Write an equation for the graph below (assuming there is no horizontal shift), and determine the amplitude, frequency, midline, and period. You must show all work to carn fall credit.
Write an equation for the graph below (assuming there is no horizontal shift), and determine the amplitude, frequency, midline, and period. You must show all work to carn fall credit.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Harmonic Motion Analysis
#### Problem Statement:
Write an equation for the graph below (assuming there is no horizontal shift), and determine the amplitude, frequency, midline, and period. You must show all work to earn full credit.
#### Graph Analysis:
The graph provided represents a sinusoidal function. Below is a detailed explanation of the key components of the graph:
- **Amplitude:** The amplitude of a sinusoidal graph is the distance from the midline to the maximum or minimum point of the graph. Observing the graph, the maximum value is 3 and the minimum value is -3. Therefore, the amplitude \( A \) is:
\[
\text{Amplitude} = 3
\]
- **Midline:** The midline is the horizontal line that passes through the middle of the graph, equidistant from the maximum and minimum values. Given the graph oscillates between -3 and 3, the midline \( M \) is:
\[
\text{Midline} = 0
\]
- **Period:** The period of a sinusoidal graph is the distance along the x-axis for the graph to complete one full cycle. Observing the graph, one full cycle occurs from \( x = 0 \) to \( x = 8 \). Thus, the period \( P \) is:
\[
\text{Period} = 8
\]
- **Frequency:** The frequency of a sinusoidal graph is the reciprocal of the period. It represents the number of cycles completed in a unit interval along the x-axis. Given the period \( P \) is 8, the frequency \( f \) is:
\[
\text{Frequency} = \frac{1}{8}
\]
#### Equation Derivation:
Given a sinusoidal function can generally be represented as:
\[
y = A \sin(B(x - C)) + D
\]
where:
- \( A \) is the amplitude.
- \( B \) is related to the period by the equation \( B = \frac{2\pi}{\text{Period}} \).
- \( C \) is the horizontal shift.
- \( D \) is the vertical shift (midline).
Since there is no horizontal shift ( \( C = 0 \) ) and the midline \( D = 0 \):
\[
y = 3 \sin\left](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05ee46cf-e5a8-42f3-bb4a-0741212ce915%2F7a0ea00d-1af1-467a-b229-487a812cd8cf%2Fep78rc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Harmonic Motion Analysis
#### Problem Statement:
Write an equation for the graph below (assuming there is no horizontal shift), and determine the amplitude, frequency, midline, and period. You must show all work to earn full credit.
#### Graph Analysis:
The graph provided represents a sinusoidal function. Below is a detailed explanation of the key components of the graph:
- **Amplitude:** The amplitude of a sinusoidal graph is the distance from the midline to the maximum or minimum point of the graph. Observing the graph, the maximum value is 3 and the minimum value is -3. Therefore, the amplitude \( A \) is:
\[
\text{Amplitude} = 3
\]
- **Midline:** The midline is the horizontal line that passes through the middle of the graph, equidistant from the maximum and minimum values. Given the graph oscillates between -3 and 3, the midline \( M \) is:
\[
\text{Midline} = 0
\]
- **Period:** The period of a sinusoidal graph is the distance along the x-axis for the graph to complete one full cycle. Observing the graph, one full cycle occurs from \( x = 0 \) to \( x = 8 \). Thus, the period \( P \) is:
\[
\text{Period} = 8
\]
- **Frequency:** The frequency of a sinusoidal graph is the reciprocal of the period. It represents the number of cycles completed in a unit interval along the x-axis. Given the period \( P \) is 8, the frequency \( f \) is:
\[
\text{Frequency} = \frac{1}{8}
\]
#### Equation Derivation:
Given a sinusoidal function can generally be represented as:
\[
y = A \sin(B(x - C)) + D
\]
where:
- \( A \) is the amplitude.
- \( B \) is related to the period by the equation \( B = \frac{2\pi}{\text{Period}} \).
- \( C \) is the horizontal shift.
- \( D \) is the vertical shift (midline).
Since there is no horizontal shift ( \( C = 0 \) ) and the midline \( D = 0 \):
\[
y = 3 \sin\left
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