How do you reflect a polygon across the y-axis in a coordinate plane?

How to Reflect a Polygon Across the y-axis

Reflecting a polygon across the y-axis is a rigid transformation (or isometry) that flips the shape over the vertical y-axis, creating a mirror image.

The Rule for Reflection

To reflect any point $(x, y)$ across the y-axis, use the coordinate rule:

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

This means the y-coordinate remains completely unchanged, while the x-coordinate changes to its opposite sign.

Step-by-Step Guide

1. Identify the Coordinates: Find the coordinates $(x, y)$ for each vertex of the original polygon.
2. Apply the Transformation: Negate each x-coordinate. For example, if a vertex is at $(3, 4)$, its reflected point (prime point) will be at $(-3, 4)$. If a vertex is at $(-2, -5)$, its reflection will be at $(2, -5)$.
3. Plot the New Vertices: Plot the new coordinates $(-x, y)$ on the coordinate plane.
4. Connect the Vertices: Draw lines connecting the new vertices in the same order as the original polygon to complete the reflected shape.

Example

Consider a triangle with vertices at $A(1, 2)$, $B(4, 3)$, and $C(2, 5)$.

Applying the rule $(x, y) \rightarrow (-x, y)$:

• $A'( -1, 2 )$
• $B'( -4, 3 )$
• $C'( -2, 5 )$

The resulting triangle $A'B'C'$ is the exact mirror image of triangle $ABC$ across the y-axis.