A 5770 consumes about 9-10 Amps from a 12 rail, a dual core CPU eats up another 5 AMps. Another 2-3 Amps for fans and such, so as long as your PSU has at least 18A you will be ok.
But I recoomand only pushing PSU to %50-60 as this keeps the heat down and increase longevity and efficiency. IN that case you want soething around 35-36A on the 12V rail, so a good 420-430W PSU will do just fine.
That sheds some light on the issue. COmpanies like Ibuypower and cyberpower will shamelessly include garbage non-name PSU with thier PC as standard to cut costs. These power supplies are as cheap as they come, I pretty sure that the 480W rating is fake as well (most of them are 300W CPU that have been re-branded).
So not only do I think you might run into problems with that PSU, it might even be a danger to the rest of your computer (and the PC company also has lackluster warranty support).
Honestly I would throw out the standard PSU and put in something good in there, like 550+ watt unit from Corsair or Antec.