Broken Power Button(Need help ASAP)

I kind of have a few issues going on right now. I have a white iPhone 5 and recently I realized I have to press my power button multiple times for it to work. Then I did a little reading and found how this problem is happening to alot of people. It's all the same, it usually works better when pressing on the left side of it. My button stopped working while I had a purple and green speck case on it. Then I told my parents about it and they said that's its pointless to replace my phone, even though we have apple care and stuff. My family is convinced that the problem is from me dropping it all of the time, but I have only dropped my phone a few times and it wasn't bad at all. I think it's the case that caused the problem, and I need my family to think that bc it's a real inconvenience having my power button only work half the time. So can someone help me prove that it's not my fault this happened then maybe they would let me get my phone replaced? Also would it be free to replace it? PS I got it last Christmas

 

Would they replace it for free?