To determine which composition of transformations maps the pre-image PQRS to the image \( P^{-} Q^{-} R^{\prime} S^{\prime} \), let's analyze each of the given options step by step.
### Understanding Transformations
1. Rotation \( R_{0, 270} \): This denotes a rotation of 270 degrees around the origin (0, 0). Generally, rotating a point \((x, y)\) by 270 degrees counterclockwise around the origin results in a new point \((y, -x)\).
2. Translation \( T_{-2, 0} \): This denotes a translation that moves every point by -2 units along the x-axis. Translating a point \((x, y)\) by \(-2\) units on the x-axis results in a new point \((x - 2, y)\).
### Analyzing the Choices
1. Option 1: \( R_{0, 270} \circ T_{-2, 0}(x, y) \)
- Apply translation first: \((x - 2, y)\)
- Then apply rotation to the translated point: \((y, -(x - 2))\)
2. Option 2: \( T_{-2, 0} \circ R_{0, 270}(x, y) \)
- Apply rotation first: \((y, -x)\)
- Then apply translation to the rotated point: \((y - 2, -x)\)
3. Option 3: \( R_{0, 270} \circ r_{y-2 x i s}(x, y) \)
- This seems to mix transformations and notations that are not standardized. Focusing on common transformations, it implies rotation applied after some reflection or custom mapping \( r_{y-2 x i s} \).
4. Option 4: \( r_{y-2 x i s} \circ R_{0, 270}(x, y) \)
- Again, involves a non-standard transformation notation, making it complex to evaluate in this context without additional context about what \( r_{y-2 x i s} \) implies.
### Conclusion
Given the transformations we typically encounter:
- Translation generally adjusts the location within the coordinate system.
- Rotation around the origin shifts the orientation of the point within the plane.
Based on these steps and transformations, the correct sequence that aligns with the common transformation order (first applying the rotation and then translation in the given instruction) is described in Option 1:
[tex]\[ \boxed{R_{0,270} \circ T_{-2,0}(x, y)} \][/tex]
