iPhone 5 Battery life issue

Over the past 2 weeks, i've noticed that the battery life on my iPhone 5 has been dwindling. I have tried just running the battery completely dead and fully chargiong it, just turning it off during the night. I have looked online for different solutions and i have all background stuff turnied off (no app upgrades, iCloud, screen brightness turned down, i even turn off LTE, and other things). The phone is just over a year old, and the battery never just ran down so quickly on my iPhone 4 when i had it. Do i need a new battery? Should i just take it to Apple and see what i should do? or Verizon because i have phone insurance with them?