Determine the period, amplitude and phase shift for each given function:
A)y = -4 cos 3x + 5
B)y = 2/3 sin (30x-90degrees)-10
c)y = -0.38 tan (x/3+pi/3)
d)y = pi cos(2x)+ pi
To determine the period, amplitude, and phase shift for each given function, we'll follow the general patterns for each type of function.
A) y = -4 cos 3x + 5:
The general form of the cosine function is y = A cos(B(x-h)) + k, where A represents the amplitude, B represents the period, h represents the horizontal shift, and k represents the vertical shift.
In this case, A = -4 since the coefficient in front of cos is the amplitude.
The period, T, is given by 2π/B, so B = 3, and the period is 2π/3.
The function has no horizontal shift (h = 0) and a vertical shift of 5 units upward (k = 5).
Therefore, for function A, the amplitude is 4, the period is 2π/3, and there is no phase shift.
B) y = 2/3 sin (30x-90 degrees)-10:
Similarly, the general form of the sine function is y = A sin(B(x-h)) + k.
In this case, A = 2/3, which is the amplitude.
The period, T, is given by 2π/B, so B = 30, and the period is 2π/30 or π/15.
The function has a horizontal shift h = 90 degrees to the right (or π/2 radians to the right) and a vertical shift of 10 units downward.
Therefore, for function B, the amplitude is 2/3, the period is π/15, and there is a phase shift of π/2 to the right.
C) y = -0.38 tan (x/3+π/3):
For the tangent function, the general form is y = A tan(B(x-h)) + k.
In this case, A = -0.38, which is the amplitude.
The period of the tangent function is π/B, so B = 1/3, and the period is π/(1/3) = 3π.
The function has a horizontal shift h = -π/3 (or π/3 radians) to the left and no vertical shift (k = 0).
Therefore, for function C, the amplitude is 0.38, the period is 3π, and there is a phase shift of π/3 to the left.
D) y = π cos(2x)+ π:
In this function, A = π, which is the amplitude.
The period of the cosine function is 2π/B, so B = 2, and the period is 2π/2 or just π.
The function has no horizontal shift (h = 0) and a vertical shift of π units upward.
Therefore, for function D, the amplitude is π, the period is π, and there is no phase shift.