To determine the rule that describes the composition of transformations mapping the pre-image [tex]\(A B C D\)[/tex] to the final image [tex]\(A^{\prime\prime} B^{\prime} C^{-1} D^{\prime}\)[/tex], we need to analyze and understand the transformations being applied.
Given the options:
1. [tex]\(r_{x-2 xis} \circ T_{-6,1}(x, y)\)[/tex]
2. [tex]\(T_{-6,1} \circ r_{x-2 xis}(x, y)\)[/tex]
3. [tex]\(R_{0,90 \circ} \circ T_{-6,1}(x, y)\)[/tex]
4. [tex]\(T_{-6,1} \odot R_{0,90}(x, y)\)[/tex]
Breakdown of notation:
- [tex]\(T_{-6,1}(x, y)\)[/tex] typically represents a translation transformation, translating the point [tex]\((x, y)\)[/tex] by [tex]\(-6\)[/tex] units in the x-direction and [tex]\(1\)[/tex] unit in the y-direction.
- [tex]\(r_{x-2 xis}(x, y)\)[/tex] seems to denote a reflection transformation, but the precise meaning isn't entirely clear from standard notation.
- [tex]\(R_{0,90 \circ}(x, y)\)[/tex] refers to a rotation, presumably by [tex]\(90^\circ\)[/tex], centered at the origin [tex]\((0,0)\)[/tex].
- [tex]\(\odot\)[/tex] symbol isn't typical in standard transformation notation and may indicate a sequential application rather than composition.
Without detailed meanings of all elements, we'll focus on combinations involving the more standard transformations (translation and some rotation/reflection). From the results, you can deduce transformations step-by-step to see which produces the final expected image.
Since we need to infer map transformations, observe rule following:
From the composition:
- Applying translation [tex]\(T_{-6, 1}\)[/tex] first means shifting the point.
- Follow by rotations or reflections (vertical axis can't be clear)
Option aligning step transformations leading:
- [tex]\(R_{90^\circ}\)[/tex] rotations visually else reflections.
Key factor is state ensuring total transformation consistency, translation followed by appropriate rotation matches composed transformations yielding final image.
Correct choice based on typically performed composition is:
[tex]\[
\boxed{R_{0,90 \circ} \circ T_{-6,1}(x, y)}
\][/tex]
