Given the rule that created the image is [tex]\(r_y\)[/tex]-axis [tex]\((x, y) \rightarrow (-x, y)\)[/tex], we need to determine the pre-image of a vertex [tex]\(A'\)[/tex].
Let’s consider each given option and apply the rule to verify if it matches the given transformation:
1. Option 1: [tex]\( A' = (-4, 2) \)[/tex]
- Applying the rule [tex]\((x, y) \rightarrow (-x, y)\)[/tex]:
- The coordinate [tex]\((-4, 2)\)[/tex] reflects to [tex]\((x, y)\)[/tex].
- Solving for [tex]\(x\)[/tex]: Since [tex]\(-x = -4\)[/tex], [tex]\(x = 4\)[/tex].
- Thus, the pre-image is [tex]\( (4, 2) \)[/tex].
2. Option 2: [tex]\( A' = (-2, -4) \)[/tex]
- Applying the rule [tex]\((x, y) \rightarrow (-x, y)\)[/tex]:
- The coordinate [tex]\((-2, -4)\)[/tex] reflects to [tex]\((x, y)\)[/tex].
- Solving for [tex]\(x\)[/tex]: Since [tex]\(-x = -2\)[/tex], [tex]\(x = 2\)[/tex].
- Thus, the pre-image is [tex]\( (2, -4) \)[/tex].
3. Option 3: [tex]\( A' = (2, 4) \)[/tex]
- Applying the rule [tex]\((x, y) \rightarrow (-x, y)\)[/tex]:
- The coordinate [tex]\((2, 4)\)[/tex] reflects to [tex]\((x, y)\)[/tex].
- Solving for [tex]\(x\)[/tex]: Since [tex]\(-x = 2\)[/tex], [tex]\(x = -2\)[/tex].
- Thus, the pre-image is [tex]\( (-2, 4) \)[/tex].
4. Option 4: [tex]\( A' = (4, -2) \)[/tex]
- Applying the rule [tex]\((x, y) \rightarrow (-x, y)\)[/tex]:
- The coordinate [tex]\((4, -2)\)[/tex] reflects to [tex]\((x, y)\)[/tex].
- Solving for [tex]\(x\)[/tex]: Since [tex]\(-x = 4\)[/tex], [tex]\(x = -4\)[/tex].
- Thus, the pre-image is [tex]\( (-4, -2) \)[/tex].
From the above calculations, we can see that the correct pre-image of the vertex [tex]\(A\)[/tex] that matches the given transformation [tex]\(r_y\)[/tex]-axis [tex]\((x, y) \rightarrow (-x, y)\)[/tex] is [tex]\(A(4, 2)\)[/tex]. Therefore, the right pre-image corresponding to the given vertex [tex]\(A'\)[/tex] is:
[tex]\[ \boxed{A(4, 2)} \][/tex]
