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(a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.) amplitude       period       horizontal shift           (b) Write an equation that represents the curve in the form y = a cos(k(x − b)).

(a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.) amplitude       period       horizontal shift           (b) Write an equation that represents the curve in the form y = a cos(k(x − b)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

(a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.)

amplitude      
period
 
 
 
horizontal shift    
 
 
 


(b) Write an equation that represents the curve in the form

y = a cos(k(x − b)).
The image displays a graph of the function \( y = \sin(2x) \). ### Graph Details: - **Axes**: The graph is plotted with the x-axis ranging from \(-\frac{\pi}{4}\) to \(\frac{3\pi}{4}\) and the y-axis ranging from \(-\frac{1}{3}\) to \(\frac{1}{3}\). - **Curve**: The curve represents the sine function \( y = \sin(2x) \), which has been adjusted to fit within the given x and y axis limits. This function oscillates between a maximum and minimum value, creating a wave-like pattern. - **Peak and Trough**: The peak of the wave occurs near the center of the graph, and the troughs occur equally around this peak, demonstrating the periodic nature of the sine function. ### Educational Context: This graph illustrates a basic transformation of the sine function, showcasing how the wave is affected by multiplying the angle \( x \) by 2, resulting in a frequency increase. This is a useful representation for understanding trigonometric transformations and the behavior of sine waves.
Transcribed Image Text:The image displays a graph of the function \( y = \sin(2x) \). ### Graph Details: - **Axes**: The graph is plotted with the x-axis ranging from \(-\frac{\pi}{4}\) to \(\frac{3\pi}{4}\) and the y-axis ranging from \(-\frac{1}{3}\) to \(\frac{1}{3}\). - **Curve**: The curve represents the sine function \( y = \sin(2x) \), which has been adjusted to fit within the given x and y axis limits. This function oscillates between a maximum and minimum value, creating a wave-like pattern. - **Peak and Trough**: The peak of the wave occurs near the center of the graph, and the troughs occur equally around this peak, demonstrating the periodic nature of the sine function. ### Educational Context: This graph illustrates a basic transformation of the sine function, showcasing how the wave is affected by multiplying the angle \( x \) by 2, resulting in a frequency increase. This is a useful representation for understanding trigonometric transformations and the behavior of sine waves.
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