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]
