I know that they price for airline tickets go up closer to the travel date, but when does that happen? I am flying Seattle to LAX in June, round trip, I have applied for a scholarship to help pay for the conference I'm attending but I won't get an answer until April. Is two months before still the lowest price?

There is no exact time that prices go up or down. It all depends on how many people are buying seats on a particular flight. If not many are buying, the price will go down. If it's popular, the price will go up. Generally if you are flexible, and willing to travel on Tuesday or Thursday, and late at night, you can get cheaper tickets.