Reflection of a Function
A reflection of a function flips its graph across an axis, creating a mirror image across the x-axis or y-axis.
Definition
A reflection flips the graph of a function to create a mirror image, across the $x$-axis, the $y$-axis, or (to find an inverse) across the line $y = x$. Reflecting $f(x)$ across the $x$-axis gives $g(x) = -f(x)$, negating all $y$-values; reflecting across the $y$-axis gives $g(x) = f(-x)$, replacing $x$ with $-x$; and reflecting across $y = x$ swaps $x$ and $y$ to produce the inverse function. Reflections are isometries (distance-preserving maps) in the Euclidean plane: the reflection across the $x$-axis is represented by the matrix $\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}$, across the $y$-axis by $\begin{bmatrix}-1 & 0\\0 & 1\end{bmatrix}$, and composing two reflections across intersecting lines produces a rotation by twice the angle between the lines.
Example
$f(x) = x^2$ opens upward, but $g(x) = -x^2$ reflects across the $x$-axis to open downward, while $h(x) = (-x)^2$ reflects across the $y$-axis (the same shape here, but different for non-symmetric functions). For $f(x) = \sqrt{x}$, domain $[0, \infty)$: reflecting across the $x$-axis gives $-\sqrt{x}$, while reflecting across the $y$-axis gives $\sqrt{-x}$ with domain $(-\infty, 0]$, so the two reflections produce different functions. In group theory, reflections generate dihedral groups $D_n$, and the reflection symmetry of a function (even: $f(-x) = f(x)$, odd: $f(-x) = -f(x)$) has implications for Fourier series: even functions have only cosine terms, odd functions only sine terms.
Key Insight
To reflect across the $x$-axis, put a negative sign in front of the whole function; to reflect across the $y$-axis, replace $x$ with $-x$ inside it. Reflecting across $y = x$ is how you find an inverse function graphically, the inverse's graph is the mirror image of the original across that diagonal line. Parity (even/odd symmetry) is a reflection property, and symmetry analysis using reflections reduces computation in physics and engineering: symmetric boundary conditions often halve the domain needed for simulation.