/*! 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 1 The amplitude \(A\) and period \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 1

The amplitude \(A\) and period \(T\) of \(f(x)=5 \sin (4 x)\) are ______ and _______.

Short Answer

Expert verified
The amplitude is 5 and the period is \(\frac{\pi}{2}\).

Step by step solution

01

Identify the Standard Form

Recall that the standard form of a sine function is given by \(f(x) = A \sin (B x)\). In this form, \(A\) represents the amplitude, and the period \(T\) is calculated using the formula \(T = \frac{2\pi}{B}\).
02

Determine the Amplitude

In the given function, \(f(x) = 5 \sin (4 x)\), compare it to the standard form to identify \(A\). Here, \(A = 5\). Therefore, the amplitude is 5.
03

Calculate the Period

Identify \(B\) from the given function, where \(B = 4\). Use the period formula \(T = \frac{2\pi}{B}\). Substituting \(B = 4\), the period \(T\) is \(T = \frac{2\pi}{4} = \frac{\pi}{2}\).

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
The amplitude of a sine function is a measure of its peak value. It tells us how far the function's graph reaches from its middle value (usually zero) to its maximum or minimum value.
To find the amplitude, we look at the coefficient in front of the sine function, which is represented by the variable 'A' in the standard form:
\[f(x) = A \, \sin(Bx)\]
In the given function \(f(x) = 5 \, \sin(4x)\), the value of 'A' is 5. This means the function's amplitude is 5. The graph of this sine function will reach up to 5 and down to -5 from its central axis.
  • Amplitude = peak value of the sine wave
  • Found by identifying the coefficient A
  • In the example, A = 5, so amplitude = 5
Period
The period of a sine function represents the length of one complete cycle of the wave. It tells us how far along the x-axis you go before the function begins to repeat. To find the period, we use the variable 'B' from the standard form and apply the formula:
\[T = \frac{2\pi}{B}\]
In the example function \(f(x) = 5 \, \sin(4x)\), 'B' is 4. Using the period formula, we get: \[T = \frac{2\pi}{4} = \frac{\pi}{2}\] This means the function completes one full cycle every \(\frac{\pi}{2}\) units along the x-axis.
  • Period = length of one cycle
  • Found by using the formula \(T = \frac{2\pi}{B}\)
  • In the example, B = 4, so period = \(\frac{\pi}{2}\)
Standard Form
The standard form of a sine function provides a clear way to identify its amplitude and period. It is written as:
\[f(x) = A \, \sin(Bx)\]
Where:
  • \(A\) is the amplitude
  • \(B\) affects the period
  • \(x\) is the input variable
The given function \(f(x) = 5 \, \sin(4x)\) matches this standard form. From this, we can easily determine the amplitude (\(A = 5\)) and calculate the period using \(B = 4\). Using the standard form helps us quickly understand the behavior of sine functions.
  • The standard form is \(f(x) = A \, \sin(Bx)\)
  • It helps identify amplitude and period easily
  • In the example, \(A = 5\) and \(B = 4\), making it easy to find amplitude and period

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 area of each triangle. Round answers to two decimal places. $$b=4, \quad c=1, \quad A=120^{\circ}$$

Find the area of each triangle. Round answers to two decimal places. $$a=6, \quad b=4, \quad C=60^{\circ}$$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(\cos \alpha=\frac{4}{5}, 0<\alpha<\frac{\pi}{2},\) find the exact value of: (a) \(\cos \frac{\alpha}{2}\) (b) \(\sin \frac{\alpha}{2}\) (c) \(\tan \frac{\alpha}{2}\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. What is the remainder when \(P(x)=2 x^{4}-3 x^{3}-x+7\) is divided by \(x+2 ?\)

State the formula for finding the area of an SAS triangle in words.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.