Question

what is the rule for the reflection? ( r_{y\text{-axis}}(x, y) \to (-x, y) ) ( r_{y\text{-axis}}(x, y) \to (x, -y) ) ( r_{x\text{-axis}}(x, y) \to (-x, y) ) ( r_{x\text{-axis}}(x, y) \to (x, -y) )

Explanation:

Step1: Recall reflection rules

Reflection over x - axis: For a point \((x,y)\), the reflection \(r_{x - axis}(x,y)\) is \((x,-y)\) (the x - coordinate stays the same, y - coordinate is negated). Reflection over y - axis: For a point \((x,y)\), the reflection \(r_{y - axis}(x,y)\) is \((-x,y)\) (the y - coordinate stays the same, x - coordinate is negated).

Step2: Analyze the graph

Looking at the graph, points like \(B(0,2)\) and \(B'(0, - 2)\), \(C(2,2)\) and \(C'(2,-2)\), \(A(-1,0)\) and \(A'(-1,0)\) (wait, no, actually \(A\) and \(A'\) – wait, looking at the symmetry, the figure is symmetric about the x - axis. So for a point \((x,y)\) above the x - axis, its reflection is \((x,-y)\) below the x - axis. So the reflection is over the x - axis, and the rule is \(r_{x - axis}(x,y)\to(x,-y)\).

Answer:

\(r_{x - axis}(x,y)\to(x,-y)\) (the last option: \(r_{x - axis}(x,y)\to(x,-y)\))