4k Monitor, 2014 Macbook air, and an all around DILEMMA I would like to get a 27 inch 4k display to hook up to my early 2014 macbook air. My Early 2014 Macbook air only has a thunderbolt 1 connection and I an aware it is not possible to get 4k quality through my thunderbolt 1 connector. 

Could I still use the display, using a thunderbolt 1 to thunderbolt 3 adaptor, and get a decent quality, perhaps 2k, on it without anything looking distorted?
Thunderbolt 1 is not "forward compatible" with Thunderbolt 3;  Thunderbolt 3 is backward compatible with Thunderbolt 1.  So, you cannot go TB1 >> TB3

That said, TB (1, 2, or 3) is a "multiplexing" of different signals - PCIe, DisplayPort, power, and USB(TB3).  What you need to do is connect the "video" signal from the Thunderbolt Port to your monitor.

So, what you need is a mini DisplayPort to DisplayPort cable to connect to your 27" monitor.  Your monitor should be able to adjust to the lower resolution while maintaining quality.