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Chapter 4: Problem 6

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. The graph is a reflected and vertically stretched version of the basic sine function.

Step by step solution

01

Identify the amplitude

The amplitude of a sine function is the absolute value of the coefficient in front of the sine. Here, the coefficient in front of \(\sin x\) is -4, so the amplitude is the absolute value of -4, which is 4.
02

Graph the given function

The function \(y = -4 \sin x\) is a downward reflection of the \(\sin x\) function scaled by a factor of 4. Start by graphing the basic sine function on the interval \([0, 2\pi]\). The sine function starts at zero, goes up to 1 at \(\pi/2\) then goes down to -1 at \(\pi\) and back to zero at \(3\pi/2\). For the function \(y = -4 \sin x\), multiply the output values of the sine function by -4, and then plot these points. The graph will start at 0, goes down to -4 at \(\pi/2\), then goes to 4 at \(\pi\), and back to 0 at \(3\pi/2\). Draw a smooth curve through these points.
03

Graph the basic sine function

Graph \(y = \sin x\) on the same set of axes. As already described, \(y = \sin x\) starts at zero, goes up to 1 at \(\pi/2\) then goes down to -1 at \(\pi\) and back to zero at \(3\pi/2\). Draw a smooth curve through these points.

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