Sections:

Transformations of Trigonometric Functions, Page 13

Phase Shift (continued)

The following problem shows how the definition of phase shift works:

Graph f(x) = cos(x - pi). What is the phase shift of this function and in which direction will you move the graph?

In this problem, c = Pi and k = 1. The phase shift is c over k equals pi over 1 equals pi.
Since c > 0, the graph will shift to the right of the basic function.

graphs of f(x) = cos(x) and f(x) = cos(x-pi)

Look at another example of how a phase shift will change the graph of a trigonometric function.

Graph f(x) = sin(x+ pi/2). What is the phase shift of this function and in which direction will you move the graph?

Sometimes it is easy to confuse the sign for c when there is an addition sign within the parenthesis. The best way to avoid making a mistake is to rewrite the equation to reflect the general equation given in your definition. Therefore, rewrite f(x) = sin(x+ pi/2) as f(x) = sin(x - (-pi/2)). Now it is easier to see that c = -pi/2.
Since k = 1 and c = -pi/2, the phase shift, c over k, is -pi/2. This graph will shift to the left of the basic function since c < 0. 

graph of f(x) = sin(x+ pi/2) and f(x) = sin(x)