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 will use the general form of the trigonometric functions and identify the corresponding values.
A) y = -4 cos 3x + 5:
1. Period: The period of a cosine function is given by the formula T = 2π/|b|, where b is the coefficient of x. In this case, b = 3, so the period is T = 2π/3.
2. Amplitude: The amplitude of a cosine function is given by the absolute value of the coefficient in front of the cosine function, which is 4 in this case. So the amplitude is 4.
3. Phase shift: There is no phase shift in this function since there is no additional term added to the x variable. Therefore, the phase shift is 0.
B) y = 2/3 sin (30x-90)-10:
1. Period: The period of a sine function is also given by the formula T = 2π/|b|, where b is the coefficient of x. In this case, b = 30, so the period is T = 2π/30 = π/15.
2. Amplitude: The amplitude of a sine function is given by the absolute value of the coefficient in front of the sine function, which is 2/3 in this case. So the amplitude is 2/3.
3. Phase shift: To determine the phase shift, we set the argument inside the sine function (30x-90) equal to zero and solve for x. 30x-90 = 0 => 30x = 90 => x = 3. So the phase shift is 3.
C) y = -0.38 tan (x/3+π/3):
1. Period: The period of a tangent function is given by the formula T = π/|b|, where b is the coefficient of x. In this case, b = 1/3, so the period is T = π/(1/3) = 3π.
2. Amplitude: The amplitude of a tangent function is not well-defined, as the tangent function does not have a maximum or minimum value.
3. Phase shift: To determine the phase shift, we set the argument inside the tangent function (x/3+π/3) equal to zero and solve for x. x/3+π/3 = 0 => x/3 = -π/3 => x = -π. So the phase shift is -π.
D) y = π cos(2x) + π:
1. Period: The period of a cosine function is given by the formula T = 2π/|b|, where b is the coefficient of x. In this case, b = 2, so the period is T = 2π/2 = π.
2. Amplitude: The amplitude of a cosine function is given by the absolute value of the coefficient in front of the cosine function, which is π in this case. So the amplitude is π.
3. Phase shift: There is no phase shift in this function since there is no additional term added to the x variable. Therefore, the phase shift is 0.