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The amplitude of a sine wave is a measure of its peak value. It represents half the distance between the maximum crest and the minimum trough of the wave. In the equation of a sine wave, which is typically written as \(y = A \sin(B(x - C)) + D\), the amplitude is denoted by the variable \(A\). Amplitude is always a positive number, even if the equation has a negative sign before the sine function. This is because amplitude reflects the wave's strength or intensity, not direction.

For example, in the given equation \(y = -3 \sin(x)\), the amplitude is \(3\), despite the negative sign. This negative sign indicates that the sine wave is reflected across the x-axis, essentially flipping it upside down—this does not affect the amplitude itself. When graphing, the peaks of the wave will reach up to 3 units above the axis of equilibrium (in this case, reflecting below due to the negative sign), and the troughs will dip 3 units below.

The period of a sine wave refers to the length of one complete cycle of the wave. It is the horizontal distance required for the wave pattern to repeat itself. The formula for finding the period is \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) in the sine function from the general form \(y = A \sin(B(x - C)) + D\).

In our example, the coefficient \(B\) is 1, as seen in the function \(y = -3 \sin(x)\), which gives us a period of \(2\pi\). Therefore, the graph will complete one full up-and-down wave cycle over a horizontal distance of \(2\pi\) units. When you graph this sine wave, you can mark off intervals of \(\pi/2\) along the x-axis to represent the quarter points of the cycle, which helps to ensure the wave is accurately portrayed.

Phase shift in a sine wave is the horizontal displacement along the x-axis. It tells us how much the wave is shifted to the left or the right from its usual starting point. The general sine function \(y = A \sin(B(x - C)) + D\) has a phase shift represented by \(C\). When \(C\) is positive, the sine wave is shifted to the right; when it is negative, the wave is shifted to the left.

In our exercise, the equation is \(y = -3 \sin(x)\), with no value subtracted from \(x\), which means \(C\) is 0 and there is no phase shift. Thus, the sine wave will start at the origin (0,0) and no horizontal shifting is necessary. If you encounter a sine wave with a non-zero phase shift, it would be necessary to adjust the starting point of your graph accordingly, moving it along the x-axis by the appropriate amount.