Answer :
To determine which of the given transformations produces the image of the triangle with the given transformed vertices, let's analyze the transformations step-by-step using the coordinates of vertex [tex]\( A(-2, -2) \)[/tex] and the resulting transformed coordinates [tex]\( A'(2, -2) \)[/tex].
Here are the steps for each transformation:
1. Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (-2, 2) \)[/tex]
- Transformed Coordinates: [tex]\( (-2, 2) \)[/tex]
This transformation does not match [tex]\( A'(2, -2) \)[/tex].
2. Transformation: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (2, -2) \)[/tex]
- Transformed Coordinates: [tex]\( (2, -2) \)[/tex]
This transformation matches [tex]\( A'(2, -2) \)[/tex]. Let's now check the other vertices for completeness:
- For vertex [tex]\( B(-1, 1) \)[/tex]:
- Applying Transformation: [tex]\( (-1, 1) \rightarrow (-1, -1) \)[/tex]
- Transformed Coordinates: [tex]\( (-1, -1) \)[/tex]
- For vertex [tex]\( C(3, 2) \)[/tex]:
- Applying Transformation: [tex]\( (3, 2) \rightarrow (-2, 3) \)[/tex]
- Transformed Coordinates: [tex]\( (-2, 3) \)[/tex]
Both vertices match their respective transformed coordinates:
- [tex]\( B'(-1, -1) \)[/tex]
- [tex]\( C'(-2, 3) \)[/tex]
3. Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (2, -2) \)[/tex]
- Transformed Coordinates: [tex]\( (2, -2) \)[/tex]
Match for [tex]\( A'(2, -2) \)[/tex], let's check for other vertices:
- For vertex [tex]\( B(-1, 1) \)[/tex]:
- Applying Transformation: [tex]\( (-1, 1) \rightarrow (1, 1) \)[/tex]
- Transformed Coordinates: [tex]\( (1, 1) \)[/tex]
- For vertex [tex]\( C(3, 2) \)[/tex]:
- Applying Transformation: [tex]\( (3, 2) \rightarrow (-3, 2) \)[/tex]
- Transformed Coordinates: [tex]\( (-3, 2) \)[/tex]
Neither of these matches the respective transformed coordinates [tex]\( B'(-1, -1)\)[/tex] and [tex]\(C'(-2, 3)\)[/tex]. So this is incorrect.
4. Transformation: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (-2, 2) \)[/tex]
- Transformed Coordinates: [tex]\( (-2, 2) \)[/tex]
This transformation does not match [tex]\( A'(2, -2) \)[/tex].
From the above steps, we can conclude that the transformation [tex]\((x, y) \rightarrow (-y, x)\)[/tex] is the one that correctly transforms the vertices of the triangle to their respective image vertices.
Thus, the answer is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
Here are the steps for each transformation:
1. Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (-2, 2) \)[/tex]
- Transformed Coordinates: [tex]\( (-2, 2) \)[/tex]
This transformation does not match [tex]\( A'(2, -2) \)[/tex].
2. Transformation: [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (2, -2) \)[/tex]
- Transformed Coordinates: [tex]\( (2, -2) \)[/tex]
This transformation matches [tex]\( A'(2, -2) \)[/tex]. Let's now check the other vertices for completeness:
- For vertex [tex]\( B(-1, 1) \)[/tex]:
- Applying Transformation: [tex]\( (-1, 1) \rightarrow (-1, -1) \)[/tex]
- Transformed Coordinates: [tex]\( (-1, -1) \)[/tex]
- For vertex [tex]\( C(3, 2) \)[/tex]:
- Applying Transformation: [tex]\( (3, 2) \rightarrow (-2, 3) \)[/tex]
- Transformed Coordinates: [tex]\( (-2, 3) \)[/tex]
Both vertices match their respective transformed coordinates:
- [tex]\( B'(-1, -1) \)[/tex]
- [tex]\( C'(-2, 3) \)[/tex]
3. Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (2, -2) \)[/tex]
- Transformed Coordinates: [tex]\( (2, -2) \)[/tex]
Match for [tex]\( A'(2, -2) \)[/tex], let's check for other vertices:
- For vertex [tex]\( B(-1, 1) \)[/tex]:
- Applying Transformation: [tex]\( (-1, 1) \rightarrow (1, 1) \)[/tex]
- Transformed Coordinates: [tex]\( (1, 1) \)[/tex]
- For vertex [tex]\( C(3, 2) \)[/tex]:
- Applying Transformation: [tex]\( (3, 2) \rightarrow (-3, 2) \)[/tex]
- Transformed Coordinates: [tex]\( (-3, 2) \)[/tex]
Neither of these matches the respective transformed coordinates [tex]\( B'(-1, -1)\)[/tex] and [tex]\(C'(-2, 3)\)[/tex]. So this is incorrect.
4. Transformation: [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
- Original Coordinates: [tex]\( A(-2, -2) \)[/tex]
- Applying Transformation: [tex]\( (-2, -2) \rightarrow (-2, 2) \)[/tex]
- Transformed Coordinates: [tex]\( (-2, 2) \)[/tex]
This transformation does not match [tex]\( A'(2, -2) \)[/tex].
From the above steps, we can conclude that the transformation [tex]\((x, y) \rightarrow (-y, x)\)[/tex] is the one that correctly transforms the vertices of the triangle to their respective image vertices.
Thus, the answer is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
