to exterminate; to eradicate; to wipe out; to put an end to
to let (fire) go out; to let die (e.g. flowers); to run out of
その地域の伝統工芸を絶やした責任は重い。
The responsibility for wiping out that region's traditional crafts is grave.
希望の火を絶やして諦めた。
He let the flame of hope die out and gave up.
笑顔を絶やさないように心がけている。
I make a point of never letting my smile disappear.