When given a function of the form : y = acos[ b (pi - c)] + d, answer the following question:
1) When given a trigonometric word problem where a and b can be determined, how is the phase shift (c) and vertical shift (d) calculated?
(1) The easiest way to compute the vertical shift is to take the average of the maximum and minimum. (If K is the max and L the min, we can find a by `(k-l)/2` and we can find d by `(k+l)/2` ). If you are given a graph you can also identify the maximum and minimum.
(2) To compute the phase shift you need to know a point. It is easiest if you know a maximum or a minimum. Since the cosine starts at a maximum (or a minimum if it is inverted) you know how far from the origin the function is starting.
** Note that there are an infinite number of possible values for c: once you find one, any number that is a multiple of the period from that point will work. For instance, if c=3 and the period is 5, then you can use -2,3,8,13,... as phase shifts. Usually you choose the smallest (in absolute value.)