To determine which rule describes the composition of transformations that maps the pre-image PQRS to the image [tex]\( P'Q'R'S' \)[/tex], we need to analyze the given options.
The correct transformation sequence that accomplishes this mapping is the third one:
[tex]\[ R_{0,270}^{\circ} \circ r_{y-2 \text{ais}}(x, y) \][/tex]
This sequence involves two transformations:
1. Rotation [tex]\( R_{0,270^{\circ}} \)[/tex]: This specifies a rotation about the origin (0,0) by 270 degrees counterclockwise. This changes the orientation of the shape.
2. Reflection [tex]\( r_{y-2 \text{ais}}(x, y) \)[/tex]: This specifies a reflection across a specific line or axis. Here "y-2 ais" appears to be reflecting over the y-axis with some horizontal translation involved.
In the context of the steps, the pre-image PQRS undergoes the following transformations:
1. Rotation by 270 degrees around the origin: This transformation will realign the points of PQRS to a new orientation.
2. Reflection: This takes the already rotated shape and reflects it across the specified line.
Hence, only one of the options fits these criteria:
[tex]\[ R_{0,270}^{\circ} \circ r_{y-2 \text{ais}}(x, y) \][/tex]
So, the correct rule for the composition of transformations that maps the pre-image PQRS to the image [tex]\( P'Q'R'S' \)[/tex] is described by the third option:
[tex]\[ \boxed{3} \][/tex]
