(Yes, I know there are already some answers about phase shifts on here, but none of them have helped me much.)
I am a Pre-Calculus student learning about trigonometric function transformations, and I am extremely confused about the definition of a phase shift (and hence very, very frustrated).
We are learning about functions of the following two general forms:
The definition of phase shift we were given was as follows: "The horizontal shift with respect to some reference wave."
We were then provided with the following graph (and given no other information beyond that it was a transformed sine or cosine function of one of the forms given above):
(Ignore my erased pencil markings in the graph!)
We were asked to find the graphed function's phase shift with respect to the cosine parent function $y=\cos(x)$. We were also asked to find the graphed function's phase shift with respect to the sine parent function $y=\sin(x)$. (Of course, our answers would be eye-balled estimates based on the graph.)
I was utterly mystified. The definition given above told me virtually nothing I wanted to know. And when I asked the teacher for an explanation, it did not end my confusion. Does anyone here have a good explanation of what a phase shift is, and perhaps also how (given solid knowledge of the definition) I could go about solving the graph problem mentioned above?
I have spent a lot of time today searching around online -- in vain -- to try to end my confusion. Nothing has helped much. A lot of people seem to talk about phase shifts in terms of where a function "starts." But a periodic function that repeats on and on forever in either direction does not seem to me to have an intuitive "start" point. Even if one were to select some period to be the "starting period," it is not clear to me how one would choose such a period. Moreover, while I understand that phase shifts have something to do with horizontal translations of the parent functions, it is not yet clear to me what precisely that relationship is -- particularly if the function we are analyzing has also been transformed in other ways (e.g. dilated and vertically translated), making things more complicated.
Thank you for any help you can provide!