When you pour baking soda on a kitchen fire and the fire goes out, this is an example of when you extinguish the fire.
When a plague kills an entire population, this is an example of how to extinguish a population.
When you find out you didn't get into the school you wanted, this is an example of a time when hope is extinguished.
When a debt is forgiven in full, this is an example of a time when the debt is extinguished.
Origin of extinguish
From American Heritage Dictionary of the English Language, 5th Edition