Reflect figures across coordinate axes

A reflection across the x-axis maps each point $(x,y)$ to $(x,-y)$ , while a reflection across the y-axis maps $(x,y)$ to $(-x,y)$ ; applying the appropriate rule to every vertex produces the image of a figure. The learner understands that reflections preserve lengths, angle measures, and shape but reverse orientation, and that reflecting across an axis changes only the corresponding coordinate, not by swapping coordinates. Reflections across arbitrary lines or in three dimensions are outside this scope.

Detailed Explanation: 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.

Learn by doing: Reflect figures across coordinate axes

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Cartesian Grid - Reflection of Point (Coordinates to Coordinates) across Axis

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