Find the period. y = 6 + Graph the function. 10 10 5 6. 2 3 3 -5- 10 6. -10 -10 Nood Holn?
Find the period. y = 6 + Graph the function. 10 10 5 6. 2 3 3 -5- 10 6. -10 -10 Nood Holn?
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Find the period.**
Given the function:
\[ y = 6 \cot \left(3x + \frac{\pi}{2}\right) \]
**Graph the function.**
The image contains two graphs of the function \( y = 6 \cot \left(3x + \frac{\pi}{2}\right) \).
- **Graph Explanation:**
- The graphs depict periodic curves typical of the cotangent function, stretched and reflected due to the coefficient and phase shift.
- Each graph spans the x-axis from \(-\frac{2\pi}{3}\) to \(\frac{2\pi}{3}\) with additional cycles repeated in this window.
- Vertical asymptotes are present where the cotangent function is undefined, indicated by dashed lines at intervals such as \(-\frac{\pi}{3}, 0, \frac{\pi}{3}\), and others symmetrically distributed.
- The y-axis is labeled from -10 to 10, reflecting the amplitude scale as modified by the coefficient 6.
**Graph Features:**
- The graphs show a repeating pattern, indicating the periodic nature of the cotangent function.
- There is a horizontal shift due to the \(\frac{\pi}{2}\) modification inside the function.
- Due to the 3x factor, the function is compressed horizontally.
**Find the period:**
The general form of the cotangent function \(\cot(bx + c)\) has a period of \(\frac{\pi}{b}\).
For the function \( y = 6 \cot \left(3x + \frac{\pi}{2}\right) \):
1. \( b = 3 \).
2. Therefore, the period = \(\frac{\pi}{3} \).
**Conclusion:**
The periodic nature of this trigonometric function makes it critical for analyzing its transformations, especially due to stretching and phase shifts.
**Need Help?**
There is a "Need Help?" section with a button labeled "Post M" below the graphs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8492a80-fb1a-4bc0-8e68-f5fde27cb209%2F4ffc236b-8312-46a9-b563-57e983f6ffb3%2F0upyys_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the period.**
Given the function:
\[ y = 6 \cot \left(3x + \frac{\pi}{2}\right) \]
**Graph the function.**
The image contains two graphs of the function \( y = 6 \cot \left(3x + \frac{\pi}{2}\right) \).
- **Graph Explanation:**
- The graphs depict periodic curves typical of the cotangent function, stretched and reflected due to the coefficient and phase shift.
- Each graph spans the x-axis from \(-\frac{2\pi}{3}\) to \(\frac{2\pi}{3}\) with additional cycles repeated in this window.
- Vertical asymptotes are present where the cotangent function is undefined, indicated by dashed lines at intervals such as \(-\frac{\pi}{3}, 0, \frac{\pi}{3}\), and others symmetrically distributed.
- The y-axis is labeled from -10 to 10, reflecting the amplitude scale as modified by the coefficient 6.
**Graph Features:**
- The graphs show a repeating pattern, indicating the periodic nature of the cotangent function.
- There is a horizontal shift due to the \(\frac{\pi}{2}\) modification inside the function.
- Due to the 3x factor, the function is compressed horizontally.
**Find the period:**
The general form of the cotangent function \(\cot(bx + c)\) has a period of \(\frac{\pi}{b}\).
For the function \( y = 6 \cot \left(3x + \frac{\pi}{2}\right) \):
1. \( b = 3 \).
2. Therefore, the period = \(\frac{\pi}{3} \).
**Conclusion:**
The periodic nature of this trigonometric function makes it critical for analyzing its transformations, especially due to stretching and phase shifts.
**Need Help?**
There is a "Need Help?" section with a button labeled "Post M" below the graphs.
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