Vertical Shift
Functions & Advanced AlgebraA vertical shift is a transformation that moves the graph of a function up or down along the y-axis.
Formula
g(x) = f(x) + k
Definition
A vertical shift moves the graph of a function straight up or straight down. Adding a number to the function moves it up; subtracting moves it down.
Example
$f(x) = x^2$ has its vertex at $(0, 0)$. $g(x) = x^2 + 4$ shifts the whole parabola $4$ units up. $h(x) = x^2 - 3$ shifts it $3$ units down.
Key Insight
Adding to the output (the whole function) shifts up; subtracting shifts down. This is simpler than horizontal shifts because the direction matches the sign.
Definition
A vertical shift by $k$ units transforms $f(x)$ into $f(x) + k$. If $k > 0$, the graph moves up $k$ units; if $k < 0$, the graph moves down $|k|$ units. The $y$-intercept and all $y$-values change by $k$.
Example
$f(x) = |x|$ passes through $(0,0)$, $(1,1)$, $(-1,1)$. $g(x) = |x| - 5$ passes through $(0,-5)$, $(1,-4)$, $(-1,-4)$. Every $y$-value decreases by $5$.
Key Insight
Vertical shifts affect the range: if $f(x)$ has range $[0, \infty)$, then $f(x) + k$ has range $[k, \infty)$. Asymptotes also shift: if $y = 2$ is a horizontal asymptote of $f$, it becomes $y = 2 + k$ for $f(x) + k$.
Definition
A vertical shift is postcomposition with the translation $T_k: y \to y + k$, acting on the range. In functional analysis, shifting a function by a constant corresponds to adding the constant function to $f$. For periodic functions, vertical shifts do not change periodicity or frequency, only the mean value.
Example
In Fourier analysis, a vertical shift by $k$ adds $k \cdot \delta(w)$ to the spectrum (a spike at frequency $0$), representing a DC offset in engineering contexts.
Key Insight
Vertical and horizontal shifts together produce a general translation. In affine geometry, translations form a subgroup of the affine group, confirming that compositions of shifts are also shifts.