From this answer:
\[\begin{align*} \|Ax\|_2 \leq \|A\|_F \|x\|_2 \end{align*}\]From this question can do better with:
\[\begin{align*} \|Ax\|_2 \leq \|A\|_2 \|x\|_2 \end{align*}\]where $|A|_2$ is the spectral norm of $A$.
Reason we can do this is that operator norms are submultiplicative, with respect to the ***
According to this answer operator norms (induced norms) are submultiplicative. This means that for any two matrices $A$ and $B$,
\[\begin{align*} \|AB\| \leq \|A\| \|B\| \end{align*}\]Some great notes on matrix norms and submultiplicativity.