Step 1: Think in terms of standardized values.
Correlation is really computed on the standardized (z-score) versions of the two variables, $z_x=\dfrac{x-\bar x}{\sigma_x}$ and similarly for $y$.
Step 2: See what shifting or scaling does to a z-score.
Adding a constant to $x$ (change of origin) shifts both $x$ and $\bar x$ equally, so it cancels out in $x-\bar x$. Multiplying $x$ by a positive constant (change of scale) scales both the numerator and $\sigma_x$ by the same factor, so it also cancels.
Step 3: Conclude for the correlation coefficient.
Since the z-scores don't change under either operation, the correlation coefficient built from them doesn't change either.
\[ \boxed{\text{Both change of scale \& origin}} \]