To find the pre-image of vertex [tex]\( A' \)[/tex] when the transformation rule [tex]\( r_y \)[/tex]-axis [tex]\( (x, y) \rightarrow (-x, y) \)[/tex] is applied, let's follow the given steps:
1. Understand the Reflection Rule: The rule [tex]\( r_y \)[/tex]-axis [tex]\( (x, y) \rightarrow (-x, y) \)[/tex] reflects a point across the y-axis. Essentially, this means that the x-coordinate of the point changes its sign while the y-coordinate remains the same.
2. Identify the Given Image Vertex [tex]\( A' \)[/tex]: According to the question, we are given that the image vertex [tex]\( A' \)[/tex] is at coordinates [tex]\( (4, -2) \)[/tex].
3. Determine the Pre-image (Original Coordinates):
- Given the rule, to find the original coordinates before the transformation, we need to reverse the operation.
- Since the rule changes the x-coordinate to its opposite sign, applying it in reverse means we change the x-coordinate back to its original sign while keeping the y-coordinate the same.
4. Reverse the Transformation for [tex]\( A' (4, -2) \)[/tex]:
- Original x-coordinate: Reverse [tex]\(-x \rightarrow -x = -4 \rightarrow x = -4\)[/tex]
- Original y-coordinate: The same [tex]\(y = -2\)[/tex]
5. Find the Pre-image Point:
- Therefore, the pre-image of vertex [tex]\( A' (4, -2) \)[/tex] is [tex]\( (-4, -2) \)[/tex].
6. Verify Among the Given Options:
- The options provided are:
- [tex]\( A(-4, 2) \)[/tex]
- [tex]\( A(-2, -4) \)[/tex]
- [tex]\( A(2, 4) \)[/tex]
- [tex]\( A(4, -2) \)[/tex]
- The pre-image [tex]\( (-4, -2) \)[/tex] matches none of the options directly, but this actual pre-image confirms that the correct pre-image must be capable of generating [tex]\( (4, -2) \)[/tex] after the reflection rule is applied.
Considering this pre-image evidence and transformation:
The pre-image [tex]\( (-4, -2) \)[/tex] matches none of the options, more accurately the correct reflection matching would involve negative axis values yielding positive values transforming between options as exactly determined is [tex]\( (-4, -2) \)[/tex].
Thus among given closest most data representing correct reflection verificationly radicalely could be more accurately determined as: revealing possible practical pre image as specified frame directly related aspect would practically return correct matching is:
Thus, the pre-image of vertex [tex]\( A' \)[/tex] at [tex]\( (4, -2) \)[/tex] after verifying practical mundane correct relational exact calculation reflection is revealed accurately matches at:
None explicitly translation as pre frame [tex]\( (-4, -2) \)[/tex]
[tex]\(A (-4, -2) practical frame equivalent correct matches candidly\)[/tex].
