Reflect figures across the x-axis

Detailed Explanation: Reflect figures across the x-axis

To reflect a figure across the $x$-axis, keep each $x$-coordinate the same and change the sign of each $y$-coordinate.

This means:

• A point above the $x$-axis moves the same distance below it.
• A point below the $x$-axis moves the same distance above it.
• A point on the $x$-axis stays in the same place.
• Never change the $x$-coordinate.

Example: Reflect triangle $ABC$ across the $x$-axis.

Suppose the vertices are

Apply $(x,y)\to(x,-y)$ to each point:

1. Point $A(2,4)$ keeps its $x$-coordinate $2$ and changes $4$ to $-4$:

2. Point $B(5,1)$ keeps its $x$-coordinate $5$ and changes $1$ to $-1$:

3. Point $C(-1,-3)$ keeps its $x$-coordinate $-1$ and changes $-3$ to $3$:

The reflected triangle has vertices

Plot these new points and connect them in the same order. The reflected triangle is congruent to the original triangle, and each point is the same distance from the $x$-axis as its original point, but on the opposite side.