To determine which reflection will produce the image of the point [tex]\((0, k)\)[/tex] at the same coordinates, let's analyze the effect of each reflection option on 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 changes the sign of the [tex]\( y \)[/tex]-coordinate. Thus, the reflection of [tex]\((0, k)\)[/tex] across the [tex]\( x \)[/tex]-axis is:
[tex]\[
(0, k) \rightarrow (0, -k)
\][/tex]
2. Reflection across the [tex]\( y \)[/tex]-axis:
The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\( y \)[/tex]-axis changes the sign of the [tex]\( x \)[/tex]-coordinate. Thus, the reflection of [tex]\((0, k)\)[/tex] across the [tex]\( y \)[/tex]-axis is:
[tex]\[
(0, k) \rightarrow (0, k)
\][/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] swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates. Thus, the reflection of [tex]\((0, k)\)[/tex] across the line [tex]\( y = x \)[/tex] is:
[tex]\[
(0, k) \rightarrow (k, 0)
\][/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] swaps the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates and changes their signs. Thus, the reflection of [tex]\((0, k)\)[/tex] across the line [tex]\( y = -x \)[/tex] is:
[tex]\[
(0, k) \rightarrow (-k, 0)
\][/tex]
By analyzing each reflection, we see that only the reflection across the [tex]\( y \)[/tex]-axis leaves the point [tex]\((0, k)\)[/tex] unchanged at its original coordinates. Therefore, the correct transformation is the reflection of the point across the [tex]\( y \)[/tex]-axis.
The answer is:
[tex]\[
\boxed{2}
\][/tex]
