To find the location of [tex]\( Z^ \)[/tex] after performing the described transformations, follow these steps:
1. Start with the initial coordinates: The point [tex]\( Z \)[/tex] has coordinates [tex]\((2, -4)\)[/tex].
2. Apply the translation rule [tex]\((x, y) \rightarrow (x-4, y+3)\)[/tex]:
- For the [tex]\( x \)[/tex]-coordinate: [tex]\( 2 - 4 = -2 \)[/tex]
- For the [tex]\( y \)[/tex]-coordinate: [tex]\( -4 + 3 = -1 \)[/tex]
After translation, the coordinates of [tex]\( Z \)[/tex] become [tex]\((-2, -1)\)[/tex].
3. Apply the 180-degree rotation:
- A 180-degree rotation about the origin will change the coordinates [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- For the [tex]\( x \)[/tex]-coordinate: [tex]\(-(-2) = 2 \)[/tex]
- For the [tex]\( y \)[/tex]-coordinate: [tex]\(-(-1) = 1 \)[/tex]
After rotation, the coordinates of [tex]\( Z^ \)[/tex] become [tex]\((2, 1)\)[/tex].
Therefore, the location of [tex]\( Z^* \)[/tex] is [tex]\((2, 1)\)[/tex]. The correct answer is not listed among the choices provided.
