To determine which reflection will produce an image of the point [tex]\((0, k)\)[/tex] at the same coordinates [tex]\((0, k)\)[/tex], we need to examine how each type of reflection affects the coordinates of the point.
1. Reflection across the [tex]$x$[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]$x$[/tex]-axis is [tex]\((x, -y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the [tex]$x$[/tex]-axis gives us the point [tex]\((0, -k)\)[/tex].
2. Reflection across the [tex]$y$[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]$y$[/tex]-axis is [tex]\((-x, y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the [tex]$y$[/tex]-axis gives us the point [tex]\((0, k)\)[/tex].
3. Reflection across the line [tex]$y = x$[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]$y = x$[/tex] is [tex]\((y, x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the line [tex]$y = x$[/tex] gives us the point [tex]\((k, 0)\)[/tex].
4. Reflection across the line [tex]$y = -x$[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]$y = -x$[/tex] is [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the line [tex]$y = -x$[/tex] gives us the point [tex]\((-k, 0)\)[/tex].
From the analysis above, it is clear that reflecting the point [tex]\((0, k)\)[/tex] across the [tex]$y$[/tex]-axis will produce an image at the same coordinates [tex]\((0, k)\)[/tex]. Therefore, the correct reflection type is:
A reflection of the point across the [tex]$y$[/tex]-axis.
