To determine which transformation correctly maps the original triangle vertices [tex]\(A, B,\)[/tex] and [tex]\(C\)[/tex] to the transformed vertices [tex]\(A', B',\)[/tex] and [tex]\(C'\)[/tex], let's analyze each provided transformation and apply it to each original vertex to see which combination gives us the target vertices.
### Original Vertices:
- [tex]\(A(-2, -2)\)[/tex]
- [tex]\(B(-1, 1)\)[/tex]
- [tex]\(C(3, 2)\)[/tex]
### Target Vertices:
- [tex]\(A'(2, -2)\)[/tex]
- [tex]\(B'(-1, -1)\)[/tex]
- [tex]\(C'(-2, 3)\)[/tex]
### Transformations:
1. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[
(-2, -(-2)) = (-2, 2)
\][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[
(-1, -1)
\][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[
(3, -2)
\][/tex]
This does not produce the desired image as [tex]\(A' \neq (-2, 2)\)[/tex].
2. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[
(-(-2), -2) = (2, -2)
\][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[
(-1, -1)
\][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[
(-2, 3)
\][/tex]
This transformation correctly produces the target vertices:
[tex]\[
A(2, -2),\ B(-1, -1),\ C(-2, 3)
\][/tex]
3. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[
(2, -2)
\][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[
(1, 1)
\][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[
(-3, 2)
\][/tex]
This does not produce the desired image as [tex]\(B' \neq (1, 1)\)[/tex] and [tex]\(C' \neq (-3, 2)\)[/tex].
4. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- For [tex]\(A(-2, -2)\)[/tex]:
[tex]\[
(-2, 2)
\][/tex]
- For [tex]\(B(-1, 1)\)[/tex]:
[tex]\[
(1, 1)
\][/tex]
- For [tex]\(C(3, 2)\)[/tex]:
[tex]\[
(2, -3)
\][/tex]
This does not produce the desired image as [tex]\(A' \neq (-2, 2)\)[/tex], [tex]\(B' \neq (1, 1)\)[/tex], and [tex]\(C' \neq (2, -3)\)[/tex].
The correct transformation is [tex]\((x, y) \rightarrow (-y, x)\)[/tex]. This transformation accurately maps the original vertices to the target vertices:
- [tex]\(A(-2, -2) \rightarrow A'(2, -2)\)[/tex]
- [tex]\(B(-1, 1) \rightarrow B'(-1, -1)\)[/tex]
- [tex]\(C(3, 2) \rightarrow C'(-2, 3)\)[/tex]
Thus, the answer is:
[tex]\[
\boxed{(x, y) \rightarrow (-y, x)}
\][/tex]
