To solve the problem of finding the transformed image of the ordered pair [tex]\((5, -4)\)[/tex] under the composition of the translation [tex]\(T_{(-1, 2)}\)[/tex] followed by a rotation [tex]\(r(180°, O)\)[/tex], we will follow these steps:
1. Translation [tex]\(T_{(-1, 2)}\)[/tex]:
We need to translate the point [tex]\((5, -4)\)[/tex] by [tex]\((-1, 2)\)[/tex]. This means we subtract 1 from the x-coordinate and add 2 to the y-coordinate.
- Translated x-coordinate: [tex]\(5 - 1 = 4\)[/tex]
- Translated y-coordinate: [tex]\(-4 + 2 = -2\)[/tex]
So, the translated point is [tex]\((4, -2)\)[/tex].
2. Rotation [tex]\(r(180°, O)\)[/tex]:
Next, we need to rotate the translated point [tex]\((4, -2)\)[/tex] by 180 degrees around the origin [tex]\(O\)[/tex]. Rotating a point [tex]\((x, y)\)[/tex] by 180 degrees around the origin results in the point [tex]\((-x, -y)\)[/tex].
- Rotated x-coordinate: [tex]\(-4\)[/tex]
- Rotated y-coordinate: [tex]\(2\)[/tex]
So, the rotated point is [tex]\((-4, 2)\)[/tex].
Combining these steps, the final transformed image of the point [tex]\((5, -4)\)[/tex] under the given composition is [tex]\((-4, 2)\)[/tex].
Thus, the correct answer is [tex]\((-4, 2)\)[/tex].
