/*! 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 17 Find the period and amplitude. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 17

Find the period and amplitude. $$y=\frac{1}{4} \sin 2 \pi x$$

Short Answer

Expert verified
The amplitude of the function \(y=\frac{1}{4}sin(2\pi x)\) is \(\frac{1}{4}\) and the period is 1.

Step by step solution

01

Identify the Amplitude

The amplitude is the coefficient of the sine function. In this case, it's \(\frac{1}{4}\). Therefore, the amplitude of the function is \(\frac{1}{4}\).
02

Identify the coefficient of x inside the sine function

The coefficient of \(x\) inside the sine function is \(2\pi\). This will be used to calculate the period.
03

Calculate the Period

The period of a sine function is given by \(T=\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\). In this case, \(b=2\pi\). Substituting \(2\pi\) into the equation for the period, we find that \(T=\frac{2\pi}{2\pi}= 1\). Therefore, the period of the sine function is 1.

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
Amplitude represents the height of the wave in a sine function, from its central position. It's important to understand this concept as it indicates how much the graph of the sine function stretches vertically.

For any function in the form \(y = a \sin(bx)\), the amplitude is the absolute value of \(a\). It tells us how far the wave reaches above and below its central axis. If you imagine the wave of the sine function, amplitude is the distance from the center line to the peak or to the trough of the wave.

  • For instance, if you have \(y = \frac{1}{4} \sin(2 \pi x)\), the amplitude is \(\left|\frac{1}{4}\right| = \frac{1}{4}\). This means the wave reaches a maximum and minimum of \(\frac{1}{4}\) units from the midline.
The amplitude does not affect how often the wave repeats or its frequency. Instead, it only influences the vertical stretching.
Period of a Function
The period of a trigonometric function is the length of one complete cycle of the wave. For a sine function, this is particularly important as it dictates how often the wave repeats over a certain interval.

The formula to find the period \(T\) of the function \(y = a \sin(bx)\) is given by \(T = \frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) in the function.

  • In the example \(y = \frac{1}{4} \sin(2 \pi x)\), \(b\) is \(2\pi\). Hence, the period is calculated as \(T = \frac{2\pi}{2\pi} = 1\).
This result means that the sine wave completes one full cycle over an interval of 1 unit. So, every 1 unit on the \(x\)-axis, the same wave pattern will start anew.
Sine Function
The sine function is one of the most fundamental trigonometric functions, often used to describe wave patterns like sound waves and tides. It's typically written as \(y = a \sin(bx + c) + d\), where each parameter modifies the wave's characteristics.

The basic sine function, \(y = \sin(x)\), creates a wave oscillating between -1 and 1. This wave has a few noticeable features:
  • The wave repeats every \(2\pi\) interval, giving a default period of \(2\pi\).
  • When modifications are applied, such as a multiplier before the \(x\) term or changes to the amplitude, the shape and properties of the wave change accordingly.
  • In \(y = \frac{1}{4} \sin(2 \pi x)\), the sine wave's amplitude has been reduced to \(\frac{1}{4}\), and the period reduced to 1, making the wave repeat more frequently than the standard sine wave.
This customization of sine function parameters allows it to fit a wide range of real-world cyclic scenarios, making it both versatile and essential.

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

Sketch a graph of the function. $$f(x)=\frac{\pi}{2}+\arctan x$$

Find a model for simple harmonic motion satisfying the specified conditions. $$\begin{array}{cc}\text{Displacement \((t=0)\)} & \text{Amplitude} & \text{Period} \\ 0& 3 \mathrm{meters}& 6 \mathrm{seconds}\end{array}$$

Find the length of the sides of a regular pentagon inscribed in a circle of radius 25 inches.

Consider the functions \(f(x)=\sin x\) and \(f^{-1}(x)=\arcsin x.\) (a) Use a graphing utility to graph the composite functions \(f \circ f^{-1}\) and \(f^{-1} \circ f.\) (b) Explain why the graphs in part (a) are not the graph of the line \(y=x\). Why do the graphs of \(f \circ f^{-1}\) and \(f^{-1} \circ f\) differ?

Fill in the blank. If not possible, state the reason. As \(x \rightarrow 1^{-},\) the value of arccos \(x \rightarrow\) \(\square\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and 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.