To find the values of [tex]\( g(x) \)[/tex] when the function [tex]\( f(x) \)[/tex] is shifted vertically down by 2 units, we need to subtract 2 from each value of [tex]\( f(x) \)[/tex].
Given the table of values for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
f(x) & 0 & 2 & 8 & 26 \\
\hline
\end{array}
\][/tex]
The corresponding [tex]\( g(x) \)[/tex] values can be calculated as follows:
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0 \\
g(0) = f(0) - 2 = 0 - 2 = -2
\][/tex]
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2 \\
g(1) = f(1) - 2 = 2 - 2 = 0
\][/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 8 \\
g(2) = f(2) - 2 = 8 - 2 = 6
\][/tex]
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 26 \\
g(3) = f(3) - 2 = 26 - 2 = 24
\][/tex]
Thus, the table of values for [tex]\( g(x) \)[/tex] is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
g(x) & -2 & 0 & 6 & 24 \\
\hline
\end{array}
\][/tex]
