Sure! Let's walk through the problem step by step.
### Step 1: Initial Coordinates of Point A
We start with the coordinates of point [tex]\(A\)[/tex]:
[tex]\[ A(0, 1) \][/tex]
### Step 2: Translation
Next, we translate point [tex]\(A\)[/tex] 1 unit to the right and 3 units down.
- Moving 1 unit to the right means adding 1 to the x-coordinate:
[tex]\[ x_{\text{new}} = 0 + 1 = 1 \][/tex]
- Moving 3 units down means subtracting 3 from the y-coordinate:
[tex]\[ y_{\text{new}} = 1 - 3 = -2 \][/tex]
So the new coordinates after the translation are:
[tex]\[ A' = (1, -2) \][/tex]
### Step 3: Rotation
Now, we rotate the translated point [tex]\(A'\)[/tex] (i.e., [tex]\( (1, -2) \)[/tex]) [tex]\(180^{\circ}\)[/tex] clockwise about the origin.
A [tex]\(180^{\circ}\)[/tex] clockwise rotation around the origin will transform any point [tex]\((x, y)\)[/tex] into [tex]\((-x, -y)\)[/tex].
Applying this rule:
[tex]\[ x_{\text{rotated}} = -1 \][/tex]
[tex]\[ y_{\text{rotated}} = 2 \][/tex]
Thus, the final coordinates of point [tex]\(A''\)[/tex] after both the translation and rotation are:
[tex]\[ A'' = (-1, 2) \][/tex]
### Conclusion
The correct coordinates of [tex]\(A''\)[/tex] are:
[tex]\[ \boxed{(-1, 2)} \][/tex]
