Answer :
To solve this problem, we need to apply transformations to the initial vertices [tex]\( B(-3, 0), C(2, -1), D(-1, 2) \)[/tex] and see which sequence leads us to [tex]\( B^{\prime \prime}(-2,1), C^{\prime \prime}(3,2), D^{\prime \prime}(0,-1) \)[/tex].
Let's check the options provided step by step.
### Option 1: [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(-3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(2, 1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(-1, -2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B'(-3, 0) \rightarrow B^{\prime \prime}(-2, 1) \)[/tex]
- [tex]\( C'(2, 1) \rightarrow C^{\prime \prime}(3, 2) \)[/tex]
- [tex]\( D'(-1, -2) \rightarrow D^{\prime \prime}(0, -1) \)[/tex]
As a result, we obtain the vertices:
- [tex]\( B^{\prime \prime}(-2, 1) \)[/tex]
- [tex]\( C^{\prime \prime}(3, 2) \)[/tex]
- [tex]\( D^{\prime \prime}(0, -1) \)[/tex]
This matches the target vertices. Therefore, Option 1 is the correct transformation sequence.
Given the correct transformation, we do not need to check the rest. But for completeness:
### Option 2: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+1, y+1)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(1, 2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B'(3, 0) \rightarrow B^{\prime \prime}(4, 1) \)[/tex]
- [tex]\( C'(-2, -1) \rightarrow C^{\prime \prime}(-1, 0) \)[/tex]
- [tex]\( D'(1, 2) \rightarrow D^{\prime \prime}(2, 3) \)[/tex]
These vertices do not match the target vertices.
### Option 3: [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+2, y+2)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(-3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(2, 1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(-1, -2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B'(-3, 0) \rightarrow B^{\prime \prime}(-1, 2) \)[/tex]
- [tex]\( C'(2, 1) \rightarrow C^{\prime \prime}(4, 3) \)[/tex]
- [tex]\( D'(-1, -2) \rightarrow D^{\prime \prime}(1, 0) \)[/tex]
These vertices do not match the target vertices.
### Option 4: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(1, 2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B'(3, 0) \rightarrow B^{\prime \prime}(5, 2) \)[/tex]
- [tex]\( C'(-2, -1) \rightarrow C^{\prime \prime}(0, 1) \)[/tex]
- [tex]\( D'(1, 2) \rightarrow D^{\prime \prime}(3, 4) \)[/tex]
These vertices do not match the target vertices.
Therefore, the correct sequence of transformations to achieve the desired vertices is:
Option 1: [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1)\)[/tex]
Let's check the options provided step by step.
### Option 1: [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(-3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(2, 1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(-1, -2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B'(-3, 0) \rightarrow B^{\prime \prime}(-2, 1) \)[/tex]
- [tex]\( C'(2, 1) \rightarrow C^{\prime \prime}(3, 2) \)[/tex]
- [tex]\( D'(-1, -2) \rightarrow D^{\prime \prime}(0, -1) \)[/tex]
As a result, we obtain the vertices:
- [tex]\( B^{\prime \prime}(-2, 1) \)[/tex]
- [tex]\( C^{\prime \prime}(3, 2) \)[/tex]
- [tex]\( D^{\prime \prime}(0, -1) \)[/tex]
This matches the target vertices. Therefore, Option 1 is the correct transformation sequence.
Given the correct transformation, we do not need to check the rest. But for completeness:
### Option 2: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+1, y+1)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(1, 2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B'(3, 0) \rightarrow B^{\prime \prime}(4, 1) \)[/tex]
- [tex]\( C'(-2, -1) \rightarrow C^{\prime \prime}(-1, 0) \)[/tex]
- [tex]\( D'(1, 2) \rightarrow D^{\prime \prime}(2, 3) \)[/tex]
These vertices do not match the target vertices.
### Option 3: [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+2, y+2)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(-3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(2, 1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(-1, -2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B'(-3, 0) \rightarrow B^{\prime \prime}(-1, 2) \)[/tex]
- [tex]\( C'(2, 1) \rightarrow C^{\prime \prime}(4, 3) \)[/tex]
- [tex]\( D'(-1, -2) \rightarrow D^{\prime \prime}(1, 0) \)[/tex]
These vertices do not match the target vertices.
### Option 4: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)\)[/tex]
First Transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow B'(3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow C'(-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow D'(1, 2) \)[/tex]
Second Transformation: [tex]\((x, y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B'(3, 0) \rightarrow B^{\prime \prime}(5, 2) \)[/tex]
- [tex]\( C'(-2, -1) \rightarrow C^{\prime \prime}(0, 1) \)[/tex]
- [tex]\( D'(1, 2) \rightarrow D^{\prime \prime}(3, 4) \)[/tex]
These vertices do not match the target vertices.
Therefore, the correct sequence of transformations to achieve the desired vertices is:
Option 1: [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1)\)[/tex]
