Question

trapezoid lmnp was reflected using the rule ( r_{y\text{-axis}}(x, y) \to (-x, y) ). which figure represents the image of the reflection? figure

Explanation:

Step1: Recall reflection over y - axis rule

The rule for reflection over the \(y\) - axis is \(r_{y - \text{axis}}(x,y)\to(-x,y)\). This means that for each point \((x,y)\) in the original figure, the \(x\) - coordinate is negated and the \(y\) - coordinate remains the same.

Step2: Analyze the original trapezoid LMNP

Looking at the original trapezoid LMNP (the purple trapezoid on the left side of the \(y\) - axis), let's consider the general position of its vertices. The original trapezoid is in the second quadrant (since \(x\) - coordinates are negative and \(y\) - coordinates are positive). After reflecting over the \(y\) - axis, the \(x\) - coordinates of the vertices will become positive (because we apply \((x,y)\to(-x,y)\) and \(x\) was negative, so \(-x\) will be positive) and the \(y\) - coordinates will remain the same.

Step3: Compare with the given figures

• Figure A: It is in the third quadrant (both \(x\) and \(y\) negative), which is not a reflection over the \(y\) - axis of the original (original has positive \(y\)).
• Figure B: It is near the origin, not a reflection of the original trapezoid.
• Figure C: It is in the first quadrant (positive \(x\) and positive \(y\)), which matches the result of reflecting a figure from the second quadrant (where LMNP is) over the \(y\) - axis (since negating negative \(x\) gives positive \(x\) and \(y\) remains positive).
• Figure D: It is in the fourth quadrant (positive \(x\), negative \(y\)), which is not a reflection over the \(y\) - axis of the original (original has positive \(y\)).

Answer:

C