Determine the period, amplitude and phase shift for each given function:
A)y = -4 cos 3x + 5
B)y = 2/3 sin (30x-90)-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 need to analyze their formulas.
A) For the function y = -4 cos 3x + 5:
The period of a cosine function is determined by the coefficient in front of "x." In this case, the coefficient is 3. The period (T) is given by T = 2π/3, which is approximately 2.09.
The amplitude of a cosine function is the absolute value of the coefficient in front of "cos". In this case, the amplitude is |-4| = 4.
There is no phase shift mentioned in the formula, so the phase shift is 0.
Therefore, for y = -4 cos 3x + 5:
- Period (T) = 2.09
- Amplitude = 4
- Phase shift = 0
B) For the function y = (2/3) sin (30x - 90) - 10:
The period of a sine function is determined by dividing 2π by the absolute value of the coefficient in front of "x" inside the sine function. In this case, the coefficient is 30. The period (T) is given by T = 2π/|30|, which is approximately 0.21.
The amplitude of a sine function is the coefficient in front of "sin," which is (2/3) in this case.
To determine the phase shift, set the argument of the sine function (in parentheses) equal to zero and solve for x:
30x - 90 = 0
30x = 90
x = 3
Therefore, for y = (2/3) sin (30x - 90) - 10:
- Period (T) = 0.21
- Amplitude = 2/3
- Phase shift = 3
C) For the function y = -0.38 tan (x/3 + π/3):
The period of a tangent function is π divided by the absolute value of the coefficient in front of "x" inside the tangent function. In this case, the coefficient is 1/3. The period (T) is given by T = π/|(1/3)|, which is approximately 9.42.
There is no amplitude for the tangent function.
To determine the phase shift, set the argument of the tangent function (in parentheses) equal to zero and solve for x:
x/3 + π/3 = 0
x/3 = -π/3
x = -π
Therefore, for y = -0.38 tan (x/3 + π/3):
- Period (T) = 9.42
- Amplitude = No amplitude
- Phase shift = -π
D) For the function y = π cos(2x) + π:
The period of a cosine function is determined by dividing 2π by the absolute value of the coefficient in front of "x" inside the cosine function. In this case, the coefficient is 2. The period (T) is given by T = 2π/|2|, which is π.
The amplitude of a cosine function is the coefficient in front of "cos," which is π in this case.
There is no phase shift mentioned in the formula, so the phase shift is 0.
Therefore, for y = π cos(2x) + π:
- Period (T) = π
- Amplitude = π
- Phase shift = 0