Answer :
To determine which transformation produces the specified image of the triangle's vertices, we need to evaluate each transformation step-by-step and check if it matches the given image vertices [tex]\( B' (-2,1), C' (3,2), D' (0,-1) \)[/tex].
1. First Transformation:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (x, -y) \)[/tex]:
- Vertex [tex]\( B (-3, 0) \)[/tex] becomes [tex]\( (-3, 0) \)[/tex].
- Vertex [tex]\( C (2, -1) \)[/tex] becomes [tex]\( (2, 1) \)[/tex].
- Vertex [tex]\( D (-1, 2) \)[/tex] becomes [tex]\( (-1, -2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+1, y+1) \)[/tex]:
- Resulting [tex]\( (-3, 0) \)[/tex] becomes [tex]\( (-2, 1) \)[/tex].
- Resulting [tex]\( (2, 1) \)[/tex] becomes [tex]\( (3, 2) \)[/tex].
- Resulting [tex]\( (-1, -2) \)[/tex] becomes [tex]\( (0, -1) \)[/tex].
The transformed vertices are [tex]\( B'(-2, 1), C'(3, 2), D'(0, -1) \)[/tex]. This matches the given image vertices.
2. Second Transformation:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (x+1, y+1) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (-x, y) \)[/tex]:
- Vertex [tex]\( B(-3, 0) \)[/tex] becomes [tex]\( (3, 0) \)[/tex].
- Vertex [tex]\( C(2, -1) \)[/tex] becomes [tex]\( (-2, -1) \)[/tex].
- Vertex [tex]\( D(-1, 2) \)[/tex] becomes [tex]\( (1, 2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+1, y+1) \)[/tex]:
- Resulting [tex]\( (3, 0) \)[/tex] becomes [tex]\( (4, 1) \)[/tex].
- Resulting [tex]\( (-2, -1) \)[/tex] becomes [tex]\( (-1, 0) \)[/tex].
- Resulting [tex]\( (1, 2) \)[/tex] becomes [tex]\( (2, 3) \)[/tex].
The transformed vertices do not match the given image vertices.
3. Third Transformation:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+2, y+2) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (x, -y) \)[/tex]:
- Vertex [tex]\( B(-3, 0) \)[/tex] remains [tex]\( (-3, 0) \)[/tex].
- Vertex [tex]\( C(2, -1) \)[/tex] becomes [tex]\( (2, 1) \)[/tex].
- Vertex [tex]\( D(-1, 2) \)[/tex] becomes [tex]\( (-1, -2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+2, y+2) \)[/tex]:
- Resulting [tex]\( (-3, 0) \)[/tex] becomes [tex]\( (-1, 2) \)[/tex].
- Resulting [tex]\( (2, 1) \)[/tex] becomes [tex]\( (4, 3) \)[/tex].
- Resulting [tex]\( (-1, -2) \)[/tex] becomes [tex]\( (1, 0) \)[/tex].
The transformed vertices do not match the given image vertices.
4. Fourth Transformation:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (x + 2, y + 2) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (-x, y) \)[/tex]:
- Vertex [tex]\( B(-3, 0) \)[/tex] becomes [tex]\( (3, 0) \)[/tex].
- Vertex [tex]\( C(2, -1) \)[/tex] becomes [tex]\( (-2, -1) \)[/tex].
- Vertex [tex]\( D(-1, 2) \)[/tex] becomes [tex]\( (1, 2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+2, y+2) \)[/tex]:
- Resulting [tex]\( (3, 0) \)[/tex] becomes [tex]\( (5, 2) \)[/tex].
- Resulting [tex]\( (-2, -1) \)[/tex] becomes [tex]\( (0, 1) \)[/tex].
- Resulting [tex]\( (1, 2) \)[/tex] becomes [tex]\( (3, 4) \)[/tex].
The transformed vertices do not match the given image vertices.
Based on the above steps, the only transformation that matches the given image vertices is:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1) \][/tex]
Thus, the correct transformation is:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1) \][/tex]
1. First Transformation:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (x, -y) \)[/tex]:
- Vertex [tex]\( B (-3, 0) \)[/tex] becomes [tex]\( (-3, 0) \)[/tex].
- Vertex [tex]\( C (2, -1) \)[/tex] becomes [tex]\( (2, 1) \)[/tex].
- Vertex [tex]\( D (-1, 2) \)[/tex] becomes [tex]\( (-1, -2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+1, y+1) \)[/tex]:
- Resulting [tex]\( (-3, 0) \)[/tex] becomes [tex]\( (-2, 1) \)[/tex].
- Resulting [tex]\( (2, 1) \)[/tex] becomes [tex]\( (3, 2) \)[/tex].
- Resulting [tex]\( (-1, -2) \)[/tex] becomes [tex]\( (0, -1) \)[/tex].
The transformed vertices are [tex]\( B'(-2, 1), C'(3, 2), D'(0, -1) \)[/tex]. This matches the given image vertices.
2. Second Transformation:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (x+1, y+1) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (-x, y) \)[/tex]:
- Vertex [tex]\( B(-3, 0) \)[/tex] becomes [tex]\( (3, 0) \)[/tex].
- Vertex [tex]\( C(2, -1) \)[/tex] becomes [tex]\( (-2, -1) \)[/tex].
- Vertex [tex]\( D(-1, 2) \)[/tex] becomes [tex]\( (1, 2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+1, y+1) \)[/tex]:
- Resulting [tex]\( (3, 0) \)[/tex] becomes [tex]\( (4, 1) \)[/tex].
- Resulting [tex]\( (-2, -1) \)[/tex] becomes [tex]\( (-1, 0) \)[/tex].
- Resulting [tex]\( (1, 2) \)[/tex] becomes [tex]\( (2, 3) \)[/tex].
The transformed vertices do not match the given image vertices.
3. Third Transformation:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+2, y+2) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (x, -y) \)[/tex]:
- Vertex [tex]\( B(-3, 0) \)[/tex] remains [tex]\( (-3, 0) \)[/tex].
- Vertex [tex]\( C(2, -1) \)[/tex] becomes [tex]\( (2, 1) \)[/tex].
- Vertex [tex]\( D(-1, 2) \)[/tex] becomes [tex]\( (-1, -2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+2, y+2) \)[/tex]:
- Resulting [tex]\( (-3, 0) \)[/tex] becomes [tex]\( (-1, 2) \)[/tex].
- Resulting [tex]\( (2, 1) \)[/tex] becomes [tex]\( (4, 3) \)[/tex].
- Resulting [tex]\( (-1, -2) \)[/tex] becomes [tex]\( (1, 0) \)[/tex].
The transformed vertices do not match the given image vertices.
4. Fourth Transformation:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (x + 2, y + 2) \][/tex]
- Apply [tex]\( (x, y) \rightarrow (-x, y) \)[/tex]:
- Vertex [tex]\( B(-3, 0) \)[/tex] becomes [tex]\( (3, 0) \)[/tex].
- Vertex [tex]\( C(2, -1) \)[/tex] becomes [tex]\( (-2, -1) \)[/tex].
- Vertex [tex]\( D(-1, 2) \)[/tex] becomes [tex]\( (1, 2) \)[/tex].
- Apply [tex]\( (x, y) \rightarrow (x+2, y+2) \)[/tex]:
- Resulting [tex]\( (3, 0) \)[/tex] becomes [tex]\( (5, 2) \)[/tex].
- Resulting [tex]\( (-2, -1) \)[/tex] becomes [tex]\( (0, 1) \)[/tex].
- Resulting [tex]\( (1, 2) \)[/tex] becomes [tex]\( (3, 4) \)[/tex].
The transformed vertices do not match the given image vertices.
Based on the above steps, the only transformation that matches the given image vertices is:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1) \][/tex]
Thus, the correct transformation is:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1) \][/tex]
