A triangle has vertices at [tex]\( B (-3,0), C (2,-1), D (-1,2) \)[/tex].

Which series of transformations would produce an image with vertices [tex]\( B' (4,1), C' (-1,0), D' (2,3) \)[/tex]?

A. [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+1, y+1)\)[/tex]
B. [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+1, y+1)\)[/tex]
C. [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+2, y+2)\)[/tex]
D. [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)\)[/tex]



Answer :

Let's determine the series of transformations needed to change the original vertices [tex]\( B(-3,0), C(2,-1), D(-1,2) \)[/tex] into the new vertices. We will break this down step by step.

The goal is to achieve the transformed vertices:

[tex]\[ B'(5,2), C'(0,1), D'(3,4) \][/tex]

The transformations to be considered are:

1. Reflection over the y-axis: This involves changing the sign of the x-coordinate.
2. Translation: This involves adjusting the coordinates by adding or subtracting a fixed value to the x and y coordinates.

Let's start by reflecting each vertex over the y-axis.

Original vertices:
- [tex]\( B(-3, 0) \)[/tex]
- [tex]\( C(2, -1) \)[/tex]
- [tex]\( D(-1, 2) \)[/tex]

Step 1: Reflect over the y-axis
- [tex]\( B(-3, 0) \rightarrow (3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow (-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow (1, 2) \)[/tex]

Now, the intermediate vertices after reflection are:
- [tex]\( B_{\text{reflected}}(3, 0) \)[/tex]
- [tex]\( C_{\text{reflected}}(-2, -1) \)[/tex]
- [tex]\( D_{\text{reflected}}(1, 2) \)[/tex]

Step 2: Translate (x, y) to (x+2, y+2)
- [tex]\( B_{\text{reflected}}(3, 0) \rightarrow (3+2, 0+2) = (5, 2) \)[/tex]
- [tex]\( C_{\text{reflected}}(-2, -1) \rightarrow (-2+2, -1+2) = (0, 1) \)[/tex]
- [tex]\( D_{\text{reflected}}(1, 2) \rightarrow (1+2, 2+2) = (3, 4) \)[/tex]

So, the final transformed vertices are:
- [tex]\( B'(5, 2) \)[/tex]
- [tex]\( C'(0, 1) \)[/tex]
- [tex]\( D'(3, 4) \)[/tex]

These are the desired vertices. Therefore, the correct series of transformations is:

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

Thus, the correct answer is:
[tex]\[ \boxed{(x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)} \][/tex]