Skill: Perform reflections

A reflection flips a figure across a line, called the line of reflection. Each point moves to the opposite side of the line, the same perpendicular distance away.

For reflections on a coordinate grid:

• Across the $x$-axis: $(x,y)\rightarrow(x,-y)$
• Across the $y$-axis: $(x,y)\rightarrow(-x,y)$

Example

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

The vertices are:

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

Step 1: Use the rule for the $x$-axis.

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

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

Step 2: Reflect each vertex.

$$\begin{aligned} A(2,4)&\rightarrow A'(2,-4)\\ B(5,4)&\rightarrow B'(5,-4)\\ C(3,1)&\rightarrow C'(3,-1) \end{aligned}$$

Step 3: Plot the new points and connect them.

Plot $A'(2,-4)$, $B'(5,-4)$, and $C'(3,-1)$, then connect the points in the same order.

The reflected triangle is the same size and shape as the original. Its points are the same distance from the $x$-axis, but they are on the opposite side.