Trigonometry : Phase Shifts

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Phase Shifts

Identify the phase shift of the following equation.

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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?

Possible Answers:

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

Correct answer:

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?

Possible Answers:

Correct answer:

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.

Possible Answers:

True 

False

Correct answer:

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 ?

Possible Answers:

Screen shot 2020 08 27 at 2.36.53 pm

Screen shot 2020 08 27 at 2.36.46 pm

Screen shot 2020 08 27 at 2.35.20 pm

Screen shot 2020 08 27 at 2.35.10 pm

Correct answer:

Screen shot 2020 08 27 at 2.35.20 pm

Explanation:

Start this problem by graphing the function of tangent.

Screen shot 2020 08 27 at 2.35.10 pm

Now we need to shift this graph  to the right.

Screen shot 2020 08 27 at 2.35.16 pm

This gives us our answer

 Screen shot 2020 08 27 at 2.35.20 pm

Example Question #7 : Phase Shifts

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

Possible Answers:

False

True 

Correct answer:

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 ?

Possible Answers:

Correct answer:

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.

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