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:

To determine the reflection, we analyze the coordinates of the endpoints before and after the transformation. The original endpoints are \((-4, -6)\) and \((-6, 4)\), and the image endpoints are \((4, -6)\) and \((6, 4)\).

• Reflection across the x - axis: Changes the sign of the \(y\) - coordinate. For a point \((x,y)\), the image is \((x, -y)\). For \((-4, -6)\), the image would be \((-4,6)\) (not \((4, -6)\)), so this is not the case.
• Reflection across the y - axis: Changes the sign of the \(x\) - coordinate. For a point \((x,y)\), the image is \((-x,y)\). For \((-4, -6)\), the image is \((4, -6)\) (since \(-(-4)=4\) and \(y=-6\) remains the same). For \((-6,4)\), the image is \((6,4)\) (since \(-(-6) = 6\) and \(y = 4\) remains the same). This matches the given image endpoints.
• Reflection across the line \(y=x\): Swaps the \(x\) and \(y\) coordinates. For \((-4, -6)\), the image would be \((-6, -4)\) (not \((4, -6)\)), so this is incorrect.
• Reflection across the line \(y=-x\): The transformation is \((x,y)\to(-y,-x)\). For \((-4, -6)\), the image would be \((6,4)\) (not \((4, -6)\)), so this is incorrect.

Answer:

B. a reflection of the line segment across the y - axis