Horizontal Shift
Functions & Advanced AlgebraA 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 to the left or right. The shape of the graph does not change, only its position.
Example
$f(x) = x^2$ has its vertex at $(0, 0)$. $g(x) = (x - 3)^2$ shifts the whole parabola $3$ units to the right. $h(x) = (x + 2)^2$ shifts it $2$ units to the left.
Key Insight
The direction of a horizontal shift seems backward: $(x - 3)$ shifts RIGHT, not left. Think of it this way: you need a larger $x$ to get the same $y$ as before, so the graph moves right.
Definition
A horizontal shift by $h$ units transforms $f(x)$ into $f(x - h)$. If $h > 0$, the graph shifts right $h$ units. If $h < 0$, the graph shifts left $|h|$ units. All key features (vertex, intercepts, asymptotes) shift by the same amount.
Example
$f(x) = \sqrt{x}$ passes through $(0,0)$, $(1,1)$, $(4,2)$. $g(x) = \sqrt{x - 4}$ passes through $(4,0)$, $(5,1)$, $(8,2)$. Every point moves $4$ units to the right.
Key Insight
Horizontal shifts affect the domain: if $f(x)$ has domain $[0, \infty)$, then $f(x - 4)$ has domain $[4, \infty)$. The inside change affects $x$-values.
Definition
A horizontal shift is a translation in the $x$-direction, a special case of an affine transformation. In the context of $f(x - h)$, it represents precomposition with the translation $T_h(x) = x - h$. For Fourier transforms, time shifts become phase shifts: if $F(f(t)) = F(w)$, then $F(f(t - h)) = e^{-iwh} \cdot F(w)$.
Example
In signal processing, a time delay of $h$ seconds 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
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.