A level maths transformations of graphs question

Hi, I have a maths question.

Suppose I want to transform the function cos(x) into cos(4x-60).
I'm gonna have to translate and stretch it, the order that I do it in makes a difference because the magnitude of the translation will be different.

So I could either translate it first (60,0) then stretch by scale factor 1/4 in x direction OR stretch it first by scale factor 1/4 in x direction but if I do it in this order then the following translation will be (15,0). (I know that these are the correct answers from an AQA a level past paper mark scheme).

I don't understand why the order matters and why the translation becomes (15,0) since i'm aiming for cos(4x-60). Please explain this in depth as I really want to understand this. I've drawn the graphs of cos(4x-60) and cos(4x) online and the only difference was that they were 60 degrees out of phase so it seems like the next step after the stretch would still be a translation (60,0).

So the two transformations we're interested in are

$f(x)\rightarrow f(x+a)$ which is a translation of a units in the horizontal direction.

and

$f(x)\rightarrow f(ax)$ which is a stretch scale factor 1/a in the horizontal direction.


Starting with $f(x)=cos(x)$ and applying a stretch scale factor 1/4 will give


$f(4x) = \cos 4x$


Call this new function g(x) so $g(x) = \cos 4x$


Then applying a translation 15 units to the right will give

$g(x-15) = \cos 4(x-15) = \cos (4x-60)$


I'll let you think about why doing it the other way around requires a translation of 60 instead of 15.