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Chapter 5: Problem 29

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-2 \sin (2 \pi x+4 \pi)$$

Short Answer

Expert verified
Amplitude = 2, Period = 1, Phase shift = -2

Step by step solution

01

Finding Amplitude

The amplitude is the absolute value of A in the standard form of the sinusoidal function. In this case, A is -2, so the amplitude is |-2| = 2.
02

Finding Period

The period is found by \(2\pi/B\). In this function B is \(2\pi\). Therefore, the period is \(2\pi / 2\pi\ = 1\), which means the function cycles once every unit length along the x-axis.
03

Finding the Phase Shift

The phase shift is found by solving for 'x' in \(Bx + C = 0\). Here, B is \(2\pi\) and C is \(4\pi\). Solving for 'x' gives \(2\pi x = -4\pi\). Dividing both sides by \(2\pi\) yields \(x = -2\). Therefore, the function is shifted 2 units to the left.
04

Graphing the Function

Plot the function by marking a complete cycle from -2 to -1 on the x-axis (since the period is 1 and it's shifted 2 units left). The amplitude is 2, so mark points at +2 and -2 on the y-axis. Draw a sinusoidal curve, descending from the maximum to the minimum since A was negative.

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