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I came across this reference to the new AVX intruction (or instructions):

Undoubtedly,the newly-added AVX is the most important improvement for Sandy Bridge,as AVX improve vector processing performance to 256bit,and floating performance increases drastically,compared with SSE,the speed get to a new stage.

I've been told that adding a discrete graphics card to a 1155 system will disable the graphics functionality of the Sandbridge CPU. Does this mean one looses the AVX functinality or does this fall under the domain of the CPU functionality?

No, AVX has nothing to do with the built in GPU, it's purely CPU. AVX basically extends SSE/2/3 to 256-bits.

Even with the GPU on-die, the GPU's vector units are not as tightly coupled to the CPU core as the CPU's own floating point units. Shuffling the data to be operated on by the AVX instructions to/from the GPU for processing would add a lot of overhead, and would likely be a disaster from a performance standpoint.