/*! 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 13 For each of the following equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 13

For each of the following equations, find the amplitude, period, horizontal shift, and midline. \(y=2 \sin (3 x-21)+4\)

Short Answer

Expert verified
Amplitude: 2, Period: \(\frac{2\pi}{3}\), Horizontal Shift: 7 units, Midline: \(y=4\).

Step by step solution

01

Identify the Standard Form

The standard form for a sine function is \( y = a \sin(bx - c) + d \). Comparing this with \( y = 2 \sin(3x - 21) + 4 \), we identify \( a = 2 \), \( b = 3 \), \( c = 21 \), and \( d = 4 \).
02

Calculate the Amplitude

The amplitude of a sine function is given by the absolute value of \( a \). Here, \( a = 2 \), so the amplitude is \( |2| = 2 \).
03

Determine the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \). With \( b = 3 \), the period is \( \frac{2\pi}{3} \).
04

Calculate the Horizontal Shift

The horizontal shift, also called phase shift, is given by \( \frac{c}{b} \). Here, \( c = 21 \) and \( b = 3 \), so the horizontal shift is \( \frac{21}{3} = 7 \). This means the graph shifts 7 units to the right.
05

Identify the Midline

The midline of a sine function is given by \( y = d \). Here, \( d = 4 \), so the midline is \( y = 4 \).

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.

Sine Function
In trigonometry, the sine function is a fundamental element with distinct features like amplitude, period, and more. It is generally represented in the format:
  • Equation Form: \( y = a \sin(bx - c) + d \)
  • Function Behavior: Repetitive oscillations that look like waves on a graph.
  • Key Parameters: \( a \) (Amplitude), \( b \) (Frequency), \( c \) (Phase Shift), \( d \) (Vertical Shift)
The sine function oscillates between -1 and 1, generating its characteristic wave shape. The particular parameters in the equation adjust how the sine wave appears on a graph, making it an essential tool in modeling periodic phenomena. In mathematics, the sine curve is used to model scenarios like sound waves, light waves, and more.
Amplitude
Amplitude in a sine function can be thought of as the height of the wave from the midline to the peak. It's the measure of the extent of the wave's oscillation.
  • Calculation: Amplitude is given by the value |\( a \)| where \( a \) is the coefficient in the equation of the sine function.
  • In the standard form equation \( y = a \sin(bx - c) + d \), the amplitude is determined by \( a \).
  • Example from Exercise: In \( y = 2 \sin(3x - 21) + 4 \), \( a = 2 \), so the amplitude is 2.
A higher amplitude means taller peaks and deeper troughs, while a lower amplitude means the waves are shorter and closer to the midline. Understanding amplitude helps you gauge the intensity or strength of the wave's oscillations.
Period
The period of a sine function refers to how long it takes for the function to complete one full cycle of its wave. The period can be influenced by the frequency of the function,
  • Formula: The period is calculated as \( \frac{2\pi}{b} \), where \( b \) is the frequency component in the sine function鈥檚 equation.
  • In the equation \( y = 2 \sin(3x - 21) + 4 \), the period is determined using \( b = 3 \). Therefore the period is \( \frac{2\pi}{3} \).
  • Interpretation: A smaller period means the waves repeat more frequently within a shorter amount of time.
Knowing the period is vital, particularly in practical applications like physics, where examining wave patterns is crucial. It allows for understanding how often fluctuations occur over a given time span.
Phase Shift
Phase shift in the sine function relates to a horizontal translation of the graph left or right along the x-axis. This is key in determining where the wave cycle starts along the x-axis.
  • Calculation: Phase shift is obtained by \( \frac{c}{b} \), where \( c \) and \( b \) originate from the transformed sine function equation.
  • From the equation \( y = 2 \sin(3x - 21) + 4 \), \( c = 21 \) and \( b = 3 \), making the phase shift \( \frac{21}{3} = 7 \).
  • Direction: A positive phase shift indicates the graph shifts to the right, while a negative shift moves it to the left.
Phase shift is essential when synchronizing signals or predicting wave actions, as it denotes where a function's cycle starts compared to a reference position on the graph. Understanding phase shifts also helps when aligning sinusoidal functions to real-world data.

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 first two positive solutions $$ 3 \cos \left(\frac{\pi}{2} x\right)=-2 $$

Evaluate the following expressions. $$ \tan ^{-1}(\sqrt{3}) $$

Find all solutions on the interval \(0 \leq \theta<2 \pi\) $$ 2 \cos (\theta)=1 $$

A population of elk oscillates 150 above and below an average of 720 during the year, hitting the lowest value in January \((t=0)\) a. Find an equation for the population, \(P\), in terms of the months since January, \(t\). b. What if the lowest value of the rabbit population occurred in March instead?

Sketch a graph of \(j(x)=\tan \left(\frac{\pi}{2} x\right)\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.