  The $\text{v}$-function of a graded filtration
$\mathcal{I}=\{I_{[k]}\}_{k\ge0}$ is introduced. Under the assumption that
$\mathcal{I}$ is Noetherian, we prove that the $\text{v}$-function
$\text{v}(I_{[k]})$ is an eventually quasi-linear function. This result applies
to several situations, including ordinary powers, and integral closures of
ordinary powers, among others. As another application, we investigate the
asymptotic behaviour of certain integer programming problems. Finally, we
present the Macaulay2 package $\texttt{VNumber}$.
