To find the pre-image of vertex [tex]\( A' \)[/tex] under the given transformation rule [tex]\( r_y \)[/tex]-axis [tex]\((x, y) \rightarrow (-x, y) \)[/tex], we need to reverse this transformation. The transformation rule [tex]\( r_y \)[/tex]-axis reflects a point across the y-axis, changing the sign of its x-coordinate while keeping the y-coordinate the same. To find the original coordinates of the point before transformation, we need to apply the same reflection rule in reverse.
Given:
[tex]\( A' \)[/tex] has coordinates [tex]\( (2, 4) \)[/tex].
The rule is [tex]\((x, y) \rightarrow (-x, y)\)[/tex]. To reverse it, we will take the coordinates of [tex]\( A' \)[/tex] and change the sign of the x-coordinate back.
1. Start with [tex]\( A' = (2, 4) \)[/tex].
2. The x-coordinate of [tex]\( A \)[/tex] will be the opposite sign of the x-coordinate of [tex]\( A' \)[/tex]. Since the x-coordinate of [tex]\( A' \)[/tex] is 2, the x-coordinate of [tex]\( A \)[/tex] will be -2.
3. The y-coordinate remains the same. Therefore, the y-coordinate of [tex]\( A \)[/tex] will be 4.
Putting these together, the pre-image coordinates [tex]\( A \)[/tex] are:
[tex]\[ A = (-2, 4). \][/tex]
Therefore, the pre-image of vertex [tex]\( A' \)[/tex] is [tex]\( \boxed{(-2, 4)} \)[/tex].
