Reflection of a Function
Functions & Advanced AlgebraA 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. You can reflect across the x-axis (flip it over the horizontal axis) or across the y-axis (flip it over the vertical axis).
Example
$f(x) = x^2$ opens upward. $g(x) = -x^2$ reflects across the $x$-axis: it opens downward. $h(x) = (-x)^2$ reflects across the $y$-axis (same shape here, but different for non-symmetric functions).
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 the function.
Definition
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 $-x$. Reflecting across the line $y = x$ gives the inverse function (swap $x$ and $y$).
Example
$f(x) = \sqrt{x}$: domain $[0, \infty)$, range $[0, \infty)$. Reflection across $x$-axis: $-\sqrt{x}$. Reflection across $y$-axis: $\sqrt{-x}$, domain $(-\infty, 0]$. The two reflections produce different functions.
Key Insight
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.
Definition
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 $\begin{bmatrix}-1 & 0\\0 & 1\end{bmatrix}$. Composing two reflections across intersecting lines produces a rotation by twice the angle between the lines.
Example
In group theory, reflections generate dihedral groups $D_n$. 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
Parity (even/odd symmetry) is a reflection property. Symmetry analysis using reflections reduces computation in physics and engineering: symmetric boundary conditions often halve the domain needed for simulation.