Question

a line segment has endpoints at (-4, -6) and (-6, 4). which reflection will produce an image with endpoints at (4, -6) and (6, 4)?
○ a reflection of the line segment across the x-axis
○ a reflection of the line segment across the y-axis
○ a reflection of the line segment across the line ( y = x )
○ a reflection of the line segment across the line ( y = -x )

Explanation:

Step1: Recall reflection rules

• Reflection across x - axis: \((x,y)\to(x, - y)\)
• Reflection across y - axis: \((x,y)\to(-x,y)\)
• Reflection across \(y = x\): \((x,y)\to(y,x)\)
• Reflection across \(y=-x\): \((x,y)\to(-y,-x)\)

Step2: Analyze the given points

Original endpoints: \((-4,-6)\) and \((-6,4)\)
Image endpoints: \((4,-6)\) and \((6,4)\)

For the first point \((-4,-6)\to(4,-6)\): The \(x\) - coordinate changes sign (\(-4\to4\)) and the \(y\) - coordinate remains the same (\(-6\to - 6\)).
For the second point \((-6,4)\to(6,4)\): The \(x\) - coordinate changes sign (\(-6\to6\)) and the \(y\) - coordinate remains the same (\(4\to4\)).

This matches the rule for reflection across the \(y\) - axis \((x,y)\to(-x,y)\) (here, since \(x\) is negative, \(-x\) becomes positive and \(y\) stays the same).

Answer:

a reflection of the line segment across the y - axis