/*! 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 30 \(29-42\) . Find the amplitude, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 30

\(29-42\) . Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=2 \sin \left(x-\frac{\pi}{3}\right) $$

Short Answer

Expert verified
Amplitude is 2, period is \(2\pi\), phase shift is \(\frac{\pi}{3}\) to the right.

Step by step solution

01

Understanding the Equation

The function we have is \( y = 2 \sin\left(x - \frac{\pi}{3}\right) \). This is a sine function of the form \( y = A \sin(Bx - C) \), where \( A \) is the amplitude, \( B \) affects the period, and \( C \) affects the phase shift.
02

Find the Amplitude

The amplitude \( A \) is the coefficient of the sine function. In \( y = 2 \sin\left(x - \frac{\pi}{3}\right) \), \( A = 2 \). Therefore, the amplitude is 2.
03

Find the Period

The period \( T \) of the sine function is given by the formula \( T = \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \) inside the sine function. Here, \( B = 1 \). So, the period \( T = \frac{2\pi}{1} = 2\pi \).
04

Find the Phase Shift

The phase shift is determined by \( \frac{C}{B} \) from \( y = A \sin(Bx - C) \). Here, \( C = \frac{\pi}{3} \) and \( B = 1 \). So, the phase shift \( \frac{\pi}{3} / 1 = \frac{\pi}{3} \). This means the graph shifts to the right by \( \frac{\pi}{3} \).
05

Sketching the Graph

To sketch one complete period, start right after the phase shift at \( x = \frac{\pi}{3} \) and go until \( x = \frac{\pi}{3} + 2\pi \). The sine curve will reach its maximum at \( y = 2 \) and its minimum at \( y = -2 \). The complete wave oscillates between those extremes over this interval.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude of the Sine Function
When studying a sine function, one of the key features we look out for is the amplitude. This lesson applies even when looking at the function \( y = 2 \sin\left(x - \frac{\pi}{3}\right) \). The amplitude is a measure of how much the wave oscillates above and below its average value, which is typically zero in the case of standard sine functions.

  • Definition: In a sine function of the form \( y = A \sin(Bx - C) \), the amplitude is the absolute value of \( A \).
  • For the example function, the coefficient \( A = 2 \). Hence, the amplitude is 2.
  • This means the sine wave will reach a maximum height of 2 units and a minimum of -2 units from the horizontal axis.
  • The wave extends equally above and below the centerline, making this characteristic easily identifiable in a graph.
By identifying the amplitude, we can understand how the sine function stretches or shrinks vertically, allowing us to visualize its peaks and valleys effectively.
Period of the Sine Function
The period of a sine function describes how long it takes for the wave to complete one full cycle. For the function \( y = 2 \sin(x - \frac{\pi}{3}) \), recognizing the period involves looking at the coefficient in front of the variable \( x \).

  • Definition: The period \( T \) of a sine wave \( y = A \sin(Bx - C) \) is calculated using the formula \( T = \frac{2\pi}{|B|} \).
  • In this case, \( B = 1 \). So, substituting into the formula gives \( T = \frac{2\pi}{1} = 2\pi \).
  • This tells us the wave repeats every \( 2\pi \) units in the horizontal direction.
  • A longer period would indicate a more stretched out wave, while a shorter period shows it is more compact.
Through understanding the period, we comprehend how frequently the oscillations occur, an essential aspect for graphing sine functions accurately.
Phase Shift of the Sine Function
Phase shift in a sine function reveals how much the graph is horizontally shifted from its standard position. This concept is crucial for drawing and interpreting the sine wave accurately for the function \( y = 2 \sin(x - \frac{\pi}{3}) \).

  • Definition: The phase shift in \( y = A \sin(Bx - C) \) can be found using \( \frac{C}{B} \), where the term \( C \) determines the horizontal shift.
  • In our equation, \( C = \frac{\pi}{3} \) and \( B = 1 \), resulting in a phase shift of \( \frac{\pi}{3} \).
  • This means the entire sine graph moves \( \frac{\pi}{3} \) units to the right.
  • The phase shift tells us where the sine wave's critical points move along the \( x \)-axis, like starting points and points of maximum and minimum values.
Mastering the concept of phase shift allows us to accurately predict where the sine cycles start in relation to the origin, ensuring the graph reflects all transformations of the function.

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

Find the values of the trigonometric functions of \(t\) from the given information. \(\sin t=-\frac{1}{4}, \quad \sec t<0\)

An initial amplitude \(k,\) damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p . )\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(19-22,\) and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises \(23-26\) (b) Graph the function. $$ k=7, \quad c=10, \quad p=\pi / 6 $$

Find the values of the trigonometric functions of \(t\) from the given information. sec \(t=3,\) terminal point of \(t\) is in Quadrant IV

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{5 \pi}{4} \quad\) (b) \(\csc \frac{\pi}{2} \quad\) (c) \(\csc \frac{3 \pi}{2}\)

An initial amplitude \(k\), damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p .)\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(19-22,\) and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises \(23-26\) (b) Graph the function. $$ k=1, \quad c=1, \quad p=1 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.