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Of course this is just an example of the intuition that I am having problems with. I understand that the period will be $2\pi/4$. This part is fine, but why must the phase shift $\pi/3$ also be divided by $4$. The hierarchy of operations is multiplication before addition. Therefore the "squeezing" of the period has already occurred, the period being $2\pi/4$. Then according to hierarchy of operations there is a phase shift of $\pi/3$, which means a shift to the left in this case, the opposite of the sign of $\pi/3$. What rule demands that we must first divide the phase shift by $4$, the same as we do to $2\pi$? This is very puzzling. If the expression read $f(x)=\sin\left(4\left(x+\frac{\pi/3}{4}\right)\right)$ then I would understand because by the hierarchy of operations. The expressions do not seem to me to have the same results on the sine function even though they are algebraically the same. My math book says to graph the function we must divide the phase shift by $4$ first after "squeezing" the period but I fail to see why.

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    Try to play with this toy example: $f(x)=\sqrt{2x+3}$, that should give you some ideas. Loosely speaking, you cannot "mix" rigid and non-rigid transformations together, you have to deal with them separately. – 2017-02-10
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    You're confusing the order of operations in an expression with graph transformations -- they are not exactly the same thing. Note that the same function can be written with different expressions. For example, aren't $f(x)=2x+6$ and $f(x)=2(x+3)$ the same thing, even though computing them would follow different operations in different order? – 2017-02-10
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    @zipirovich ...You are exactly understanding the situation ..but that is why I am asking the question. ! I agree with you but mathematics does not! The textbooks say otherwise. They divide the phase by 4...I do not understand why. – 2017-02-10
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    I think I got it. the phase shift is the one associated with original x. my mistake. by pulling the x it makes it clear. Since multiplication takes priority it confuses the matter when the phase shift appears in the parenthesis but normally the input is associated with the normal 2pi period so you have to scale it. They are functionally the same once you know it. It should be disallowed not to have a parenthesis when a phase shift occurs that includes only x with scalar out front! so you have to take it out from the expression of the phase shift. – 2017-02-10

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