what is the purpose of "as" here?

I don't have the documentation to support my answer, but if it helps I believe there is an implied pronoun:

Use the Pythagorean Theorem to find the missing side, being careful to mark the value as positive or negative as it makes sense in that quadrant.

It's similar to saying something like, "please stand as able" for a ceremony, implying "please stand as you are able," which contains not only an implied pronoun but an implied verb.

It's being used as a conjunction, and it seems most similar to this definition

Used to indicate by comparison the way that something happens or is done:
they can do as they wish
she kissed him goodbye, as usual

In your quote, it actually means something more like whichever.

It is being used as a conjunction in this case to indicate the reason or time for doing something. In this case, it is a subordinating conjunction with the clause:

...makes sense in that quadrant.

In other words, your values may have different signs depending on the quadrant.