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) )
(graph with points m(-5,4), m(5,4), l(-6,2), n(-3,2), n(3,2), l(6,2) on a coordinate plane)

Explanation:

Step1: Analyze coordinates before and after reflection

Take point \( M(-5, 4) \) and its image \( M'(5, 4) \). The \( x \)-coordinate changes sign (\(-5\) becomes \(5\)), \( y \)-coordinate remains same (\(4\) stays \(4\)). Similarly, \( L(-6, 2) \to L'(6, 2) \) and \( N(-3, 2) \to N'(3, 2) \) show \( x \)-coordinate sign flips, \( y \)-coordinate unchanged.

Step2: Recall reflection rules

Reflection over \( y \)-axis: \( r_{y\text{-axis}}(x, y) \to (-x, y) \) (flips \( x \)-coordinate sign, \( y \) same). Reflection over \( x \)-axis: \( r_{x\text{-axis}}(x, y) \to (x, -y) \) (flips \( y \)-coordinate sign, \( x \) same). Here, \( x \)-coordinate signs flip, \( y \) same, so it's \( y \)-axis reflection.

Answer:

\( r_{y\text{-axis}}(x, y) \to (-x, y) \)