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Sagot :
To determine which transformation sequence would produce the image with vertices [tex]\( B^{\prime \prime}(-2,1), C^{\prime \prime}(3,2), D^{\prime \prime}(0,-1) \)[/tex], we need to consider each option step-by-step and see if it aligns with the given transformed vertices.
Let's analyze each option systematically:
### Option 1:
1. Transform [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 1, y + 1)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (-3, 0)\)[/tex]
- Second transformation: [tex]\((-3, 0) \rightarrow (-2, 1)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (2, 1)\)[/tex]
- Second transformation: [tex]\((2, 1) \rightarrow (3, 2)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (-1, -2)\)[/tex]
- Second transformation: [tex]\((-1, -2) \rightarrow (0, -1)\)[/tex]
This sequence correctly maps to [tex]\( B^{\prime \prime}(-2,1), C^{\prime \prime}(3,2), D^{\prime \prime}(0,-1) \)[/tex].
### Option 2:
1. Transform [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 2, y + 2)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (3, 0)\)[/tex]
- Second transformation: [tex]\((3, 0) \rightarrow (5, 2)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (-2, -1)\)[/tex]
- Second transformation: [tex]\((-2, -1) \rightarrow (0, 1)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (1, 2)\)[/tex]
- Second transformation: [tex]\((1, 2) \rightarrow (3, 4)\)[/tex]
This option does not provide the required vertices.
### Option 3:
1. Transform [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 2, y + 2)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (-3, 0)\)[/tex]
- Second transformation: [tex]\((-3, 0) \rightarrow (-1, 2)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (2, 1)\)[/tex]
- Second transformation: [tex]\((2, 1) \rightarrow (4, 3)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (-1, -2)\)[/tex]
- Second transformation: [tex]\((-1, -2) \rightarrow (1, 0)\)[/tex]
This option also does not provide the required vertices.
### Option 4:
1. Transform [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 1, y + 1)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (3, 0)\)[/tex]
- Second transformation: [tex]\((3, 0) \rightarrow (4, 1)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (-2, -1)\)[/tex]
- Second transformation: [tex]\((-2, -1) \rightarrow (-1, 0)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (1, 2)\)[/tex]
- Second transformation: [tex]\((1, 2) \rightarrow (2, 3)\)[/tex]
This option does not provide the required vertices.
Based on the analysis, the transformation that produces the image vertices [tex]\( B^{\prime \prime}(-2,1), C^{\prime \prime}(3,2), D^{\prime \prime}(0,-1) \)[/tex] is:
[tex]\[ (x, y) \rightarrow (x, -y), (x, y) \rightarrow (x + 1, y + 1) \][/tex]
Hence, the correct transformation sequence is:
[tex]\[ (x, y) \rightarrow (x, -y), (x, y) \rightarrow (x + 1, y + 1) \][/tex]
Let's analyze each option systematically:
### Option 1:
1. Transform [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 1, y + 1)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (-3, 0)\)[/tex]
- Second transformation: [tex]\((-3, 0) \rightarrow (-2, 1)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (2, 1)\)[/tex]
- Second transformation: [tex]\((2, 1) \rightarrow (3, 2)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (-1, -2)\)[/tex]
- Second transformation: [tex]\((-1, -2) \rightarrow (0, -1)\)[/tex]
This sequence correctly maps to [tex]\( B^{\prime \prime}(-2,1), C^{\prime \prime}(3,2), D^{\prime \prime}(0,-1) \)[/tex].
### Option 2:
1. Transform [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 2, y + 2)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (3, 0)\)[/tex]
- Second transformation: [tex]\((3, 0) \rightarrow (5, 2)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (-2, -1)\)[/tex]
- Second transformation: [tex]\((-2, -1) \rightarrow (0, 1)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (1, 2)\)[/tex]
- Second transformation: [tex]\((1, 2) \rightarrow (3, 4)\)[/tex]
This option does not provide the required vertices.
### Option 3:
1. Transform [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 2, y + 2)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (-3, 0)\)[/tex]
- Second transformation: [tex]\((-3, 0) \rightarrow (-1, 2)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (2, 1)\)[/tex]
- Second transformation: [tex]\((2, 1) \rightarrow (4, 3)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (-1, -2)\)[/tex]
- Second transformation: [tex]\((-1, -2) \rightarrow (1, 0)\)[/tex]
This option also does not provide the required vertices.
### Option 4:
1. Transform [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
2. Transform [tex]\((x, y) \rightarrow (x + 1, y + 1)\)[/tex]
- Starting with [tex]\( B(-3, 0) \)[/tex]:
- First transformation: [tex]\((-3, 0) \rightarrow (3, 0)\)[/tex]
- Second transformation: [tex]\((3, 0) \rightarrow (4, 1)\)[/tex]
- Starting with [tex]\( C(2, -1) \)[/tex]:
- First transformation: [tex]\((2, -1) \rightarrow (-2, -1)\)[/tex]
- Second transformation: [tex]\((-2, -1) \rightarrow (-1, 0)\)[/tex]
- Starting with [tex]\( D(-1, 2) \)[/tex]:
- First transformation: [tex]\((-1, 2) \rightarrow (1, 2)\)[/tex]
- Second transformation: [tex]\((1, 2) \rightarrow (2, 3)\)[/tex]
This option does not provide the required vertices.
Based on the analysis, the transformation that produces the image vertices [tex]\( B^{\prime \prime}(-2,1), C^{\prime \prime}(3,2), D^{\prime \prime}(0,-1) \)[/tex] is:
[tex]\[ (x, y) \rightarrow (x, -y), (x, y) \rightarrow (x + 1, y + 1) \][/tex]
Hence, the correct transformation sequence is:
[tex]\[ (x, y) \rightarrow (x, -y), (x, y) \rightarrow (x + 1, y + 1) \][/tex]
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