# 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 ?