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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 characterizethe 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 forcomputing the Ap\'ery set of $M_n$ in this case, and prove that severalnumerical monoid invariants, such as the genus and Frobenius number, areeventually quasipolynomial as a function of $n$.

Ap\'ery sets of shifted numerical monoids