A reflection over the line [tex]\( y = x \)[/tex] involves swapping the [tex]\( x \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate of each point. Essentially, for a point [tex]\( (x, y) \)[/tex], its reflection over the line [tex]\( y = x \)[/tex] would be [tex]\( (y, x) \)[/tex].
Let's consider the given transformations and see which one correctly represents this:
1. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]: This transformation changes the sign of the [tex]\( x \)[/tex]-coordinate while keeping the [tex]\( y \)[/tex]-coordinate unchanged. This is a reflection over the [tex]\( y \)[/tex]-axis, not over the line [tex]\( y = x \)[/tex].
2. [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]: This transformation changes the signs of both the [tex]\( x \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate. This is a 180-degree rotation around the origin or a point reflection over the origin, not a reflection over the line [tex]\( y = x \)[/tex].
3. [tex]\((x, y) \rightarrow (y, x)\)[/tex]: This transformation swaps the [tex]\( x \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate. This directly matches the reflection over the line [tex]\( y = x \)[/tex], as it takes a point [tex]\( (x, y) \)[/tex] and reflects it to [tex]\( (y, x) \)[/tex].
4. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]: This transformation swaps the coordinates and changes the sign of the new [tex]\( y \)[/tex]-coordinate (which was originally the [tex]\( x \)[/tex]-coordinate). This is not a reflection over the line [tex]\( y = x \)[/tex] but a rotation combined with a reflection over the [tex]\( y \)[/tex]-axis.
Thus, the correct transformation that represents a reflection over the line [tex]\( y = x \)[/tex] is:
[tex]\[
(x, y) \rightarrow (y, x)
\][/tex]
