To determine the pre-image of the vertex [tex]\( A' \)[/tex] given the rule of reflection across the y-axis [tex]\( r_{\text{y-axis}}(x, y) \rightarrow (-x, y) \)[/tex]:
1. Understand the Reflection Rule: When reflecting a point across the y-axis, the x-coordinate changes sign while the y-coordinate remains the same. Thus, for a point [tex]\((x, y)\)[/tex] under reflection, the image becomes [tex]\((-x, y)\)[/tex].
2. Identify the Given Image [tex]\( A' \)[/tex]:
Suppose the coordinates of the image vertex [tex]\( A' \)[/tex] are [tex]\( (4, 2) \)[/tex].
3. Use the Rule to Find Pre-image:
- According to [tex]\( r_{\text{y-axis}}(x, y) \rightarrow (-x, y) \)[/tex], given the image [tex]\( A' = (4, 2) \)[/tex]:
- Let the pre-image be [tex]\((x, y)\)[/tex]. For the point [tex]\((4, 2)\)[/tex] to be the reflected image, we reverse the sign change transformation:
[tex]\[
-x = 4 \implies x = -4
\][/tex]
The y-coordinate remains unchanged:
[tex]\[
y = 2
\][/tex]
4. Combine the Coordinates and Verify:
Thus, the pre-image [tex]\( A \)[/tex] is [tex]\((-4, 2)\)[/tex].
Therefore, the pre-image of the vertex [tex]\( A' \)[/tex] is [tex]\( \boxed{(-4, 2)} \)[/tex].
