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Chapter 5: Problem 28

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \sin (2 \pi x+4)$$

Short Answer

Expert verified
The amplitude of the function \(y=3 \sin (2 \pi x+4)\) is 3, the period is 1, and the phase shift is approximately -0.64 or -64% of the period.

Step by step solution

01

Determine the Amplitude

The amplitude of the function is the absolute value of the coefficient A of the sine function. In this case, given the function \(y=3 \sin (2 \pi x+4)\), the amplitude is \(|3|\) which equals 3.
02

Determine the Period

The period of the function is the reciprocal of the coefficient B in front of x inside the sine function. For a sine function, the period is usually \(2pi\) divided by the absolute value of B. Thus, for the given function \(y=3 \sin (2 \pi x+4)\), the period is \(2\pi / |2\pi|\) which is 1.
03

Determine the Phase Shift

The phase shift of the function is the value of C divided by B, then flip the sign. Given the function \(y=3 \sin (2 \pi x+4)\), the phase shift is \(-4 / (2\pi)\), which equals approximately -0.64 or -64% of the period.
04

Graph the Function

The graph of one period of the function can now be sketched based on the amplitude, period, and phase shift. Start by drawing a sine wave that goes up to 3 and down to -3 (amplitude), over a distance of 1 (period). Then shift the wave to the right by the phase shift of approximately -0.64. It's recommended to use a graphing tool for a precise representation.

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Most popular questions from this chapter

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Will help you prepare for the material covered in the next section. a. Graph \(y=-3 \cos \frac{x}{2}\) for \(-\pi \leq x \leq 5 \pi\) b. Consider the reciprocal function of \(y=-3 \cos \frac{x}{2}\) namely, \(y=-3 \sec \frac{k}{2} .\) What does your graph from part (a) indicate about this reciprocal function for \(x=-\pi, \pi, 3 \pi,\) and \(5 \pi ?\)

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Solve: \(\frac{x}{x-3}=\frac{3}{x-3}-\frac{3}{4}\)
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