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In certain contexts (such as linear algebra), linear refers to the fact that the mapping must cross the origin.

This follows from the constraint that for a linear map f, f(a + b) = f(a) + f(b), which is not true for the farenheit-celsius example.

Also: a linear function plus a constant is said to be affine!

I just learned something. Is it correct to call both of the first order? Or is that term really only to be used for the underlying polynomials?

I think of nth order as mostly just applying to approximations. But I guess first order would be ok here.

If f is linear, then f(a)=f(a+0)=f(a)+f(0). So f(0)=0.