Sketch the graph of the given equation over the interval [- 2n, 2n]. y = sin (8x) Use the graphing tool to graph the equation. Type pi to insert n as needed.
Sketch the graph of the given equation over the interval [- 2n, 2n]. y = sin (8x) Use the graphing tool to graph the equation. Type pi to insert n as needed.
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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![## 5.4.15
### Instruction:
Sketch the graph of the given equation over the interval \( [ -2\pi, 2\pi ] \).
\[ y = \sin( 8x ) \]
Use the graphing tool to graph the equation. Type `pi` to insert \( \pi \) as needed.
### Graph Explanation:
The provided function is \( y = \sin( 8x ) \), which is a sine function with a frequency 8 times higher than the standard sine function \( y = \sin( x ) \).
1. **Amplitude**: The amplitude of the function remains constant at 1, which is the coefficient of the sine function.
2. **Period**: The period of the sine function \( y = \sin( kx ) \) is given by \( \frac{2\pi}{k} \). Therefore, for the given function \( y = \sin( 8x ) \), the period is \( \frac{2\pi}{8} = \frac{\pi}{4} \).
3. **Interval**: The graphing interval is \( [ -2\pi, 2\pi ] \). This means you will need to plot the graph from -2π to 2π on the x-axis.
By following these points, you will obtain a graph with recurring waves. Each wave will cycle every \( \frac{\pi}{4} \) units along the x-axis, displaying a total of \( \frac{4\pi}{\pi/4} = 16 \) cycles between \( -2\pi \) and \( 2\pi \). Use the graphing tool to visualize this sine wave properly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1668550e-16d3-480e-a17e-3f1198f7538c%2Fe62ea73b-25e8-4be4-a7f6-c82bbe211932%2Fijt4fxf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## 5.4.15
### Instruction:
Sketch the graph of the given equation over the interval \( [ -2\pi, 2\pi ] \).
\[ y = \sin( 8x ) \]
Use the graphing tool to graph the equation. Type `pi` to insert \( \pi \) as needed.
### Graph Explanation:
The provided function is \( y = \sin( 8x ) \), which is a sine function with a frequency 8 times higher than the standard sine function \( y = \sin( x ) \).
1. **Amplitude**: The amplitude of the function remains constant at 1, which is the coefficient of the sine function.
2. **Period**: The period of the sine function \( y = \sin( kx ) \) is given by \( \frac{2\pi}{k} \). Therefore, for the given function \( y = \sin( 8x ) \), the period is \( \frac{2\pi}{8} = \frac{\pi}{4} \).
3. **Interval**: The graphing interval is \( [ -2\pi, 2\pi ] \). This means you will need to plot the graph from -2π to 2π on the x-axis.
By following these points, you will obtain a graph with recurring waves. Each wave will cycle every \( \frac{\pi}{4} \) units along the x-axis, displaying a total of \( \frac{4\pi}{\pi/4} = 16 \) cycles between \( -2\pi \) and \( 2\pi \). Use the graphing tool to visualize this sine wave properly.
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