To Kill Off
To Let Die Out
transitive verb, godan verb
絶える is "to die out." 絶やす is "to kill off." You know this because if something is to die off, you'd say "no." But if you're the one doing the killing off, you'd say yes (やす).
The reading comes from 絶える.
夜の間、火を絶やさないように気をつけるんだぞ。
Make sure the fire doesn’t go out during the night.