To find the pre-image of vertex [tex]\( A' \)[/tex] given the transformation rule [tex]\( r_{y \text{-axis}}(x, y) \rightarrow(-x, y) \)[/tex], follow these steps:
1. Understand the transformation rule: The rule [tex]\( r_{y \text{-axis}}(x, y) \rightarrow(-x, y) \)[/tex] means that when a point [tex]\( (x, y) \)[/tex] is reflected over the y-axis, the x-coordinate is negated, while the y-coordinate remains unchanged.
2. Identify the coordinates of the image point [tex]\( A' \)[/tex]: The coordinates of the image point are given as [tex]\( A'(-4, 2) \)[/tex].
3. Apply the inverse transformation: To find the pre-image, we need to reverse the transformation. Since the rule is to negate the x-coordinate and keep the y-coordinate the same, the inverse transformation will also negate the x-coordinate back:
- The x-coordinate of the pre-image will be the negation of [tex]\(-4\)[/tex], which is [tex]\(4\)[/tex].
- The y-coordinate of the pre-image remains the same, which is [tex]\(2\)[/tex].
4. Determine the pre-image coordinates: Therefore, the pre-image of [tex]\( A'(-4, 2) \)[/tex] is at the coordinates [tex]\( (4, 2) \)[/tex].
Hence, the pre-image of vertex [tex]\( A' \)[/tex] is [tex]\( \boxed{(4, 2)} \)[/tex].
