To find the pre-image of vertex [tex]\( A' \)[/tex] under the given rule, let’s carefully analyze each step:
1. Understanding the Rule: The rule [tex]\( r_{y\text{-axis}} \)[/tex] indicates a reflection across the y-axis. In mathematical terms, the transformation is defined as:
[tex]\[
(x, y) \rightarrow (-x, y)
\][/tex]
2. Given Image Coordinates: We are provided with the coordinates of the image vertex [tex]\( A' \)[/tex], which are [tex]\( (-4, 2) \)[/tex].
3. Applying the Reflection Rule:
- For the x-coordinate: Since the transformation rule states to negate the x-coordinate, we take the given [tex]\( x \)[/tex]-coordinate of [tex]\( A' \)[/tex] (which is [tex]\( -4 \)[/tex]) and negate it. Hence, the x-coordinate of the pre-image [tex]\( A \)[/tex] becomes:
[tex]\[
-(-4) = 4
\][/tex]
- For the y-coordinate: The transformation rule indicates that the y-coordinate remains the same, so the y-coordinate of the pre-image [tex]\( A \)[/tex] will be:
[tex]\[
y = 2
\][/tex]
4. Result: Therefore, the coordinates of the pre-image vertex [tex]\( A \)[/tex] after applying the transformation are:
[tex]\[
(4, 2)
\][/tex]
So, the pre-image of [tex]\( A' \)[/tex] under the transformation [tex]\( r_{y\text{-axis}} \)[/tex] is [tex]\( (4, 2) \)[/tex].
Thus, the answer is:
[tex]\[
A(4, 2)
\][/tex]
