State the amplitude, period, and phase shift of the function y=-3cos(3x+ (3pi/2))
Amplitude: 3
Period: 2pi/3
Phase Shift: -pi/2
Am I right?
yes
Although I would personally like to see it written as
y = -3cos 3(x + pi/2)
that way the phase shift sort of hits you in the face.
Yes, you are correct!
The given function is in the form y = A*cos(Bx + C), where A is the amplitude, B is the frequency (or 2pi/period), and C is the phase shift.
In this case:
Amplitude = 3 (the coefficient of the cosine function).
Frequency = 3 (the coefficient in front of x, which is also equal to B).
Period = 2pi/B = 2pi/3 (since B = 3, meaning the graph completes 3 cycles in 2pi).
Phase Shift = -C/B = -(3pi/2) / 3 = -pi/2 (since C = (3pi/2), and B = 3).
Therefore, the amplitude is 3, the period is 2pi/3, and the phase shift is -pi/2.
To determine the amplitude, period, and phase shift of the function y = -3cos(3x + (3pi/2)), we can use the general form of the cosine function, which is y = A*cos(Bx + C).
1. Amplitude (A): The amplitude represents the maximum distance the graph of the function extends vertically from the midline. In this case, the coefficient of the cosine function is -3, indicating that the amplitude is 3. So, you are correct in stating that the amplitude is 3.
2. Period (P): The period of the function represents the length of one complete cycle. To find the period, we use the formula P = (2pi)/B, where B is the coefficient inside the cosine function. In this case, B = 3, so the period is P = (2pi)/3. Therefore, the period is (2pi)/3 or 2pi/3. You are correct in stating that the period is 2pi/3.
3. Phase Shift (C/B): The phase shift tells us how the graph of the function is horizontally shifted to the right or left. In the given equation, the phase shift can be found by setting the argument inside the cosine function to 0 and solving for x. So, 3x + (3pi/2) = 0, and by solving this equation, we get x = -pi/2. Therefore, the phase shift is -pi/2. You are correct in stating that the phase shift is -pi/2.
Therefore, your answers are correct:
Amplitude: 3
Period: 2pi/3
Phase Shift: -pi/2