 2: [tex]\((-1, 2)\)[/tex]
Calculate the change in [tex]\(y\)[/tex] ([tex]\(\Delta y\)[/tex]):
[tex]\[
\Delta y = y_2 - y_1 = 2 - 8 = -6
\][/tex]
Calculate the change in [tex]\(x\)[/tex] ([tex]\(\Delta x\)[/tex]):
[tex]\[
\Delta x = x_2 - x_1 = -1 - (-2) = -1 + 2 = 1
\][/tex]
Now, we can find the slope by dividing the change in [tex]\(y\)[/tex] by the change in [tex]\(x\)[/tex]:
[tex]\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{-6}{1} = -6
\][/tex]
Thus, the slope of the function is:
[tex]\[
-6
\][/tex]
Among the given options, the correct answer is:
[tex]\[
\boxed{-6}
\][/tex]
