You first must determine if the battery is failing or if there are apps that are draining the battery. If you take it to an Apple store the Genius Bar can test the battery. If it needs to be replaced Apple will replace it for $79. Make an appointment first at https://www.apple.com/retail and click on Genius Bar.
It's also worth considering the age of the phone. The battery should last 2-3 years at least.
Well, it is an older phone so there is that issue too, no way to know...
Who is your provider?
You can talk to Apple about it, but the battery replacement really shouldn't cost that much, but then again we're talking about apple.
Weigh the costs of getting a refurb iPhone 4S, it's a great phone and I'm very happy with it.