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
Find the equation from the graph.
![The image features a graph on a coordinate plane with the horizontal axis labeled as \( w \) and the vertical axis labeled as \( f(w) \). The graph shows a periodic function, likely a sine or cosine function, with peaks at regular intervals.
The key characteristics of the graph are:
1. **Periodicity**: The function exhibits periodic behavior, repeating its pattern at regular intervals along the \( w \)-axis. The period appears to be around 2 units, indicating the full cycle of the function.
2. **Amplitude**: The maximum and minimum values of \( f(w) \) are 6 and -6, respectively. This suggests the function has an amplitude of 6.
3. **Vertical Shift**: The function is centered around the horizontal axis \( f(w) = 0 \), with no apparent vertical shift.
4. **Symmetry**: The function is symmetric with respect to the vertical centerline of each cycle, suggesting it could be a cosine function.
5. **Reflection**: The graph appears to be an even function since it is symmetric around the y-axis, which is characteristic of a cosine function.
**Question**:
Find an equation for the graph above.
**Answer Box**:
[Space for input]
Based on these observations, you could consider a function of the form \( f(w) = A \cos(Bw) \).
- Amplitude (\( A \)) is 6.
- To determine \( B \), note that \( B = \frac{2\pi}{\text{period}} \).
An example equation based on this would be \( f(w) = 6 \cos(\pi w) \), assuming the period is 2 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6039e07c-487e-4ea5-bef7-8c3c22691e12%2Fdf7be520-eaf1-4245-9889-cacc786caeb0%2Ff64bg3n_processed.png&w=3840&q=75)
Transcribed Image Text:The image features a graph on a coordinate plane with the horizontal axis labeled as \( w \) and the vertical axis labeled as \( f(w) \). The graph shows a periodic function, likely a sine or cosine function, with peaks at regular intervals.
The key characteristics of the graph are:
1. **Periodicity**: The function exhibits periodic behavior, repeating its pattern at regular intervals along the \( w \)-axis. The period appears to be around 2 units, indicating the full cycle of the function.
2. **Amplitude**: The maximum and minimum values of \( f(w) \) are 6 and -6, respectively. This suggests the function has an amplitude of 6.
3. **Vertical Shift**: The function is centered around the horizontal axis \( f(w) = 0 \), with no apparent vertical shift.
4. **Symmetry**: The function is symmetric with respect to the vertical centerline of each cycle, suggesting it could be a cosine function.
5. **Reflection**: The graph appears to be an even function since it is symmetric around the y-axis, which is characteristic of a cosine function.
**Question**:
Find an equation for the graph above.
**Answer Box**:
[Space for input]
Based on these observations, you could consider a function of the form \( f(w) = A \cos(Bw) \).
- Amplitude (\( A \)) is 6.
- To determine \( B \), note that \( B = \frac{2\pi}{\text{period}} \).
An example equation based on this would be \( f(w) = 6 \cos(\pi w) \), assuming the period is 2 units.
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