Composite transformations intuition

Question

I'm struggling to understand composite transformations. If we let $f(x) = \sin x$, and then draw the graph $f(2x)$. Why is it that when we translate $f(2x)$ by $\pi/4$ units to the right, one of the $x$ intercepts to the graph becomes $- \pi / 4$. I understand why this happens algebraically, but can someone please help me understand the graphical intuition behind this?