To transform a table of data to represent the reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex], follow these steps:
1. Identify the original points on the table:
The table provided is as follows:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-2 & -31 \\
-1 & 0 \\
1 & 2 \\
2 & 33 \\
\end{array}
\][/tex]
2. Understand the reflection rule over the line [tex]\( y = x \)[/tex]:
To reflect a function [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex], you need to swap the [tex]\( x \)[/tex]-values and the [tex]\( y \)[/tex]-values. This means each coordinate point [tex]\((x, y)\)[/tex] becomes [tex]\((y, x)\)[/tex].
3. Apply the rule to each point:
[tex]\[
\begin{array}{c|c|c|c}
\text{Original} & \text{Transformed} \\
x & y & (x, y) & (y, x) \\
\hline
-2 & -31 & (-2, -31) & (-31, -2) \\
-1 & 0 & (-1, 0) & (0, -1) \\
1 & 2 & (1, 2) & (2, 1) \\
2 & 33 & (2, 33) & (33, 2) \\
\end{array}
\][/tex]
4. Summarize the transformation:
After swapping the [tex]\( x \)[/tex]-values and [tex]\( y \)[/tex]-values, the transformed points are:
[tex]\[
\begin{array}{c|c}
y & x \\
\hline
-31 & -2 \\
0 & -1 \\
2 & 1 \\
33 & 2 \\
\end{array}
\][/tex]
Hence, the rule to transform the table of data to represent the reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex] is to swap the [tex]\( x \)[/tex]-values and the [tex]\( y \)[/tex]-values. Therefore, the correct transformation action for the reflection is not provided in the multiple choice options A, B, or C. But, based on the process, we understand that the operation needed is: Swapping the [tex]\( x \)[/tex]-values and [tex]\( y \)[/tex]-values.
