  Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function
with $F"\delta\subseteq\delta$ and $\GCH$ holds, then there is a
cofinality-preserving forcing extension in which $2^\gamma= F(\gamma)$ for each
regular cardinal $\gamma<\delta$, and in which $\delta$ remains Woodin.
  I also present a new example in which forcing a certain behavior of the
continuum function on the regular cardinals, while preserving a given large
cardinal, requires large cardinal strength beyond that of the original large
cardinal under consideration. Specifically, I prove that the existence of a
$\lambda$-supercompact cardinal $\kappa$ such that $\GCH$ fails at $\lambda$ is
equiconsistent with the existence of a cardinal $\kappa$ that is
$\lambda$-supercompact and $\lambda^{++}$-tall.
  I generalize a theorem on measurable cardinals due to Levinski, which says
that given a measurable cardinal, there is a forcing extension preserving the
measurability of $\kappa$ in which $\kappa$ is the least regular cardinal at
which $\GCH$ holds. Indeed, I show that Levinski's result can be extended to
many other large cardinal contexts. This work paves the way for many additional
results, analogous to the results stated above for Woodin cardinals and
partially supercompact cardinals.
