Find the amplitude, period, and phase shift of the function defined by each of the following equations, and sketch the graph of one complete period.
(3.1) [tex]$y=4-2 \sin (x)$[/tex]
(3.2) [tex]$y=\frac{1}{2} \cos \left(2 x-\frac{\pi}{3}\right)-1$[/tex]
Answer (2)
For y = 4 − 2 sin ( x ) , the amplitude is 2, the period is 2 π , and the phase shift is 0.
For y = 2 1 cos ( 2 x − 3 π ) − 1 , rewrite as y = 2 1 cos ( 2 ( x − 6 π )) − 1 . The amplitude is 2 1 , the period is π , and the phase shift is 6 π .
The vertical shift for y = 4 − 2 sin ( x ) is 4, and for y = 2 1 cos ( 2 x − 3 π ) − 1 is -1.
The final answers are: Amplitude = 2, Period = 2 π , Phase Shift = 0 for the first equation, and Amplitude = 2 1 , Period = π , Phase Shift = 6 π for the second equation. A m pl i t u d e = 2 , P er i o d = 2 π , P ha se S hi f t = 0 ; A m pl i t u d e = 2 1 , P er i o d = π , P ha se S hi f t = 6 π
Explanation
Problem Overview We are asked to find the amplitude, period, and phase shift of two given trigonometric functions and then sketch one complete period of each.
Analysis of the First Equation (3.1) The given equation is y = 4 − 2 sin ( x ) . We can rewrite this as y = − 2 sin ( x ) + 4 . This is a sine function that has been vertically shifted, amplitude changed, and reflected across the x-axis.
Finding the Amplitude The amplitude of the function is the absolute value of the coefficient of the sine function, which is ∣ − 2∣ = 2 .
Determining the Period The period of the standard sine function sin ( x ) is 2 π . Since the argument of the sine function here is simply x , the period remains 2 π .
Identifying the Phase Shift There is no phase shift in this function because there is no horizontal translation in the argument of the sine function.
Finding the Vertical Shift The vertical shift is given by the constant term, which is 4. This means the entire graph is shifted upward by 4 units.
Analysis of the Second Equation (3.2) The given equation is y = 2 1 cos ( 2 x − 3 π ) − 1 . We can rewrite this as y = 2 1 cos ( 2 ( x − 6 π )) − 1 . This is a cosine function with amplitude change, period change, phase shift, and vertical shift.
Finding the Amplitude The amplitude of this function is the absolute value of the coefficient of the cosine function, which is ∣ 2 1 ∣ = 2 1 .
Determining the Period The period of the standard cosine function cos ( x ) is 2 π . However, the argument of the cosine function is 2 x , which means the period is compressed by a factor of 2. Therefore, the period is 2 2 π = π .
Identifying the Phase Shift The phase shift can be found by looking at the horizontal translation inside the cosine function. We have 2 ( x − 6 π ) , which means the phase shift is 6 π to the right.
Finding the Vertical Shift The vertical shift is given by the constant term, which is -1. This means the entire graph is shifted downward by 1 unit.
Final Answer Summary In summary: (3.1) For y = 4 − 2 sin ( x ) :
Amplitude: 2
Period: 2 π
Phase Shift: 0 (3.2) For y = 2 1 cos ( 2 x − 3 π ) − 1 :
Amplitude: 2 1
Period: π
Phase Shift: 6 π
Examples
Understanding the characteristics of trigonometric functions, such as amplitude, period, and phase shift, is crucial in many real-world applications. For example, in electrical engineering, these parameters help describe alternating current (AC) waveforms. The amplitude represents the peak voltage or current, the period determines the frequency of the AC signal, and the phase shift indicates the relative timing between different signals. By analyzing and manipulating these parameters, engineers can design and optimize circuits for various applications, such as power transmission, signal processing, and communication systems. Similarly, in acoustics, these parameters are used to describe sound waves, where amplitude corresponds to loudness, period to frequency (pitch), and phase shift to the relative timing of sound waves from different sources.
The first function, y = 4 − 2 sin ( x ) , has an amplitude of 2, a period of 2 π , and no phase shift. The second function, y = 2 1 cos ( 2 x − 3 π ) − 1 , has an amplitude of 2 1 , a period of π , and a phase shift of 6 π to the right.
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