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
![**Title: Analyzing Trigonometric Functions**
**Objective: Determine the amplitude, period, and phase shift (if any) of the given function.**
---
**Function:**
\[ y = -5\cos{x} \]
**Question:**
The amplitude is [ ]
---
**Explanation:**
1. **Amplitude:** The amplitude of a trigonometric function like \(\cos{x}\) is the absolute value of the coefficient in front of the cosine. For the function \(y = -5\cos{x}\), the coefficient is -5. Therefore, the amplitude is \(|-5| = 5\).
2. **Period:** The period of a cosine function \(\cos(bx)\) is calculated as \(\frac{2\pi}{|b|}\). In this case, since there is no coefficient attached to \(x\), \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\).
3. **Phase Shift:** The phase shift of a cosine function \(y = a\cos(bx - c)\) is calculated as \(\frac{c}{b}\). Since there is no horizontal shift in \( -5\cos{x} \), there is no phase shift.
This exercise helps in understanding how to find key attributes of trigonometric functions like amplitude, period, and phase shift, all of which are crucial for graphing these functions accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff08af0c6-4dde-44ec-9e21-243b109290d6%2Fe55fa47e-1661-4a99-8d44-dce29e443229%2F0dm08yp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Trigonometric Functions**
**Objective: Determine the amplitude, period, and phase shift (if any) of the given function.**
---
**Function:**
\[ y = -5\cos{x} \]
**Question:**
The amplitude is [ ]
---
**Explanation:**
1. **Amplitude:** The amplitude of a trigonometric function like \(\cos{x}\) is the absolute value of the coefficient in front of the cosine. For the function \(y = -5\cos{x}\), the coefficient is -5. Therefore, the amplitude is \(|-5| = 5\).
2. **Period:** The period of a cosine function \(\cos(bx)\) is calculated as \(\frac{2\pi}{|b|}\). In this case, since there is no coefficient attached to \(x\), \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\).
3. **Phase Shift:** The phase shift of a cosine function \(y = a\cos(bx - c)\) is calculated as \(\frac{c}{b}\). Since there is no horizontal shift in \( -5\cos{x} \), there is no phase shift.
This exercise helps in understanding how to find key attributes of trigonometric functions like amplitude, period, and phase shift, all of which are crucial for graphing these functions accurately.
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