/*! 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 24 Describe the relationship betwee... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitude, period, and shifts. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=\sin 3 x \end{aligned}$$

Short Answer

Expert verified
The amplitude of both functions is 1. However, while the period of \( f(x) = \sin x \) is \( 2\pi \), \( g(x) = \sin 3x \) has a period of \( \frac{2\pi}{3} \). There are no phase shifts in either function.

Step by step solution

01

Amplitude Comparison

The amplitude of a function is its peak positive or negative value. For both \( f(x) \) and \( g(x) \), the amplitude is 1 as, regardless of the coefficient of \( x \) in sin functions, the output will always vary from -1 to 1.
02

Period Comparison

The period of a function is the interval over which its shape repeats. For \( f(x) = \sin x \), the period is \( 2\pi \), as this is the interval over which the sine function typically repeats. When we switch to \( g(x) = \sin 3x \), however, the period changes. The coefficient in front of \( x \) in the sine function acts as frequency multiplier: the period becomes \( \frac{2\pi}{3} \), three times faster than the period of \( f(x) \).
03

Phase Shift Comparison

The phase shift of a function is a horizontal shift. For both \( f(x) = \sin x \) and \( g(x) = \sin 3x \), there are no horizontal or vertical shifts since there are no additional constants modifying the functions.

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

Finding Arc Length Find the length of the are on a circle of radius \(r\) intercepted by a central angle \(\boldsymbol{\theta}\). $$r=3 \text { meters, } \theta=150^{\circ}$$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$h(x)=x \sin \frac{1}{x}$$

The normal monthly high temperatures \(H\) (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $$H(t)=56.94-20.86 \cos \left(\frac{\pi t}{6}\right)-11.58 \sin \left(\frac{\pi t}{6}\right)$$ and the normal monthly low temperatures \(L\) are approximated by $$L(t)=41.80-17.13 \cos \left(\frac{\pi t}{6}\right)-13.39 \sin \left(\frac{\pi t}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January (see figure). (Source: National Climatic Data Center (GRAPH CANNOT COPY). (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June \(21,\) but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow \infty, \text { the value of } \arctan x \rightarrow\text { _____ } .$$

Geometry Find the length of the sides of a regular pentagon inscribed in a circle of radius 25 inches.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.