  A numerical monoid is an additive submonoid of the non-negative integers.
Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$
obtained by adding $n$ to each generator of $S$. In this paper, we characterize
the Ap\'ery set of $M_n$ in terms of the Ap\'ery set of the base monoid $S$
when $n$ is sufficiently large. We give a highly efficient algorithm for
computing the Ap\'ery set of $M_n$ in this case, and prove that several
numerical monoid invariants, such as the genus and Frobenius number, are
eventually quasipolynomial as a function of $n$.
