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Chapter 4: Problem 10

Determine the amplitude and period of each function. Then graph one period of the function. $$y=2 \sin \frac{1}{4} x$$

Short Answer

Expert verified
The amplitude of the function \(y = 2\sin\left(\frac{1}{4}x\right)\) is 2 and the period is \(8\pi\). In the graph, the function starts at y=0 at x=0, reaches 2 at \(2\pi\), returns to 0 at \(4\pi\), goes to -2 at \(6\pi\), and finally back to 0 at \(8\pi\).

Step by step solution

01

Determine the Amplitude

The amplitude of the function is typically the coefficient in front of the sine function. So here, the amplitude of the function \(y = 2\sin\left(\frac{1}{4}x\right)\) is 2.
02

Determine the Period

The period of the sine function is given by \(2\pi/B\). In this case, B equals to 1/4, so the calculation gives \(2\pi/\frac{1}{4} = 8\pi\). So, the period of the function \(y = 2\sin\left(\frac{1}{4}x\right)\) is \(8\pi\).
03

Graph the Function

To graph the function, first set up the x-axis from 0 to \(8\pi\), as this represents one period of the function. Mark the midpoint of the period at \(4\pi\), and the quarter points at \(2\pi\) and \(6\pi\). The y-values will range from -2 to 2 (negative to positive amplitude). The graph starts at y=0 at x=0 (origin), reaches the maximum amplitude at \(2\pi\), returns to 0 at \(4\pi\), goes to the minimum amplitude at \(6\pi\), and finally back to 0 at \(8\pi\), completing one period.

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