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Chapter 5: Problem 101

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

Short Answer

Expert verified
The graph of the function \(y=3\sin(2x+\pi)\) varies 3 units above and below the centerline, completes one full cycle from 0 to \(\pi\) and another from \(\pi\) to \(2\pi\) and is shifted left by \(\pi/2\) units.

Step by step solution

01

Identify the amplitude

The amplitude of the function \(y = a \sin(bx + c) + d\) is given by the absolute value of \(a\). For our function \(y = 3 \sin (2x + \pi)\), the amplitude is \(|3|\), which is 3. This means that the graph varies 3 units above and below the midline.
02

Identify the period

The period of the function is given by \(2\pi/\abs{b}\). For our function, \(b = 2\), so the period of the function is \(2\pi/\abs{2} = \pi\). This means one complete cycle of the graph is completed in the interval of \(\pi\). Since we are asked for two periods, the graph has to cover the interval of \(2\pi\).
03

Identify the phase shift

The phase shift of the function is given by \(-c/b\). For our function, \(c = \pi\), hence the phase shift is \(-\pi/2\). The graph will thus be shifted to the left by \(\pi/2\) units.
04

Graph the function

Using the information gathered from steps 1 to 3, it's now possible to sketch the graph. It will rise and fall 3 units from the centreline. The function will complete one full cycle from 0 to \(\pi\) and the other complete cycle from \(\pi\) to \(2\pi\). The graph will also be shifted to the left by \(\pi/2\) units. With all this information in hand, the function is graphed using the graphing tool.

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Most popular questions from this chapter

Exercises \(127-129\) will help you prepare for the material covered in the next section. Determine the amplitude and period of \(y=10 \cos \frac{\pi}{6} x\).

What is an angle?

a. Graph the restricted secant function, \(y=\sec x,\) by restricting \(x\) to the intervals \(\left[0, \frac{\pi}{2}\right)\) and \(\left(\frac{\pi}{2}, \pi\right]\) b. Use the horizontal line test to explain why the restricted secant function has an inverse function. c. Use the graph of the restricted secant function to graph \(y=\sec ^{-1} x\).

In Exercises \(113-116\), use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. In Exercises \(113-114\), convert each angle to a decimal in degrees. Round your answer to two decimal places. $$ 65^{\circ} 45^{\prime} 20^{\prime \prime} $$

Use a graphing utility to graph two periodsof the function. Use a graphing utility to graph \(y=\sin x\) and \(y=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}\) in a \(\left[-\pi, \pi, \frac{\pi}{2}\right]\) by \([-2,2,1]\) viewing rectangle. How do the graphs compare?
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