We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Find the Phase Shift of a Sine or Cosine Function

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Find the phase shift of F(t)=3sin(2t+4) .

Possible Answers:

Correct answer:

Explanation:

In the formula,

F(t)=Asin(Bt-C)+D .

$\frac{C}{B}$ represents the phase shift.

Plugging in what we know gives us:

$\frac{C}{B}=\frac{-4}{2}$ .

Simplified, the phase is then .

Report an Error

Example Question #2 : Find The Phase Shift Of A Sine Or Cosine Function

Please choose the best answer from the following choices.

Describe the phase shift of the following function:

$y=sin(x+\pi )$

Possible Answers:

Vertical stretch by $\pi$ radians.

Shift left by $\pi$ radians.

Shift down by $\pi$ radians.

Shift up by $\pi$ radians.

Correct answer:

Shift left by $\pi$ radians.

Explanation:

Since $\pi$ is being added inside the parentheses, there will be a horizontal shift. The goal is to maintain zero within the parentheses so you will shift left $\pi$ radians.

Report an Error

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Which equation would produce this graph?

Phase shift 1

Possible Answers:

$y=sin\left(x+\frac{\pi}{2}\right)$

$y=sin(x-\pi)$

$y=sin\left(x-\frac{\pi}{3}\right)$

$y=sin\left(\frac{\pi}{2}x\right)$

$y=sin\left(x-\frac{\pi}{2}\right)$

Correct answer:

$y=sin\left(x-\frac{\pi}{2}\right)$

Explanation:

This is the graph of sine, but shifted to the right $\frac{\pi}{2}$ units. To reflect this shift, $\frac{\pi}{2}$ should be subtracted from x.

Thus resulting in

$y=sin\left(x-\frac{\pi}{2}\right)$ .

Report an Error

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Which equation would produce this sine graph?

Phase shift 2

Possible Answers:

$y=2sin\left(x-\frac{\pi}{2}\right)-1$

$y=sin\left(2x-\frac{\pi}{4}\right)$

y=sin(2x)-3

$y=sin\left(x-\frac{3\pi}{4}\right)-1$

$y=2sin\left(x-\frac{\pi}{4}\right)-1$

Correct answer:

$y=2sin\left(x-\frac{\pi}{4}\right)-1$

Explanation:

The graph has an amplitude of 2 but has been shifted down 1:

Phase shift 2 dots

In terms of the equation, this puts a 2 in front of sin, and -1 at the end.

This makes it easier to see that the graph starts [is at 0] where $x=\frac{\pi}{4}$ .

The phase shift is $\frac{\pi}{4}$ to the right, or $x-\frac{\pi}{4}$ .

Report an Error

Example Question #5 : Find The Phase Shift Of A Sine Or Cosine Function

Write the equation for a sine graph with a maximum at $\left(\frac{ \pi }{8}, 3\right)$ and a minimum at $\left(\frac{ 9 \pi }{8 } , -1 \right)$ .

Possible Answers:

$y = 2 \sin (x + \frac{ 3 \pi }{ 8 } ) + 1$

$y = 2 \sin ( \frac{1}{2}( x - \frac{ \pi }{8})) + 1$

$y = 3 \sin (\frac{ 1}{2} x )$

$y = 2 \sin (x - \frac{ \pi }{ 8 }) + 1$

$y = 2 \sin (x - \frac{ 3 \pi }{8 } ) - 1$

Correct answer:

$y = 2 \sin (x + \frac{ 3 \pi }{ 8 } ) + 1$

Explanation:

To write this equation, it is helpful to sketch a graph:

Trig graph 1

Indicating the maximum and minimum points, we can see that this graph has been shifted up 1, and it has an amplitude of 2.

The distance from the maximum to the minimum point is half the wavelength. In this case, the wavelength is $\frac{ 9 \pi }{ 8 } - \frac{ \pi }{8 } = \frac{ 8 \pi }{8 } = \pi$ . That means the full wavelength is $2 \pi$ , and the frequency is 1.

This sketch shows that the graph starts to the left of the y-axis. To figure out exactly where, subtract $\frac{ \pi }{2}$ from the maximum x-coordinate, $\frac{ \pi }{ 8 }$ :

$\frac{ \pi }{ 8 } - \frac{ \pi }{2} = \frac{ \pi }{8 } - \frac{ 4 \pi }{ 8 } = \frac{ - 3 \pi }{ 8 }$ .

Our equation will be in the form $y = A \cdot \sin ( f (x - h )) + k$ where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

This graph has an equation of

$y = 2 \sin ( x + \frac{ 3 \pi }{ 8 } ) + 1$ .

Report an Error

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Write the equation for a cosine graph with a maximum at $\left( \frac{ \pi }{3} , 3 \right)$ and a minimum at $\left(\frac{ 4 \pi }{3} , -3 \right)$ .

Possible Answers:

$y = 2 \cos (\frac{ 1}{2} ( x + \frac{ \pi }{3} )) + 1$

$y = 2 \cos (x + \frac{ \pi }{3} )+ 1$

$y = 3 \cos (x - \frac{ \pi }{3} )$

$y = 3 \cos (\frac{ 1}{2} ( x - \frac{ \pi }{3} ))$

$y = -3 \cos (x - \frac{ \pi } {3 })$

Correct answer:

$y = 3 \cos (x - \frac{ \pi }{3} )$

Explanation:

In order to write this equation, it is helpful to sketch a graph:

Trig graph 2

The dotted line is at $x = \frac{ \pi }{3}$ , where the maximum occurs and therefore where the graph starts. This means that the graph is shifted to the right $\frac{ \pi }{3}$ .

The distance from the maximum to the minimum is half the entire wavelength. Here it is $\frac{ 4 \pi }{3} - \frac{ \pi }{3} = \frac{ 3 \pi }{3} = \pi$ .

Since half the wavelength is $\pi$ , that means the full wavelength is $2 \pi$ so the frequency is just 1.

The amplitude is 3 because the graph goes symmetrically from -3 to 3.

The equation will be in the form $y = A \cdot \cos (f (x - h)) + k$ where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

This equation is

$y = 3 \cos (x - \frac{ \pi } {3 })$ .

Report an Error

Example Question #7 : Find The Phase Shift Of A Sine Or Cosine Function

Write the equation for a sine function with a maximum at $\left(\frac{ 7 \pi }{10} , 7 \right)$ and a minimum at $\left(\frac{ 17 \pi }{10 } , 1\right)$ .

Possible Answers:

$y = 7 \sin (x + \frac{ \pi }{5} ) - 3$

$y = 3 \sin (\frac{ 1}{2} (x - \frac{ 7 \pi }{10 })) + 4$

$y = \sin (x - \frac{ \pi}{10}) + 7$

$y = 3 \sin (x - \frac{ \pi }{5 }) + 4$

$y = 7 \sin (x - \frac{\pi }{10}) + 1$

Correct answer:

$y = 3 \sin (x - \frac{ \pi }{5 }) + 4$

Explanation:

The equation will be in the form $y = A \sin (f (x - h)) + k$ where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

To write the equation, it is helpful to sketch a graph:

Trig graph 3

From plotting the maximum and minimum, we can see that the graph is centered on y = 4 with an amplitude of 3.

The distance from the maximum to the minimum is half the wavelength. For this graph, this distance is $\frac{ 17 \pi }{10 } - \frac{ 7 \pi }{10 } = \frac{ 10 \pi }{10 } = \pi$ .

This means that the total wavelength is $2 \pi$ and the frequency is 1.

The graph starts $\frac{ \pi }{2}$ behind the maximum point. To determine this x value, subtract $\frac{ \pi }{2 }$ from the x-coordinate of the maximum:

$\frac{ 7 \pi }{10 } - \frac{ \pi }{2} = \frac{ 7 \pi }{10 } - \frac{ 5 \pi }{10 } = \frac{ 2 \pi }{10 } = \frac{ \pi }{5}$

Our equation is:

$y = 3 \sin (x - \frac{ \pi }{5} ) + 4$ .

Report an Error

View Pre-Calculus Tutors

Evgenii
Certified Tutor

View Pre-Calculus Tutors

Shubham
Certified Tutor

Jaypee University of Information and Technology, Bachelor of Technology, Computer Science.

View Pre-Calculus Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Denver, ACT Tutors in San Francisco-Bay Area, Calculus Tutors in Los Angeles, Math Tutors in Miami, SSAT Tutors in Chicago, GMAT Tutors in San Francisco-Bay Area, Calculus Tutors in Miami, SAT Tutors in Atlanta, English Tutors in Dallas Fort Worth, ISEE Tutors in New York City

Popular Courses & Classes

SAT Courses & Classes in Seattle, GMAT Courses & Classes in Phoenix, GRE Courses & Classes in New York City, ACT Courses & Classes in Phoenix, GMAT Courses & Classes in Philadelphia, GMAT Courses & Classes in San Diego, MCAT Courses & Classes in Los Angeles, ISEE Courses & Classes in Seattle, GRE Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

ACT Test Prep in Denver, LSAT Test Prep in Denver, GMAT Test Prep in Phoenix, SAT Test Prep in Los Angeles, GMAT Test Prep in Dallas Fort Worth, LSAT Test Prep in Dallas Fort Worth, ACT Test Prep in Dallas Fort Worth, SAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Philadelphia, GMAT Test Prep in San Diego

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Find the Phase Shift of a Sine or Cosine Function

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Example Question #2 : Find The Phase Shift Of A Sine Or Cosine Function

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Example Question #5 : Find The Phase Shift Of A Sine Or Cosine Function

Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Example Question #7 : Find The Phase Shift Of A Sine Or Cosine Function

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: