The Ubuntu forums are currently down but there is a thread over there titled: AMD Radeon 7470M. In that thread someone asked the following question:
Question: Has anybody successfully run linux on a laptop with a AMD
Radeon HD 7470M?
Background: HP Pavilion dv7 with and AMD Radeon HD 7470M graf card
To which someone answered that they were able to use Linux Mint successfully with that hardware.
Also in the Linux Mint forums there was this thread titled: Installing AMD Radeon Graphics Driver. In this thread a user reported that they were able to use the "beta" version of the Catalyst drivers and they worked fine.
I also have a Radeon video card from ATI (specifically, 7470M). I
downloaded the lastest stable driver and the beta one. I ended up
using the beta driver since it works seamlessly compared to the STABLE
one who literally screwed up my display until I removed it and
installed the beta driver.
If you go to the AMD website you can find the "beta" drivers available here, on a page titled: AMD Catalyst™ 13.6 LINUX Beta Driver. The article is dated 5/29/2013 and is article #: RN-LN-B13.6.