Question

a point has the coordinates (0, k).
which reflection of the point will produce an image at the same coordinates, (0, k)?
○ a reflection of the point across the x - axis
○ a reflection of the point across the y - axis
○ a reflection of the point across the line y = x
○ a reflection of the point across the line y = -x

Explanation:

1. For reflection across the x - axis: The rule is \((x,y)\to(x, - y)\). For the point \((0,k)\), after reflection across the x - axis, it becomes \((0, - k)\), which is not the same as the original point.
2. For reflection across the y - axis: The rule is \((x,y)\to(-x,y)\). For the point \((0,k)\), substituting \(x = 0\) into the rule, we get \((- 0,k)=(0,k)\), so the image has the same coordinates as the original point.
3. For reflection across the line \(y=x\): The rule is \((x,y)\to(y,x)\). For the point \((0,k)\), after reflection, it becomes \((k,0)\), which is not the same as the original point (unless \(k = 0\), but the question is general for any \(k\)).
4. For reflection across the line \(y=-x\): The rule is \((x,y)\to(-y,-x)\). For the point \((0,k)\), after reflection, it becomes \((-k,0)\), which is not the same as the original point.

Answer:

B. a reflection of the point across the y - axis