To solve this problem, we need to use the transformation rule involving reflection across the y-axis. The rule is given as:
[tex]\[ r_{y \text{-axis}}(x, y) \rightarrow (-x, y) \][/tex]
This transformation means that any point [tex]\((x, y)\)[/tex] will be reflected across the y-axis to become [tex]\((-x, y)\)[/tex]. Our goal is to find which of the given points is the pre-image of the vertex [tex]\( A' \)[/tex].
Given:
[tex]\[ A' = (4, 2) \][/tex]
We need to check each of the given points to see which one, when the transformation rule is applied, results in the coordinates [tex]\((4, 2)\)[/tex].
Let's examine each point one by one:
1. For [tex]\( A = (-4, 2) \)[/tex]:
[tex]\[ r_{y \text{-axis}}(-4, 2) \rightarrow (-(-4), 2) = (4, 2) \][/tex]
This matches [tex]\( A' \)[/tex].
2. For [tex]\( A = (-2, -4) \)[/tex]:
[tex]\[ r_{y \text{-axis}}(-2, -4) \rightarrow (-(-2), -4) = (2, -4) \][/tex]
This does not match [tex]\( A' \)[/tex].
3. For [tex]\( A = (2, 4) \)[/tex]:
[tex]\[ r_{y \text{-axis}}(2, 4) \rightarrow (-(2), 4) = (-2, 4) \][/tex]
This does not match [tex]\( A' \)[/tex].
4. For [tex]\( A = (4, -2) \)[/tex]:
[tex]\[ r_{y \text{-axis}}(4, -2) \rightarrow (-(4), -2) = (-4, -2) \][/tex]
This does not match [tex]\( A' \)[/tex].
Only the point [tex]\( (-4, 2) \)[/tex], when transformed, produces the vertex [tex]\( A' \)[/tex] with coordinates [tex]\( (4, 2) \)[/tex].
Therefore, the pre-image of vertex [tex]\( A' \)[/tex] is:
[tex]\[ A(-4, 2) \][/tex]
