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Chapter 4: Problem 45

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

Short Answer

Expert verified
The amplitude is 3, the period is \(\pi\), and the phase shift is \(\pi / 2\) to the right.

Step by step solution

01

Determine Amplitude

The amplitude is given by the absolute value of the coefficient of the cosine function, which in this case is 3.
02

Determine Period

The period of a cosine function is given by \(2\pi\) divided by the absolute value of the coefficient of \(x\). In this case, the coefficient of \(x\) is 2, hence the period of the function is \(\pi\).
03

Determine Phase Shift

The cosine function is shifted by \(\phi\) units to the right if it is of the form \(y = \cos (x - \phi)\). In this case, it is \(\cos (2x - \pi)\), so it is shifted by \(\pi / 2\) to the right. The phase shift is positive as the direction of shift is to right.
04

Plot the Function

Plot the cosine function taking into account the amplitude of 3, the period of \(\pi\), and the phase shift of \(\pi / 2\). This would result in a cosine wave that starts from \(\pi / 2\), goes up till 3, comes down through 0 at \(3\pi/2\), goes down till -3 at \(2\pi\), then goes back to 0 at \(5\pi/2\) completing one cycle or period of \(\pi\).

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Most popular questions from this chapter

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x+\pi)$$

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, \(y=-3 \sec \frac{x}{2} .\) What does your graph from part (a) indicate about this reciprocal function for \(x=-\pi, \pi, 3 \pi,\) and \(5 \pi ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After using the four-step procedure to graph \(y=-\cot \left(x+\frac{\pi}{4}\right),\) I checked my graph by verifying it was the graph of \(y=\cot x\) shifted left \(\frac{\pi}{4}\) unit and reflected about the \(x\) -axis.

Assuming Earth to be a sphere of radius 4000 miles, how many miles north of the Equator is Miami, Florida, if it is \(26^{\circ}\) north from the Equator? Round your answer to the nearest mile.

The number of hours of daylight in Boston is given by $$ y=3 \sin \frac{2 \pi}{365}(x-79)+12 $$ where \(x\) is the number of days after January 1 a. What is the amplitude of this function? b. What is the period of this function? c. How many hours of daylight are there on the longest day of the year? d. How many hours of daylight are there on the shortest day of the year? e. Graph the function for one period, starting on January 1
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