To transform a table of data to represent the reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex], we need to understand that reflecting a function over the line [tex]\( y = x \)[/tex] involves swapping the [tex]\( x \)[/tex]-coordinates and [tex]\( y \)[/tex]-coordinates. The reason for this is that reflection over the line [tex]\( y = x \)[/tex] essentially interchanges the roles of the [tex]\( x \)[/tex]-axis and [tex]\( y \)[/tex]-axis.
Given the table for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & -31 \\
-1 & 0 \\
1 & 2 \\
2 & 33 \\
\hline
\end{array}
\][/tex]
To find the reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex], we apply the following transformation rule:
- Switch the [tex]\( x \)[/tex]-values and [tex]\( y \)[/tex]-values in the table.
Applying this transformation to the table, we get:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-31 & -2 \\
0 & -1 \\
2 & 1 \\
33 & 2 \\
\hline
\end{array}
\][/tex]
This transformed table represents the reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex].
Thus, the correct rule to use is:
- A. Switch the [tex]\( x \)[/tex]-values and [tex]\( y \)[/tex]-values in the table.
