Find the amplitude, period, and phase shift of the functions:
f(x)=4sec(x+3pi/2)
f(x)=tan((1/3t)-pi)
f(x)=-5sin(4t+3pi)
To find the amplitude, period, and phase shift of each function, we can use the general forms of trigonometric functions:
1. For f(x) = 4sec(x + 3π/2):
Amplitude: The amplitude of the secant function is always equal to the absolute value of the reciprocal of the coefficient of the cosine function. In this case, the coefficient of the cosine function is 4, so the amplitude is 1/4.
Period: The period of the secant function is 2π. There is no transformation affecting the period in this case, so the period remains the same.
Phase Shift: The phase shift of the secant function is equal to the opposite of the value inside the cosine function. In this case, the phase shift is -3π/2.
2. For f(x) = tan((1/3)x - π):
Amplitude: The amplitude of the tangent function is not defined since it is an unbounded function.
Period: The period of the tangent function is π. Since the coefficient of x is 1/3, the period is multiplied by 3 to give a final period of 3π.
Phase Shift: The phase shift of the tangent function is equal to the opposite of the value inside the tangent function. In this case, the phase shift is π.
3. For f(x) = -5sin(4x + 3π):
Amplitude: The amplitude of the sine function is always equal to the absolute value of the coefficient in front of the sine function. In this case, the coefficient is 5, so the amplitude is 5.
Period: The period of the sine function is 2π divided by the coefficient in front of the x. In this case, the coefficient is 4, so the period is 2π/4, which simplifies to π/2.
Phase Shift: The phase shift of the sine function is equal to the opposite of the value inside the sine function. In this case, the phase shift is -3π.
So, the amplitude, period, and phase shift are as follows:
1. Amplitude = 1/4
Period = 2π
Phase shift = -3π/2
2. Amplitude: Not defined for tangent function
Period = 3π
Phase shift = π
3. Amplitude = 5
Period = π/2
Phase shift = -3π
To find the amplitude, period, and phase shift of the given functions, we need to understand the general form of these trigonometric functions and how they relate to these properties.
1. For the function f(x) = 4sec(x + 3π/2):
- Amplitude: The amplitude of the secant function is the absolute value of the reciprocal of the coefficient of the x-term, which is |4| = 4.
- Period: The period of the secant function is 2π, which means that the whole function repeats itself every 2π units.
- Phase Shift: The phase shift of the secant function is in the form (x - h), where h determines the horizontal shift. In this case, x + 3π/2 implies a phase shift of 3π/2 units to the left.
2. For the function f(x) = tan((1/3t) - π):
- Amplitude: The amplitude of the tangent function is not applicable since it does not have a bounded range.
- Period: The period of the tangent function is π, which means that the function repeats after π units.
- Phase Shift: The phase shift of the tangent function is in the form (x - h), where h determines the horizontal shift. In this case, (1/3)t - π implies a phase shift of π/3 units to the right.
3. For the function f(x) = -5sin(4t + 3π):
- Amplitude: The amplitude of the sine function is the absolute value of the coefficient of the trigonometric term, which is |-5| = 5.
- Period: The period of the sine function is calculated by dividing 2π by the coefficient of the t-term, which gives us 2π / 4 = π/2 units.
- Phase Shift: The phase shift of the sine function is in the form (t - h), where h determines the horizontal shift. In this case, 4t + 3π implies a phase shift of -3π/4 units to the left.
Remember, amplitude represents the maximum value of the function, period is the length of one complete cycle, and phase shift represents the horizontal shift.