To determine the pre-image of a vertex \( A' \) given the rule \( r_{y \text{-axis}} (x, y) \rightarrow (-x, y) \), we need to understand that this rule reflects points over the y-axis. This reflection changes the sign of the x-coordinate while leaving the y-coordinate unchanged.
Now, let's determine the pre-image of each of the possible given points:
1. Given point \( A(-4, 2) \):
- If this is a reflection of a point over the y-axis, then we start with the point before reflection, which would be \((x, y) = (4, 2)\).
2. Given point \( A(-2, -4) \):
- Applying the reflection rule in reverse, we change the sign of the x-coordinate, so the pre-image would be \((x, y) = (2, -4)\).
3. Given point \( A(2, 4) \):
- Applying the reflection rule in reverse, we change the sign of the x-coordinate, so the pre-image would be \((x, y) = (-2, 4)\).
4. Given point \( A(4, -2) \):
- Applying the reflection rule in reverse, we change the sign of the x-coordinate, so the pre-image would be \((x, y) = (-4, -2)\).
Thus, the pre-images for the given points are:
- For \( A(-4, 2) \), the pre-image is \( (4, 2) \).
- For \( A(-2, -4) \), the pre-image is \( (2, -4) \).
- For \( A(2, 4) \), the pre-image is \( (-2, 4) \).
- For \( A(4, -2) \), the pre-image is \( (-4, -2) \).
So the pre-images are:
[tex]\[
\begin{aligned}
& (4, 2), \\
& (2, -4), \\
& (-2, 4), \\
& (-4, -2).
\end{aligned}
\][/tex]
