Skill: 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.

$$(x,y)\longrightarrow (x,-y)$$

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

$$A(2,4),\qquad B(5,1),\qquad C(-1,-3).$$

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

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

$$A'(2,-4)$$

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

$$B'(5,-1)$$

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

$$C'(-1,3)$$

The reflected triangle has vertices

$$\boxed{A'(2,-4),\ B'(5,-1),\ C'(-1,3)}.$$

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.