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Chapter 4: Problem 1

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\). $$y=4 \sin x$$

Short Answer

Expert verified
The amplitude of the function \(y=4\sin x\) is 4. Graphs of the functions \(y=\sin x\) and \(y=4\sin x\) in the range \(0 \leq x \leq 2 \pi\) would show similar pattern but the later one will be vertically stretched by a factor of 4 which is due to change in amplitude.

Step by step solution

01

Find the Amplitude

The amplitude of a trigonometric function like \(\sin(x)\) is given by the coefficient in front of the function. Here, the coefficient in front of \(sin(x)\) is 4, therefore, the amplitude of the function \(y=4 \sin x\) is 4.
02

Graph y=\(sin(x)\)

On a graph, plot the function \(y=\sin(x)\) from \(0 \leq x \leq 2 \pi\). The pattern of \(sin(x)\) starts from 0 at x=0, reaches its highest point which is 1 at \(x = \pi/2\), goes again to zero at \(x = \pi\), reaches its lowest point which is -1 at \(x = 3\pi/2\), and finally goes back to 0 at \(x = 2\pi\).
03

Graph y=4\(sin(x)\)

Now, on the same graph, plot the function \(y=4\sin(x)\). The pattern remains the same but all the y-values are multiplied by 4 because of the amplitude. It starts from 0 at x=0, reaches its highest point which is 4 at \(x=\pi/2\), again goes to zero at \(x=\pi\), gets to its lowest point which is -4 at \(x=3\pi/2\), and then returns back to 0 at x=\(2\pi\).
04

Compare the Graphs

You should see that the \(4sin(x)\) graph is the same as the \(sin(x)\) graph, but stretched vertically by a factor of 4. This confirms the earlier finding regarding the amplitude. In other words, the amplitude affects the 'height' of the sinusoidal function on a graph.

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