/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Determine the amplitude, period,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 23

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=\frac{1}{2} \sin \left(x+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The amplitude is \(\frac{1}{2}\), the period is 1, and the function is phase shifted to the right \(\frac{\pi}{2}\) units.

Step by step solution

01

Calculation of Amplitude

Identify the variable A in our provided function \(y=\frac{1}{2} \sin \left(x+\frac{\pi}{2}\right)\). This is denoted by the coefficient of sin, which in our case is \(\frac{1}{2}\). The amplitude is then the absolute value of A, or \(|\frac{1}{2}|\), which equals \(\frac{1}{2}\).
02

Calculation of the Period

The period is found by taking the reciprocal of the absolute value of B where B is the coefficient of x. Since there is no explicit coefficient in x in our function, we understand that B is 1, and thus the period is \(|1|^{-1} = 1\).
03

Calculation of Phase Shift

The phase shift can be calculated as \(-C/B\). Here, C is the term that is added or subtracted inside the sine function. In our case, C = \(\frac{\pi}{2}\). So, our phase shift is given by \(-C/B = -\frac{\pi}{2} / 1 = -\frac{\pi}{2}\). This means the graph will shift to the right by \(\frac{\pi}{2}\) units.
04

Graphing the Function

To graph the function \(y=\frac{1}{2} \sin \left(x+\frac{\pi}{2}\right)\), we start at the phase shift, which is \(-\frac{\pi}{2}\), and plot the sine wave, noting that at every whole integer multiple of the period the function cycles. This is a sine wave with an amplitude of \(\frac{1}{2}\) (meaning it reaches a maximum height of \(\frac{1}{2}\) and minimum of \(-\frac{1}{2}\)), a period of 1, and shifted right by \(\frac{\pi}{2}\) units.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a graphing utility to graph two periodsof the function. Use a graphing utility to graph Use a graphing utility to graph \( y=\sin x-\frac{\sin 3 x}{9}+\frac{\sin 5 x}{25} \) in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by \([-2,2,1]\) viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?

Graph \(y=\sin ^{-1} x+\cos ^{-1} x\) in a \([-2,2,1]\) by \([0,3,1]\) viewing rectangle. What appears to be true about the sum of the inverse sine and inverse cosine for values between \(-1\) and \(1,\) inclusive?

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \cos \left(\sin ^{-1} 2 x\right) $$

Find the slant asymptote of $$f(x)=\frac{2 x^{2}-7 x-1}{x-2}$$

Determine the domain and the range of each function. $$ f(x)=\sin ^{-1}(\sin x) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.