MacBook performance while on battery vs plugged in

Yes, sort of. Intel uses a technology called SpeedStep. This technology allows the CPU (and in your case, shared GPU) to adjust its clock speed on the fly, in order to balance performance with power consumption.

When your MacBook Air (MBA) is plugged in (on AC), it is fed the energy it needs, and thus the CPU doesn't have to concern itself with conservation. Conversely, when your MBA is running on battery power, conservation is quite important. In this case, the system will balance performance needs and power consumption in an effort to elongate battery life.

Naturally, your MBA won't be "slower," but the CPU cycles will be allocated in a much more sparingly.

A classic case of software and hardware working together to enhance the user experience (and just another way for Apple to maximize up time).

While I have not personally tried this software, CoolBook claims to have the ability to disable SpeedStep (among other features). Though on their site, they do list that a lack of support for "i3/i5/i7 CPUs", which is unfortunately what most modern Macs are using.