Buy used... you should be able to get an old 6870 or the like for ~$50ish and it'll perform better than the low end stuff. You might even be able to game at 1440p if you play at lowest settings.
I guess 1440p is 2K .. I have tested my R9 280 (hd7950) to 4K but it only has HDMI 1.4 which can only do 30Hz @ 4K and I need HDMI 2.0 to push 60Hz but I don't know if that is the same issues with 2K or not.