People tend to confuse these two adjectives. Alternate means going back and forth between two things, such as when things occur by turns (odds and evens).
Example: We have alternate Fridays off as part of our 9/80 work week.
Alternative means another option, such as when a choice is offered.
Example: We will need to find an alternative supplier if AcmeThermo cannot get our thermocombobulator here by August 15.
Back in the Cretaceous when I was a cheerleader in high school, there were seven regular cheerleaders, and I was the “alternate” cheerleader. This was not a correct use of the term, as I did not cheer every other game, but only as another option if one of the regulars happened to be sick or “benched” for some reason. I was really the “alternative” cheerleader. But I did not point that out to anybody, because that would have been too nerdy for words. I thought my being a cheerleader would make the smartest girl in school popular, but NOOOOO. Apparently nobody was brave enough to date Coach’s daughter. Dad had a big wooden paddle hanging on the wall in his office in the boys’ gym that said “Board of Education,” and he was not afraid to use it. I’m glad he never brought it home when he retired!