Skill: Reflect figures across coordinate axes

A reflection flips a figure like a mirror image across a coordinate axis. Apply the reflection rule to every vertex of the figure.

• Across the $x$-axis:

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

The $x$-coordinate stays the same, and the $y$-coordinate changes sign.

• Across the $y$-axis:

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

The $y$-coordinate stays the same, and the $x$-coordinate changes sign.

Example

Triangle $(ABC)$ has vertices

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

Reflect the triangle across the $x$-axis.

Since the reflection is across the $x$-axis, use

$$(x,y)\rightarrow(x,-y).$$

Apply the rule to each vertex:

$$\begin{aligned} A(-2,1)&\rightarrow A'(-2,-1)\\ B(3,1)&\rightarrow B'(3,-1)\\ C(1,4)&\rightarrow C'(1,-4) \end{aligned}$$

So the reflected triangle has vertices

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

Notice that only the $y$-coordinates changed. The $x$-coordinates stayed the same, so the coordinates were not swapped. The reflected triangle has the same side lengths, angle measures, and shape as the original, but its orientation is reversed.