To determine which reflection of the point [tex]\((0, k)\)[/tex] will produce an image at the same coordinates [tex]\((0, k)\)[/tex], let's analyze the effect of reflecting the point across each given axis and line.
1. Reflection across the [tex]\( x \)[/tex]-axis:
- For a point [tex]\((x, y)\)[/tex], reflecting it across the [tex]\( x \)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- For our specific point [tex]\((0, k)\)[/tex], reflecting it across the [tex]\( x \)[/tex]-axis would yield the point [tex]\((0, -k)\)[/tex], which is not the same as the original point unless [tex]\( k = 0 \)[/tex]. Therefore, this option does not generally give us [tex]\((0, k)\)[/tex].
2. Reflection across the [tex]\( y \)[/tex]-axis:
- For a point [tex]\((x, y)\)[/tex], reflecting it across the [tex]\( y \)[/tex]-axis results in the point [tex]\((-x, y)\)[/tex].
- For our specific point [tex]\((0, k)\)[/tex], reflecting it across the [tex]\( y \)[/tex]-axis would yield the point [tex]\((0, k)\)[/tex], which is exactly the same as the original point. Thus, this reflection maintains the coordinates.
3. Reflection across the line [tex]\( y = x \)[/tex]:
- For a point [tex]\((x, y)\)[/tex], reflecting it across the line [tex]\( y = x \)[/tex] results in the point [tex]\((y, x)\)[/tex].
- For our specific point [tex]\((0, k)\)[/tex], reflecting it across the line [tex]\( y = x \)[/tex] would yield the point [tex]\((k, 0)\)[/tex], which is different from the original point unless [tex]\( k = 0 \)[/tex]. Hence, this option does not generally give us [tex]\((0, k)\)[/tex].
4. Reflection across the line [tex]\( y = -x \)[/tex]:
- For a point [tex]\((x, y)\)[/tex], reflecting it across the line [tex]\( y = -x \)[/tex] results in the point [tex]\((-y, -x)\)[/tex].
- For our specific point [tex]\((0, k)\)[/tex], reflecting it across the line [tex]\( y = -x \)[/tex] would yield the point [tex]\((-k, 0)\)[/tex], which is once again different from the original point unless [tex]\( k = 0 \)[/tex]. So, this option also does not generally give us [tex]\((0, k)\)[/tex].
Upon examining all the reflections, the only reflection that results in the point [tex]\((0, k)\)[/tex] remaining unchanged is the reflection across the [tex]\( y \)[/tex]-axis.
Therefore, the correct reflection is across the [tex]\( y \)[/tex]-axis, which corresponds to the second option.
