To determine which reflection will produce an image at the same coordinates [tex]\((0, k)\)[/tex], let's analyze each option carefully:
1. Reflection across the x-axis:
- For any point [tex]\((x, y)\)[/tex], reflecting it across the x-axis results in the new point [tex]\((x, -y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflection across the x-axis yields [tex]\((0, -k)\)[/tex].
2. Reflection across the y-axis:
- For any point [tex]\((x, y)\)[/tex], reflecting it across the y-axis results in the new point [tex]\((-x, y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflection across the y-axis yields [tex]\((0, k)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- For any point [tex]\((x, y)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] results in the new point [tex]\((y, x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflection across the line [tex]\(y = x\)[/tex] yields [tex]\((k, 0)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- For any point [tex]\((x, y)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] results in the new point [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflection across the line [tex]\(y = -x\)[/tex] yields [tex]\((-k, 0)\)[/tex].
Of all these transformations, only the reflection across the y-axis results in the point retaining its original coordinates [tex]\((0, k)\)[/tex].
Therefore, the correct answer is:
- a reflection of the point across the y-axis
