Apple will give you a new one because it seems like a hardware issue. Since your covered you wont have to pay.
Thanks for asking
- Reset the iOS device. Nothing will be lost
Reset iOS device: Hold down the On/Off button and the Home button at the same time for at
least ten seconds, until the Apple logo appears. Already done
- Reset all settings
Go to Settings > General > Reset and tap Reset All Settings.
All your preferences and settings are reset. Information (such as contacts and calendars) and media (such as songs and videos) aren’t affected.
- Restore from backup. See:
- Restore to factory settings/new iOS device.
If still problem, make an appointment at the Genius Bar of an Apple store since it appears you have a hardware problem. Other uses have reported the problem both before and during updating to iOS 7
make sure to run a back up and perform a restore.
Back up: settings>icloud>storage and black up>back up
after you back up you can preform a restore.
this will delete all content and not the content you have selected on icould.
You can view the stuff thats on icloud by going to settings>icloud and it shows which apps are on.
You can restore by going to settings>general>reset>erase all content and settings.