  Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with
coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number
of the powers of $I$ is investigated. Natural lower and upper bounds which are
linear functions in $k$ are determined for $\text{v}(I^k)$. We call
$\text{v}(I^k)$ the $\text{v}$-function of $I$. We prove that $\text{v}(I^k)$
is a linear function in $k$ for $k$ large enough, of the form
$\text{v}(I^k)=\alpha(I)k+b$, where $\alpha(I)$ is the initial degree of $I$,
and $b\in\mathbb{Z}$ is a suitable integer. For this aim, we construct new
blowup algebras associated to graded ideals. Finally, for a monomial ideal in
two variables, we compute explicitly its $\text{v}$-function.
