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)?
options:

• a reflection of the line segment across the line y = x
• 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 x - axis

Explanation:

Step1: Recall reflection rules

For reflection across the \( y \)-axis: \((x,y)\to(-x,y)\). For reflection across the \( x \)-axis: \((x,y)\to(x,-y)\). For reflection across \( y = x \): \((x,y)\to(y,x)\). For reflection across \( y=-x \): \((x,y)\to(-y,-x)\).

Step2: Analyze original and image endpoints

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

Check reflection across \( y \)-axis: For \((-4,-6)\), applying \( (x,y)\to(-x,y) \) gives \( (4,-6) \). For \((-6,4)\), applying \( (x,y)\to(-x,y) \) gives \( (6,4) \). This matches the image endpoints.

Check other reflections:

• Across \( x \)-axis: \((-4,-6)\to(-4,6)\) (not matching \((4,-6)\)).
• Across \( y = x \): \((-4,-6)\to(-6,-4)\) (not matching).
• Across \( y=-x \): \((-4,-6)\to(6,4)\), \((-6,4)\to(-4,6)\) (not matching).

Answer:

a reflection of the line segment across the \( y \)-axis