To determine which reflection of a point with coordinates \((0, k)\) will produce an image at the same coordinates \((0, k)\), let's analyze each given reflection option:
1. Reflection across the [tex]$x$[/tex]-axis:
- When a point \((x,y)\) is reflected across the [tex]$x$[/tex]-axis, its new coordinates become \((x, -y)\).
- For the point \((0, k)\), reflecting it across the [tex]$x$[/tex]-axis results in \((0, -k)\).
- This is not the same as \((0,k)\).
2. Reflection across the [tex]$y$[/tex]-axis:
- When a point \((x,y)\) is reflected across the [tex]$y$[/tex]-axis, its new coordinates become \((-x, y)\).
- For the point \((0, k)\), reflecting it across the [tex]$y$[/tex]-axis results in \((0, k)\).
- This is the same as \((0,k)\).
3. Reflection across the line [tex]$y=x$[/tex]:
- When a point \((x,y)\) is reflected across the line [tex]$y=x$[/tex], its new coordinates become \((y, x)\).
- For the point \((0, k)\), reflecting it across the line [tex]$y=x$[/tex] results in \((k, 0)\).
- This is not the same as \((0,k)\).
4. Reflection across the line [tex]$y=-x$[/tex]:
- When a point \((x,y)\) is reflected across the line [tex]$y=-x$[/tex], its new coordinates become \((-y, -x)\).
- For the point \((0, k)\), reflecting it across the line [tex]$y=-x$[/tex] results in \((-k, 0)\).
- This is not the same as \((0,k)\).
Therefore, the reflection that produces an image at the same coordinates, \((0, k)\), is a reflection across the [tex]$y$[/tex]-axis.
The correct answer is:
a reflection of the point across the [tex]$y$[/tex]-axis
