Final answer:
The student is asking about geometric transformations, specifically reflecting a figure over the line y=x and rotating it 180 degrees. After reflecting, the coordinates (x, y) become (y, x), and a 180-degree rotation inverts them to (-y, -x).
Step-by-step explanation:
The student's question pertains to the transformation of geometric figures in the plane. Specifically, they are asking about the effect of reflecting a figure over the line y=x followed by a 180-degree rotation. Reflecting over the line y=x swaps the x- and y-coordinates of each point in the figure. This transformation results in a figure that is a mirror image across the line y=x. A subsequent 180-degree rotation can be understood as flipping the figure upside down around the origin, or in other words, changing the sign of both the x- and y-coordinates of the new points.
The overall effect on the coordinates of any point (x, y) after these two transformations can be described by the equations:
- Reflection over y=x: The original point (x, y) becomes (y, x).
- 180-degree rotation: The point (y, x) becomes (-y, -x).
These transformations are important in understanding concepts such as slope and y-intercept, and how changing these values affects the position and orientation of lines in the coordinate plane.