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A triangle has vertices at [tex]\( B (-3,0), C (2,-1), D (-1,2) \)[/tex]. Which transformation would produce an image with vertices [tex]\( B''(-2,1), C''(3,2), D''(0,-1) \)[/tex]?

A. [tex]\( (x, y) \rightarrow (x,-y), (x, y) \rightarrow (x+1, y+1) \)[/tex]

B. [tex]\( (x, y) \rightarrow (-x, y), (x, y) \rightarrow (x+2, y+2) \)[/tex]

C. [tex]\( (x, y) \rightarrow (x,-y), (x, y) \rightarrow (x+2, y+2) \)[/tex]

D. [tex]\( (x, y) \rightarrow (-x, y), (x, y) \rightarrow (x+1, y+1) \)[/tex]



Answer :

GinnyAnswer
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]

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