To determine the rule that describes the composition of transformations mapping pre-image [tex]\(ABCD\)[/tex] to the final image [tex]\(A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}\)[/tex], let’s analyze each of the transformations described and the order in which they should be applied.
1. [tex]\(r_{x-2x i 5}\)[/tex]: This appears to denote a rotation transformation. "5" could represent an angle, but the exact rotation center isn’t specified in typical notation.
2. [tex]\(T_{-6,1}\)[/tex]: This denotes a translation transformation by the vector [tex]\((-6, 1)\)[/tex], meaning every point [tex]\((x, y)\)[/tex] is moved to [tex]\((x - 6, y + 1)\)[/tex].
Given the options:
1. [tex]\(r_{x-2 x i 5}\)[/tex] OT [tex]\(T_{-6,1}(x, y)\)[/tex]: This appears to denote a rotation followed by translation, but the notation "OT" is unclear.
2. [tex]\(T_{-6,1} \circ r_{x-2 x i s}(x, y)\)[/tex]: This means first apply the rotation [tex]\(r_{x-2 x i s}\)[/tex] and then apply the translation [tex]\(T_{-6,1}\)[/tex].
3. [tex]\(R_{0,500}\)[/tex] OT [tex]\(-6,1(x, y)\)[/tex]: This is ambiguous due to non-standard notation, but even so, the numbers “500” and “-6,1” do not clearly relate to the specified transformations.
4. [tex]\(T_{-6,1} \circ R_{0,900}(x, y)\)[/tex]: This means first apply the rotation [tex]\(R_{0,900}\)[/tex], then the translation [tex]\(T_{-6,1}\)[/tex], but the "900" is not specified clearly how it correlates to the given transformations.
The correct rule for describing the given transformations in the sequence indicated (first a rotation, then a translation) is:
[tex]\[
T_{-6,1} \circ r_{x-2 x i s}(x, y)
\][/tex]
Here, [tex]\(T_{-6,1}\)[/tex] denotes the translation and [tex]\(\circ\)[/tex] denotes the composition of the two transformations, applying the rotation first followed by the translation.
