A few of the existing solutions were close, but not quite exactly what I wanted. This function works with both positive or negative values and handles boundary cases.
function addDays(date, days) {
return new Date(
date.getFullYear(),
date.getMonth(),
date.getDate() + days,
date.getHours(),
date.getMinutes(),
date.getSeconds(),
date.getMilliseconds()
);
}