Reflection
A reflection flips a figure across a line of reflection, producing a mirror image that is congruent to the original.
Formula
\text{Reflection across x-axis: } (x, y) \to (x, -y)
Definition
A reflection flips a shape across a line, called the line of reflection or mirror line, so every point is flipped to the opposite side of the line while staying the same distance from it, like a mirror image. Reflecting across the x-axis maps $(x,y) \to (x,-y)$; across the y-axis, $(x,y) \to (-x,y)$; across $y=x$, $(x,y) \to (y,x)$; across $y=-x$, $(x,y) \to (-y,-x)$. Reflections are isometries that reverse orientation, and a reflection across line $l$ maps each point $P$ to $P'$ such that $l$ is the perpendicular bisector of $PP'$. Formally, reflection across a line through the origin with unit direction vector $u = (\cos\theta, \sin\theta)$ is given by the matrix $M = 2uu^T - I = \begin{bmatrix}\cos(2\theta) & \sin(2\theta)\\\sin(2\theta) & -\cos(2\theta)\end{bmatrix}$; reflections are involutions ($M^2 = I$) with determinant $-1$, distinguishing them from rotations ($\det = +1$), and every isometry of $\mathbb{R}^2$ is a composition of at most $3$ reflections.
Example
Reflecting the point $(3, 4)$ across the x-axis gives $(3, -4)$ (x-coordinate unchanged, y-coordinate's sign flips), and reflecting across the y-axis gives $(-3, 4)$. Reflecting a triangle with vertices $A(1,2)$, $B(4,2)$, $C(3,5)$ across the y-axis gives $A'(-1,2)$, $B'(-4,2)$, $C'(-3,5)$, congruent to the original but a mirror image with reversed orientation. Composing reflections across lines at angle $\theta$ apart produces a rotation by $2\theta$ about their intersection, so reflecting across the x-axis then the y-axis (angle $\pi/2$ apart) gives rotation by $\pi$ ($180$ degrees).
Key Insight
If you write a word on paper and hold it up to a mirror, the mirror shows a reflection: left and right are swapped, but up and down stay the same. Reflections reverse orientation, a clockwise-oriented triangle becomes counterclockwise after reflection, and two reflections compose to a rotation (if the lines intersect) or a translation (if the lines are parallel), which is why every rotation can be decomposed into two reflections. The minimal number of reflections needed to build any isometry ($1$, $2$, or $3$) classifies it: $1$ for reflections, $2$ for rotations and translations, $3$ for glide reflections.