To determine the [tex]\( y \)[/tex]-intercepts of the function given in the table, we need to identify the points where the function value [tex]\( f(x) = 0 \)[/tex]. The function values are essentially the [tex]\( y \)[/tex]-coordinates, and the y-intercept occurs whenever the function crosses the y-axis, i.e., [tex]\( f(x) = 0 \)[/tex].
From the given data:
[tex]\[
\begin{array}{|c|c|}
\hline x & f(x) \\
\hline -2 & -8 \\
\hline -1 & 0 \\
\hline 0 & 0 \\
\hline 1 & -2 \\
\hline 2 & 0 \\
\hline 3 & 12 \\
\hline
\end{array}
\][/tex]
Let's go through the table step-by-step and identify the points where [tex]\( f(x) = 0 \)[/tex]:
1. When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -8 \)[/tex] (not 0).
2. When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (this is a y-intercept).
3. When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (this is a y-intercept).
4. When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -2 \)[/tex] (not 0).
5. When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (this is a y-intercept).
6. When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 12 \)[/tex] (not 0).
Therefore, the points where the function has [tex]\( f(x) = 0 \)[/tex] are:
[tex]\[
(-1, 0), (0, 0), (2, 0)
\][/tex]
So, the list of all [tex]\( y \)[/tex]-intercepts of the function is:
[tex]\[
(-1, 0), (0, 0), (2, 0)
\][/tex]
The correct option is:
[tex]\[
(-1,0), (0,0), (2,0)
\][/tex]
