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.

The rule is:

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

• 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.

Example

Reflect triangle $(ABC)$ across the $x$-axis.

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

Apply the rule to each point:

1. Point $(A(2,4))$:

$$(2,4)\rightarrow(2,-4)$$

So, $(A'=(2,-4))$.

2. Point $(B(5,1))$:

$$(5,1)\rightarrow(5,-1)$$

So, $(B'=(5,-1))$.

3. Point $(C(-3,-2))$:

$$(-3,-2)\rightarrow(-3,2)$$

So, $(C'=(-3,2))$.

The reflected triangle has vertices

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

Notice that the $x$-coordinates stayed the same, while each $y$-coordinate changed sign. Plot the new points and connect them in the same order to draw the reflected figure.