To determine the pre-image of vertex [tex]\( A' \)[/tex] under the rule [tex]\( r_{\cdot}(x, y) \rightarrow (-x, y) \)[/tex], we need to reverse this transformation. The rule given changes the sign of the x-coordinate while leaving the y-coordinate unchanged.
Given the image [tex]\( A'(4, -2) \)[/tex], we need to find the original coordinates (pre-image) that would map to [tex]\( A'(4, -2) \)[/tex] using the given transformation rule [tex]\( r_{\cdot}(x, y) \rightarrow (-x, y) \)[/tex].
### Step-by-Step Solution
1. The transformation rule is [tex]\( r_{\cdot}(x, y) \rightarrow (-x, y) \)[/tex]. This means that:
- The x-coordinate of the image is the negative of the x-coordinate of the pre-image.
- The y-coordinate of the image is the same as the y-coordinate of the pre-image.
2. To find the pre-image, we need to reverse the transformation:
- If the image has coordinates [tex]\( (4, -2) \)[/tex], then the rule [tex]\( r_{\cdot}(x, y) \rightarrow (-x, y) \)[/tex] implies that:
- The x-coordinate of the pre-image should be the negative of the x-coordinate of the image. Therefore, if the image's x-coordinate is 4, the pre-image's x-coordinate should be:
[tex]\[
x = -4
\][/tex]
- The y-coordinate of the pre-image should be the same as the y-coordinate of the image. Therefore, if the image's y-coordinate is -2, the pre-image's y-coordinate should be:
[tex]\[
y = -2
\][/tex]
3. Combining these results, the pre-image coordinates before transformation would be:
[tex]\[
A(-4, -2)
\][/tex]
Thus, the correct pre-image of [tex]\( A' \)[/tex] under the given rule [tex]\( r_{\cdot}(x, y) \rightarrow (-x, y) \)[/tex] is [tex]\( A(-4, -2) \)[/tex].
### Conclusion
The pre-image of vertex [tex]\( A' \)[/tex] is [tex]\( \boxed{A(-4, -2)} \)[/tex].
