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.

\(\displaystyle y=-3sin(\frac{x}{2}+\frac{\pi}{4})+1\)

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle \frac{\pi}{2}\)

\(\displaystyle \frac{1}{2}\)

\(\displaystyle -3\)

\(\displaystyle \frac{\pi}{4}\)

Correct answer:

\(\displaystyle \frac{\pi}{2}\)

Explanation:

If we use the standard form of a sine function

\(\displaystyle y=Asin(Bx-C)+D\)

the phase shift can be calculated by \(\displaystyle \frac{C}{B}\).  Therefore, in our case, our phase shift is

\(\displaystyle \frac\frac{}{}{}{}\)\(\displaystyle \frac{\frac{\pi}{4}}{\frac{1}{2}}=\frac{\pi}{4}\cdot2=\frac{\pi}{2}\)

Example Question #1 : Phase Shifts

Which of the following is equivalent to \(\displaystyle 2\sin(2x-3\pi)\)

Possible Answers:

\(\displaystyle -2\cos(\frac{x}{2}-\frac{\pi}{2})\)

\(\displaystyle -2\cos(2x-\frac{\pi}{4})\)

\(\displaystyle 2\sin(2x)\)

\(\displaystyle 2\cos(x)\)

\(\displaystyle -2\cos(2(x-\frac{\pi}{4}))\)

Correct answer:

\(\displaystyle -2\cos(2(x-\frac{\pi}{4}))\)

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 \(\displaystyle -2\sin(2x)\) 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 \(\displaystyle -2\cos(2(x-\frac{\pi}{4}))\).

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

Example Question #132 : Trigonometry

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

Possible Answers:

The distance a function is shifted vertically from the general position

The distance a function is shifted horizontally from the general position

The distance a function is shifted diagonally from the general position

A measure of the length of a function between vertical asymptotes

Correct answer:

The distance a function is shifted horizontally from the general position

Explanation:

Take the function \(\displaystyle cos(x)\) for example.  The graph for \(\displaystyle cos(x)\)is

 

 

If we were to change the function to \(\displaystyle cos(x + \pi)\), our phase shift is \(\displaystyle -\pi\).  This means we need to shift our entire graph \(\displaystyle \pi\) units to the left.

 

 

Our new graph \(\displaystyle cos(x + \pi)\) is the following

 

 

 

Example Question #12 : Trigonometric Graphs

Consider the function \(\displaystyle 4sec(5(x+\frac{\pi}{3})) +10\).  What is the phase shift of this function?

Possible Answers:

\(\displaystyle 4\)

\(\displaystyle -\frac{\pi}{3}\)

\(\displaystyle 10\)

\(\displaystyle \frac{\pi}{3}\)

Correct answer:

\(\displaystyle -\frac{\pi}{3}\)

Explanation:

The general form for the secant transformation equation is \(\displaystyle D + Asec(B(x - C))\)\(\displaystyle C\) represents the phase shift of the function.  When considering \(\displaystyle 4sec(5(x+\frac{\pi}{3})) +10\) we see that \(\displaystyle C = -\frac{\pi}{3}\).  So our phase shift is \(\displaystyle -\frac{\pi}{3}\) and we would shift this function \(\displaystyle \frac{\pi}{3}\) units to the left of the original secant function’s graph.

 

 

Example Question #13 : Trigonometric Graphs

True or False: If the function \(\displaystyle sin(x)\) has a phase shift of \(\displaystyle n2\pi\), then the graph will not be changed.

Possible Answers:

False

True 

Correct answer:

True 

Explanation:

This is true because the graph \(\displaystyle sin(x)\) has a period of \(\displaystyle 2\pi\), meaning it repeats itself every \(\displaystyle 2\pi\) units.  So if \(\displaystyle sin(x)\) 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 \(\displaystyle sin(x)\)and in purple is the graph of  \(\displaystyle sin(x + 4\pi)\).

 

 

 

Example Question #2 : Phase Shifts

Which of the following is the graph of \(\displaystyle tan(x)\)  with a phase shift of \(\displaystyle \frac{\pi}{2}\)?

Possible Answers:

Screen shot 2020 08 27 at 2.35.10 pm

Screen shot 2020 08 27 at 2.35.20 pm

Screen shot 2020 08 27 at 2.36.53 pm

Screen shot 2020 08 27 at 2.36.46 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 \(\displaystyle \frac{\pi}{2}\) 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 #1 : Phase Shifts

True or False: The function \(\displaystyle csc(x + \pi)\) has a phase shift of  \(\displaystyle \pi\).

Possible Answers:

True 

False

Correct answer:

False

Explanation:

The form of the general cosecant function is \(\displaystyle D + Acsc(B(x - C))\).  So if we have \(\displaystyle csc(x + \pi)\) then \(\displaystyle C\), which represents the phase shift, is equal to \(\displaystyle -\pi\).  This gives us a phase shift of \(\displaystyle -\pi\).

Example Question #61 : Trigonometric Functions And Graphs

Which of the following is the phase shift of the function \(\displaystyle 5 - cot(6x - \pi)\)?

Possible Answers:

\(\displaystyle -\pi\)

\(\displaystyle \pi\)

\(\displaystyle \frac{\pi}{6}\)

\(\displaystyle -\frac{\pi}{6}\)

Correct answer:

\(\displaystyle \frac{\pi}{6}\)

Explanation:

The general form of the cotangent function is \(\displaystyle D + cot(B(x - C))\).  So first we need to get \(\displaystyle cot(6x - \pi)\)  into the form \(\displaystyle cot(B(x - c))\).

 

\(\displaystyle cot(6x - \pi)\)

\(\displaystyle cot(6(x - \frac{\pi}{6}))\)


From this we see that \(\displaystyle C = \frac{\pi}{6}\) giving us our answer.

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