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Chapter 4: Problem 4

Determine the amplitude of each function. Then graph the function and \(y=\sin x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\). $$y=\frac{1}{4} \sin x$$

Short Answer

Expert verified
The amplitude of the function \(y = \frac{1}{4} \sin(x)\) is 1/4, and its graph oscillates between -1/4 and 1/4 over the interval 0 to \(2\pi\). Compared to the graph of the standard sine function \(y = \sin(x)\), which has an amplitude of 1, the former graph is 'shorter' but has the same trend and period.

Step by step solution

01

Identify the Amplitude of the Function

The amplitude of a sine function is the absolute value of the coefficient of the sine term. In the function, \(y = \frac{1}{4} \sin(x)\), the coefficient of the sine term is 1/4. So, the amplitude of the function is \(\left| \frac{1}{4} \right| = \frac{1}{4}\).
02

Graph the Function \(y = \frac{1}{4} \sin(x)\) within \(0 \leq x \leq 2 \pi\)

The sine function typically oscillates between -1 and 1 with a period of \(2\pi\). However, given our amplitude is 1/4, our function will oscillate between -1/4 and 1/4. Draw a sinusoidal wave starting from the origin (0, 0) to (2蟺, 0), making sure that the peak and trough of the wave are at 1/4 and -1/4 respectively on the y-axis.
03

Graph the Function \(y = \sin(x)\) within \(0 \leq x \leq 2 \pi\)

This is a standard sine function with amplitude 1 and no phase shift. Draw a sinusoidal wave starting from the origin (0, 0) to (2蟺, 0), with the peak and trough at 1 and -1 respectively on the y-axis. This wave should appear taller than the first one since its amplitude is greater.
04

Compare the Two Graphs

Notice that both functions have the same trend and period. Every high point on the \(y = \sin(x)\) graph corresponds to a high point on the \(y = \frac{1}{4} \sin(x)\) graph, and every low point also corresponds. However, the amplitude or maximum value of \(y = \sin(x)\) is larger than the amplitude of \(y = \frac{1}{4} \sin(x)\), which reflects the comparison of their coefficients of 1 and 1/4 respectively.

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