Answer
$\frac{1}{4}$
Work Step by Step
We know that if $A$ is a real number and $B\gt0$, then the graphs of $y=Asin(Bx)$ and $y=Acos(Bx)$ will have an amplitude of $|A|$.
Therefore, the amplitude of the graph of $y=-\frac{1}{4}cos x$ is $|-\frac{1}{4}|=\frac{1}{4}$.