Determine the number of cycles the following sine function has in the interval from 0 to 2π. Find the amplitude and period of the function.
y = -6 sin π0
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The given sine function is y = -6 sin(πx).
To determine the number of cycles in the interval from 0 to 2π, we need to find the number of times the sine function completes a full oscillation within this range.
The general form of the sine function is y = A sin(Bx + C), where A represents the amplitude, B represents the frequency (related to the period), and C represents any phase shift.
In this case, our function is y = -6 sin(πx). Notice that there is no phase shift (C = 0) and B = π.
The amplitude of the function is represented by the coefficient A, which is 6 in this case (negative because of the negative sign before the sine function). Therefore, the amplitude is 6.
The period of the function is given by T = (2π)/|B|, where B is the frequency. In this case, the frequency is π. Therefore, the period is T = (2π)/π = 2.
So, the number of cycles in the interval from 0 to 2π is 1, the amplitude is 6, and the period is 2.