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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![### Graphing Trigonometric Functions
**Objective:**
Graph two full periods of the function \( f(x) = \cos(6x) \) and determine the amplitude and period of the function.
**Instructions:**
1. **Graph the Function:**
- Graph \( f(x) = \cos(6x) \) over a span of two full periods.
2. **Determine the Amplitude and Period:**
- Enter the exact values for amplitude and period.
- For the number \( \pi \), either:
- Choose \( \pi \) from the drop-down menu.
- Or type in Pi (with a capital P).
**Amplitude:**
\[ A = \, \text{[Number input field]} \]
**Period:**
\[ P = \, \text{[Equation input field with various mathematical symbols and functions available]} \]
**Helpful Symbols Provided:**
- \( a^b \)
- \( \frac{a}{b} \)
- \( \sqrt{a} \)
- \( |a| \)
- \( \pi \)
- \( \sin(a) \)
**Explanation of the Graph and Input Fields:**
The graphing area isn't displayed in the provided image, but typically in a trigonometric graphing exercise, you would see a coordinate system where the cosine wave is periodically oscillating. The function \( \cos(6x) \) will have six times the frequency of the basic cosine function \( \cos(x) \), thus making its period shorter.
**Details:**
- **Amplitude (A):** The amplitude of a trigonometric function like cosine is the maximum value the function reaches from its mean position. For \( f(x) = \cos(6x) \), the amplitude is \( 1 \) since the coefficient of \( \cos \) is \( 1 \).
- **Period (P):** The period of a function \( \cos(bx) \) is calculated as \( \frac{2\pi}{b} \). Here \( b = 6 \), so the period \( P \) is \( \frac{2\pi}{6} \) or \( \frac{\pi}{3} \).
***In summary, input the amplitude as 1 and the period as \( \frac{\pi}{3} \).***](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e8b5006-f396-4197-9ccc-50e519f20a0a%2F0a29c34f-7c09-4992-86cf-578c3bd1efb5%2Fq7lsnvt_processed.png&w=3840&q=75)
Transcribed Image Text:### Graphing Trigonometric Functions
**Objective:**
Graph two full periods of the function \( f(x) = \cos(6x) \) and determine the amplitude and period of the function.
**Instructions:**
1. **Graph the Function:**
- Graph \( f(x) = \cos(6x) \) over a span of two full periods.
2. **Determine the Amplitude and Period:**
- Enter the exact values for amplitude and period.
- For the number \( \pi \), either:
- Choose \( \pi \) from the drop-down menu.
- Or type in Pi (with a capital P).
**Amplitude:**
\[ A = \, \text{[Number input field]} \]
**Period:**
\[ P = \, \text{[Equation input field with various mathematical symbols and functions available]} \]
**Helpful Symbols Provided:**
- \( a^b \)
- \( \frac{a}{b} \)
- \( \sqrt{a} \)
- \( |a| \)
- \( \pi \)
- \( \sin(a) \)
**Explanation of the Graph and Input Fields:**
The graphing area isn't displayed in the provided image, but typically in a trigonometric graphing exercise, you would see a coordinate system where the cosine wave is periodically oscillating. The function \( \cos(6x) \) will have six times the frequency of the basic cosine function \( \cos(x) \), thus making its period shorter.
**Details:**
- **Amplitude (A):** The amplitude of a trigonometric function like cosine is the maximum value the function reaches from its mean position. For \( f(x) = \cos(6x) \), the amplitude is \( 1 \) since the coefficient of \( \cos \) is \( 1 \).
- **Period (P):** The period of a function \( \cos(bx) \) is calculated as \( \frac{2\pi}{b} \). Here \( b = 6 \), so the period \( P \) is \( \frac{2\pi}{6} \) or \( \frac{\pi}{3} \).
***In summary, input the amplitude as 1 and the period as \( \frac{\pi}{3} \).***
Transcribed Image Text:# Function Graph Identification
## Select the correct graph of the function \( f(x) = \cos(6x) \).
Below are five different graphs, each representing a trigonometric function. Your task is to identify which one accurately represents the function \( f(x) = \cos(6x) \).
**Graph 1:**
The graph showcases a trigonometric wave with a high frequency, oscillating between y-values of -1 and 1. The complete cycle appears to be compressed multiple times within a standard period.
**Graph 2:**
This graph illustrates a smoother, lower-frequency wave, with one complete cycle fitting within the standard period range on the x-axis. The amplitude again ranges between -1 and 1.
**Graph 3:**
Similar to Graph 1 but with the wave peaks and troughs occurring at different intervals. This graph also depicts high frequency, oscillating between -1 and 1.
**Graph 4:**
Another high-frequency wave, like Graph 1 and 3, but with different phase shifts. It also oscillates between the y-values of -1 and 1.
**Graph 5:**
This graph presents a wave of even lower frequency than the others, fitting fewer cycles within the same period. The amplitude remains within the -1 to 1 range.
### Explanation:
The function \( f(x) = \cos(6x) \) describes a cosine wave with a frequency factor of 6, meaning the wave should complete six full cycles within a standard period. The correct graph should depict a relatively high-frequency wave consistent with this characteristic.
### Selection:
Based on the description, **Graph 1** is the correct representation of the function \( f(x) = \cos(6x) \) as it most accurately depicts the high-frequency oscillation expected from the cosine function with a frequency factor of 6.
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