So wolframalpha gives me this: wolfram's result for inverse fourier transform
but I have this relation in mind: fourier transform rule I'm using
and combining this with the shift theorem I would expect that $$f(x+a)\mapsto F(\omega)*e^{ia\omega}$$ not $$e^{-ia\omega}$$ which is what I think I'm getting here. I'm sure I'm wrong and I think I intuitively understand why, because of the negative sign in the function of $x$, in the exponent, but I can't analytically prove why this is the case. So far as I know, a function of the form $x^2$ would become a function of the form $(x+a)^2$, even if there's a minus sign out the front.