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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?

Ricky
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Laksh Sharma
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    Better for me to respond to the (underlying) broader question. Suppose that $~f:\Bbb{R}\to\Bbb{R},~$ and you take the graph of $~f(x)~$ and translate it $~k~$ units to the right $~: ~k \in \Bbb{R^+}.~$ What are you then graphing ? Answer : Assuming that by translate you mean that the entire graph is being shifted $~k~$ units to the right, you are (in effect) now graphing the function $~g(x) = f(x-k).$ That is, the height of the graph $~g(x)~$ at $~[x = (x_0 + k)]~$ equals the height of the graph $~f(x)~$ at $~[x = (x_0)].$ – user2661923 Sep 14 '23 at 11:55

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