To find the pre-image of vertex [tex]\( A' \)[/tex] using the transformation rule [tex]\( r_{y \text{-axis}}(x, y) \rightarrow (-x, y) \)[/tex], we need to reverse the transformation. The transformation reflects points across the y-axis, which means for every image point [tex]\( (x', y') \)[/tex], the corresponding pre-image point [tex]\( (x, y) \)[/tex] is found by negating the x-coordinate of the image point.
Given the list of image points:
1. [tex]\( A'(-4, 2) \)[/tex]
2. [tex]\( A'(-2, -4) \)[/tex]
3. [tex]\( A'(2, 4) \)[/tex]
4. [tex]\( A'(4, -2) \)[/tex]
We will apply the reverse transformation to each point, which means we change the sign of the x-coordinate for each image point:
1. For [tex]\( A'(-4, 2) \)[/tex]:
[tex]\[ A = (-(-4), 2) = (4, 2) \][/tex]
2. For [tex]\( A'(-2, -4) \)[/tex]:
[tex]\[ A = (-(-2), -4) = (2, -4) \][/tex]
3. For [tex]\( A'(2, 4) \)[/tex]:
[tex]\[ A = (-(2), 4) = (-2, 4) \][/tex]
4. For [tex]\( A'(4, -2) \)[/tex]:
[tex]\[ A = (-(4), -2) = (-4, -2) \][/tex]
Thus, the pre-image points corresponding to the given image points are:
1. [tex]\( (4, 2) \)[/tex]
2. [tex]\( (2, -4) \)[/tex]
3. [tex]\( (-2, 4) \)[/tex]
4. [tex]\( (-4, -2) \)[/tex]
These are the original points before the transformation was applied.
