Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Determine the A amplitude , P period and Ph phase shift .
![### Trigonometric Function Equations
Below are a set of trigonometric equations involving sine and cosine functions. Each equation features distinct transformations that can include amplitude changes, phase shifts, and frequency variations.
1. **Equation 1:**
\[
y = 2 \cos\left(\frac{1}{2}x - \frac{\pi}{2}\right)
\]
- **Description:** The amplitude is 2, with a horizontal stretch factor of 2 (due to the coefficient \(\frac{1}{2}\) with \(x\)). The function is shifted right by \(\frac{\pi}{2}\).
2. **Equation 2:**
\[
y = 4 \sin\left(\frac{1}{4}x + \frac{\pi}{8}\right)
\]
- **Description:** The amplitude is 4. The function has a horizontal stretch factor of 4 and a phase shift to the left by \(\frac{\pi}{8}\).
3. **Equation 3:**
\[
y = \frac{1}{2} \sin\left(2x + \frac{\pi}{2}\right)
\]
- **Description:** The amplitude is \(\frac{1}{2}\). The frequency is doubled (due to coefficient 2 with \(x\)), and the function shifts left by \(\frac{\pi}{2}\).
4. **Equation 4:**
\[
y = -3 \cos(4x - \pi)
\]
- **Description:** The amplitude is 3, and the negative sign denotes a vertical reflection. The frequency is quadrupled, and there is a right phase shift by \(\pi\).
5. **Equation 5:**
\[
y = 2 + 3 \cos(\pi x)
\]
- **Description:** This represents a cosine function with an amplitude of 3. The frequency is \(\pi\), and there is a vertical shift upward by 2 units.
Each equation showcases different ways to manipulate sine and cosine functions through changes in amplitude, frequency, phase shifts, and reflections.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92755273-dfea-4585-b73b-42dc69649a75%2F64ec256e-65bc-403b-a695-dcfc531c3efc%2F76d217i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Function Equations
Below are a set of trigonometric equations involving sine and cosine functions. Each equation features distinct transformations that can include amplitude changes, phase shifts, and frequency variations.
1. **Equation 1:**
\[
y = 2 \cos\left(\frac{1}{2}x - \frac{\pi}{2}\right)
\]
- **Description:** The amplitude is 2, with a horizontal stretch factor of 2 (due to the coefficient \(\frac{1}{2}\) with \(x\)). The function is shifted right by \(\frac{\pi}{2}\).
2. **Equation 2:**
\[
y = 4 \sin\left(\frac{1}{4}x + \frac{\pi}{8}\right)
\]
- **Description:** The amplitude is 4. The function has a horizontal stretch factor of 4 and a phase shift to the left by \(\frac{\pi}{8}\).
3. **Equation 3:**
\[
y = \frac{1}{2} \sin\left(2x + \frac{\pi}{2}\right)
\]
- **Description:** The amplitude is \(\frac{1}{2}\). The frequency is doubled (due to coefficient 2 with \(x\)), and the function shifts left by \(\frac{\pi}{2}\).
4. **Equation 4:**
\[
y = -3 \cos(4x - \pi)
\]
- **Description:** The amplitude is 3, and the negative sign denotes a vertical reflection. The frequency is quadrupled, and there is a right phase shift by \(\pi\).
5. **Equation 5:**
\[
y = 2 + 3 \cos(\pi x)
\]
- **Description:** This represents a cosine function with an amplitude of 3. The frequency is \(\pi\), and there is a vertical shift upward by 2 units.
Each equation showcases different ways to manipulate sine and cosine functions through changes in amplitude, frequency, phase shifts, and reflections.
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