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Chapter 4: Problem 5

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\). $$y=-3 \sin x$$

Short Answer

Expert verified
The amplitude of the function \(y=-3 \sin x\) is 3.

Step by step solution

01

Find the Amplitude

The function provided is \(y=-3 \sin x\). The amplitude of a sine function is the absolute value of its coefficient. In this case, the coefficient is -3. Thus, the amplitude is \(|-3|=3\).
02

Plot the Provided Function

To graph \(y=-3 \sin x\), remember that a negative coefficient inverts the basic sine curve. Therefore, instead of beginning and ending at 0 (with a peak at \(\pi/2\) and a trough at \(3\pi/2\)), the graph will now begin and end at 0 (having a trough at \(\pi/2\) and a peak at \(3\pi/2\)). Note that the amplitude - the maximum reach from the midline (x-axis) is 3.
03

Plot the Standard Sine Function

The standard function \(y=\sin x\) starts and ends at 0, reaches a maximum of 1 at \(\pi/2\), and a minimum of -1 at \(3\pi/2\). Plot this function alongside \(y=-3 \sin x\).
04

Compare the Graphs

Observe that \(y=-3 \sin x\) is an upside-down and stretched version of \(y=\sin x\). Its peaks and troughs are three times as far from the x-axis as those of \(y=\sin x\).

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Most popular questions from this chapter

Graph \(y=\sin \frac{1}{x}\) in a [-0.2,0.2,0.01] by [-1.2,1.2,0.01] viewing rectangle. What is happening as \(x\) approaches 0 from the left or the right? Explain this behavior.

Graph one period of each function. $$y=-\left|2 \sin \frac{\pi x}{2}\right|$$

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When an angle's measure is given in terms of \(\pi,\) I know that it's measured using radians.

Use a vertical shift to graph one period of the function. $$y=2 \cos \frac{1}{2} x+1$$
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