Write the equation of the trigonometric graph.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Analyzing and Writing the Equation of the Trigonometric Graph
#### Problem Statement:
Write the equation of the trigonometric graph.
#### Graph Description:
The graph presented is a smooth, continuous curve resembling a sine wave, likely representing a trigonometric function. It displays periodic behavior and specific characteristics such as amplitude and phase shift.
#### Graph Analysis:
1. **Axes and Intercepts**:
- The graph is plotted on a coordinate system with the x-axis and y-axis intersecting at the origin (0,0).
- The x-axis is labeled with intervals at \(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\).
- The y-axis is labeled with intervals from -5 to 5.
2. **Period**:
- The graph completes one full cycle from \( -\pi \) to \( 2\pi \), indicating a period of \(3\pi\).
3. **Amplitude**:
- The highest point on the graph is at y = 3, and the lowest is at y = -3. This suggests an amplitude of 3.
4. **Midline**:
- The midline of the graph appears to be at y = 0, indicating no vertical shift from the origin.
5. **Phase Shift**:
- The graph does not appear to be horizontally shifted, as the waveform appears centered around the origin.
#### Writing the Equation:
Based on the analysis, the trigonometric function resembles a sine function of the form:
\[ y = A \sin(Bx) \]
where:
- \( A \) is the amplitude.
- \( B \) affects the period of the function.
Given that:
- Amplitude (\( A \)) is 3.
- The period is \( 3\pi \), which can be represented as \( \frac{2\pi}{B} = 3\pi \). Solving for \( B \), we find \( B = \frac{2}{3} \).
Thus, the equation of the graph is:
\[ y = 3 \sin\left( \frac{2}{3}x \right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F109ac5c0-1ab9-40e2-b229-fad4b311f706%2F13fb6166-372c-4518-8ff3-67d396ecfe21%2F4nzv3e_processed.png&w=3840&q=75)
Transcribed Image Text:### Analyzing and Writing the Equation of the Trigonometric Graph
#### Problem Statement:
Write the equation of the trigonometric graph.
#### Graph Description:
The graph presented is a smooth, continuous curve resembling a sine wave, likely representing a trigonometric function. It displays periodic behavior and specific characteristics such as amplitude and phase shift.
#### Graph Analysis:
1. **Axes and Intercepts**:
- The graph is plotted on a coordinate system with the x-axis and y-axis intersecting at the origin (0,0).
- The x-axis is labeled with intervals at \(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\).
- The y-axis is labeled with intervals from -5 to 5.
2. **Period**:
- The graph completes one full cycle from \( -\pi \) to \( 2\pi \), indicating a period of \(3\pi\).
3. **Amplitude**:
- The highest point on the graph is at y = 3, and the lowest is at y = -3. This suggests an amplitude of 3.
4. **Midline**:
- The midline of the graph appears to be at y = 0, indicating no vertical shift from the origin.
5. **Phase Shift**:
- The graph does not appear to be horizontally shifted, as the waveform appears centered around the origin.
#### Writing the Equation:
Based on the analysis, the trigonometric function resembles a sine function of the form:
\[ y = A \sin(Bx) \]
where:
- \( A \) is the amplitude.
- \( B \) affects the period of the function.
Given that:
- Amplitude (\( A \)) is 3.
- The period is \( 3\pi \), which can be represented as \( \frac{2\pi}{B} = 3\pi \). Solving for \( B \), we find \( B = \frac{2}{3} \).
Thus, the equation of the graph is:
\[ y = 3 \sin\left( \frac{2}{3}x \right) \]
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