Question

a line segment has endpoints at (-1, 4) and (4, 1). which reflection will produce an image with endpoints at (-4, 1) and (-1, -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

The rule for reflecting a point $(x,y)$ across the $x$-axis is $(x,-y)$.
The rule for reflecting a point $(x,y)$ across the $y$-axis is $(-x,y)$.
The rule for reflecting a point $(x,y)$ across the line $y = x$ is $(y,x)$.
The rule for reflecting a point $(x,y)$ across the line $y=-x$ is $(-y,-x)$.

Step2: Apply rules to endpoints

For the point $(-1,4)$:

• Across $x$-axis: $(-1, - 4)$
• Across $y$-axis: $(1,4)$
• Across $y = x$: $(4,-1)$
• Across $y=-x$: $(-4,1)$

For the point $(4,1)$:

• Across $x$-axis: $(4,-1)$
• Across $y$-axis: $(-4,1)$
• Across $y = x$: $(1,4)$
• Across $y=-x$: $(-1,-4)$

Answer:

D. a reflection of the line segment across the line $y=-x$