To determine the location of the point [tex]\( Z^ \)[/tex] after the given transformations, we proceed step-by-step:
1. Initial Coordinates of [tex]\( Z \)[/tex]:
Start with the initial coordinates of point [tex]\( Z \)[/tex], which are [tex]\((-1, -9)\)[/tex].
2. Applying the Translation Rule:
The translation rule provided is [tex]\((x, y) \rightarrow (x-4, y+3)\)[/tex].
- For [tex]\( x \)[/tex], we subtract 4 from the initial [tex]\( x \)[/tex]-coordinate:
[tex]\[ x_{\text{translated}} = -1 - 4 = -5 \][/tex]
- For [tex]\( y \)[/tex], we add 3 to the initial [tex]\( y \)[/tex]-coordinate:
[tex]\[ y_{\text{translated}} = -9 + 3 = -6 \][/tex]
So, after translation, the coordinates of [tex]\( Z \)[/tex] are [tex]\((-5, -6)\)[/tex].
3. Applying the 180-Degree Rotation:
A 180-degree rotation about the origin changes the coordinates [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- For [tex]\( x \)[/tex], we negate the translated [tex]\( x \)[/tex]-coordinate:
[tex]\[ x_{\text{rotated}} = -(-5) = 5 \][/tex]
- For [tex]\( y \)[/tex], we negate the translated [tex]\( y \)[/tex]-coordinate:
[tex]\[ y_{\text{rotated}} = -(-6) = 6 \][/tex]
Therefore, after the 180-degree rotation, the coordinates of [tex]\( Z^ \)[/tex] are [tex]\((5, 6)\)[/tex].
Thus, the location of [tex]\( Z^* \)[/tex] after the translation and the 180-degree rotation is [tex]\(\boxed{(5, 6)}\)[/tex].
