Does a function $f$ exists such that: $f(n-k) \ne \Theta(f(n))$ for some constant $k\geq1$?
Counterexample: $f(n)=n!$ As $f(n)$ is $n$ times bigger  than $f(n-1)$, it is clear that $f(n-1) \neq \Theta(f(n))$.