Trigonometry : Phase Shifts

Example Questions

Example Question #1 : Phase Shifts

Identify the phase shift of the following equation.       Explanation:

If we use the standard form of a sine function the phase shift can be calculated by .  Therefore, in our case, our phase shift is  Example Question #2 : Phase Shifts

Which of the following is equivalent to       Explanation:

The first zero can be found by plugging 3π/2 for x, and noting that it is a double period function, the zeros are every π/2, count three back and there is a zero at zero, going down.

A more succinct form for this answer is but that was not one of the options, so a shifted cosine must be the answer.

The first positive peak is at π/4 at -1, so the cosine function will be shifted π/4 to the right and multiplied by -1. The period and amplitude still 2, so the answer becomes .

To check, plug in π/4 for x and it will come out to -2.

Example Question #3 : Phase Shifts

Which of the following is the correct definition of a phase shift?

The distance a function is shifted horizontally from the general position

A measure of the length of a function between vertical asymptotes

The distance a function is shifted vertically from the general position

The distance a function is shifted diagonally from the general position

The distance a function is shifted horizontally from the general position

Explanation:

Take the function for example.  The graph for is If we were to change the function to , our phase shift is .  This means we need to shift our entire graph units to the left. Our new graph is the following Example Question #4 : Phase Shifts

Consider the function .  What is the phase shift of this function?     Explanation:

The general form for the secant transformation equation is  represents the phase shift of the function.  When considering we see that .  So our phase shift is and we would shift this function units to the left of the original secant function’s graph.

Example Question #5 : Phase Shifts

True or False: If the function has a phase shift of , then the graph will not be changed.

True

False

True

Explanation:

This is true because the graph has a period of , meaning it repeats itself every units.  So if has a phase shift of any multiple of , then it will just overlay the original graph.  This is shown below.  In orange is the graph of and in purple is the graph of . Example Question #6 : Phase Shifts

Which of the following is the graph of with a phase shift of ?     Explanation:

Start this problem by graphing the function of tangent. Now we need to shift this graph to the right.  Example Question #7 : Phase Shifts

True or False: The function has a phase shift of .

False

True

False

Explanation:

The form of the general cosecant function is .  So if we have then , which represents the phase shift, is equal to .  This gives us a phase shift of .

Example Question #8 : Phase Shifts

Which of the following is the phase shift of the function ?     Explanation:

The general form of the cotangent function is .  So first we need to get into the form .  From this we see that giving us our answer. 