To determine which transformation represents a reflection over the [tex]\( y \)[/tex]-axis, let's analyze the impact of reflecting a point [tex]\((x, y)\)[/tex] over the [tex]\( y \)[/tex]-axis.
Reflection over the [tex]\( y \)[/tex]-axis means flipping the point across the vertical [tex]\( y \)[/tex]-axis. The [tex]\( y \)[/tex]-coordinate of the point remains unchanged, but the [tex]\( x \)[/tex]-coordinate changes its sign.
Therefore, a point [tex]\((x, y)\)[/tex], when reflected over the [tex]\( y \)[/tex]-axis, will transform to [tex]\((-x, y)\)[/tex].
Now, let’s match this with the given options:
1. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
2. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
3. [tex]\((x, y) \rightarrow (y, x)\)[/tex]
4. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
From the explanation above, the transformation [tex]\((x, y) \rightarrow (-x, y)\)[/tex] correctly represents a reflection over the [tex]\( y \)[/tex]-axis.
Hence, the correct transformation is [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
