Vertical Shift
A 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 down: transforming $f(x)$ into $f(x) + k$ moves the graph up $k$ units if $k > 0$ and down $|k|$ units if $k < 0$, changing the $y$-intercept and every $y$-value by $k$. It 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 $f$, and for periodic functions a vertical shift changes only the mean value, not the period or frequency.
Example
$f(x) = x^2$ has vertex $(0, 0)$; $g(x) = x^2 + 4$ shifts it $4$ units up and $h(x) = x^2 - 3$ shifts it $3$ units down. $f(x) = |x|$ passes through $(0,0)$, $(1,1)$, $(-1,1)$, while $g(x) = |x| - 5$ passes through $(0,-5)$, $(1,-4)$, $(-1,-4)$, every $y$-value decreasing by $5$. 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
Adding to the whole function's output shifts the graph up while subtracting shifts it down, which is simpler than horizontal shifts because the direction matches the sign; a vertical shift also affects the range, if $f(x)$ has range $[0, \infty)$, then $f(x) + k$ has range $[k, \infty)$, and any horizontal asymptote $y = c$ becomes $y = c + k$. Vertical and horizontal shifts together produce a general translation, and in affine geometry, translations form a subgroup of the affine group, confirming that compositions of shifts are also shifts.