Reflection in the Cartesian Plane

What is Reflection?

Reflection is a type of transformation that flips a point or shape over a line, creating a mirror image. In the Cartesian plane, we often reflect over the x-axis, y-axis, or lines like x = k or y = k.

Reflection Over the Axes

• Reflection over the x-axis: If a point \$(x, y) \\$ is reflected over the x-axis, its image is \$(x, -y) \\$.
• Reflection over the y-axis: If a point \$(x, y) \\$ is reflected over the y-axis, its image is \$(-x, y) \\$.

Reflection Over Other Lines

• Reflection over the line \$x = k \\$: The image of a point \$(x, y) \\$ is \$(2k - x, y) \\$.
• Reflection over the line \$y = k \\$: The image of a point \$(x, y) \\$ is \$(x, 2k - y) \\$.

Examples

Example 1: Reflecting a Point Over the x-axis

Problem: Reflect the point \$(3, 4) \\$ over the x-axis.

Example 2: Reflecting a Point Over the Line \$y = 2 \\$

Problem: Reflect the point \$(5, 1) \\$ over the line \$y = 2 \\$.