Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.