Question
which transformation results in the same image as a composition of reflections across the x-axis and then the y-axis?
Answers
GPT-4o mini
A composition of reflections across the x-axis followed by the y-axis will result in a transformation equivalent to a rotation of 180 degrees around the origin.
Here's a breakdown of the process:
1. **Reflection across the x-axis**: This transformation changes the coordinates \((x, y)\) to \((x, -y)\).
2. **Reflection across the y-axis**: This further transforms the new coordinates \((x, -y)\) to \((-x, -y)\).
Thus, starting with the original point \((x, y)\), after both reflections, you end up with the point \((-x, -y)\).
This point can also be visualized as a rotation of 180 degrees, which makes the two transformations equivalent. Therefore, the combined effect of the two reflections results in the same image as a single rotation of 180 degrees around the origin.
Here's a breakdown of the process:
1. **Reflection across the x-axis**: This transformation changes the coordinates \((x, y)\) to \((x, -y)\).
2. **Reflection across the y-axis**: This further transforms the new coordinates \((x, -y)\) to \((-x, -y)\).
Thus, starting with the original point \((x, y)\), after both reflections, you end up with the point \((-x, -y)\).
This point can also be visualized as a rotation of 180 degrees, which makes the two transformations equivalent. Therefore, the combined effect of the two reflections results in the same image as a single rotation of 180 degrees around the origin.