Horizontal Shift
A horizontal shift is a transformation that moves the graph of a function left or right along the x-axis.
Formula
g(x) = f(x - h)
Definition
A horizontal shift slides the graph of a function left or right without changing its shape. Transforming $f(x)$ into $f(x - h)$ shifts the graph right $h$ units if $h > 0$ and left $|h|$ units if $h < 0$, with all key features (vertex, intercepts, asymptotes) shifting by the same amount. It is a translation in the $x$-direction, a special case of an affine transformation, representing precomposition with $T_h(x) = x - h$; in Fourier analysis, time shifts become phase shifts, if $F(f(t)) = F(w)$, then $F(f(t - h)) = e^{-iwh} \cdot F(w)$.
Example
$f(x) = x^2$ has vertex $(0, 0)$, while $g(x) = (x - 3)^2$ shifts the parabola $3$ units right and $h(x) = (x + 2)^2$ shifts it $2$ units left. $f(x) = \sqrt{x}$ passes through $(0,0)$, $(1,1)$, $(4,2)$, and $g(x) = \sqrt{x - 4}$ passes through $(4,0)$, $(5,1)$, $(8,2)$, every point moved $4$ units right. In signal processing, a time delay of $h$ seconds applied to a signal $f(t)$ produces $f(t - h)$, which has the same frequency content but a phase shift of $-wh$ at each frequency $w$.
Key Insight
The direction of a horizontal shift seems backward, $(x - 3)$ shifts RIGHT not left, because you need a larger $x$ to get the same $y$ as before, so the graph moves right; this also affects the domain, if $f(x)$ has domain $[0, \infty)$, then $f(x - 4)$ has domain $[4, \infty)$. Phase shifts in quantum mechanics and signal processing are horizontal shifts in the time or position domain, showing how the transformation has physical interpretability far beyond elementary graphing.