3. f(x)=1/3sin(2/3x-π/4)+4
Find the amplitude, period, phase shift, and vertical shift of the function please.
Take a look at this,and memorize it.
f(x)=Amplitude*sin(2PI*time/period-shiftToRight)+vertical shift
I am bothered by the use of x as a variable. Normally, we use t for time. If x here represents distance, then period has NO meaning. Period applies to time varying signals.
http://www.regentsprep.org/Regents/math/algtrig/ATT7/phaseshift.htm
To find the amplitude, period, phase shift, and vertical shift of the function f(x)=1/3sin(2/3x-π/4)+4, you can use the general form of a sinusoidal function: f(x) = A*sin(B(x - C)) + D.
In this case, the equation f(x)=1/3sin(2/3x-π/4)+4 is already written in the general form, so we can simply identify the values of A, B, C, and D:
Amplitude (A): The amplitude of the function is the absolute value of the coefficient of the sine term. In this case, the amplitude is |1/3| = 1/3.
Period (P): The period of the function can be found using the formula P = 2π/|B|. In this case, B = 2/3, so the period is P = 2π/|(2/3)| = 2π/(2/3) = 3π.
Phase Shift (C): The phase shift of the function is given by the expression (C/B). In this case, C = π/4 and B = 2/3, so the phase shift is (π/4) / (2/3) = (3π/4) / (2/3) = 3π/8.
Vertical Shift (D): The vertical shift is the value of the term D, which is added to the function. In this case, D = 4.
Therefore, the amplitude is 1/3, the period is 3π, the phase shift is 3π/8, and the vertical shift is 4.
To find the amplitude, period, phase shift, and vertical shift of the function f(x) = 1/3sin(2/3x - π/4) + 4, you can use the standard form of a sinusoidal function:
f(x) = A*sin(Bx - C) + D
In this form, A represents the amplitude, B represents the frequency (which affects the period), C represents the phase shift, and D represents the vertical shift.
1. Amplitude (A):
The amplitude of a sinusoidal function is the absolute value of the coefficient of the sine function, which is the number in front of sin(Bx - C). In this case, A = 1/3, so the amplitude is 1/3.
2. Period:
To find the period, you need to identify the value of B. In this case, B = 2/3. The period of a sinusoidal function is found using the formula T = 2π/|B|. Therefore, the period is T = 2π/|2/3| = 2π*3/2 = 3π.
3. Phase Shift:
The phase shift can be determined by setting the argument of the sine function (Bx - C) equal to zero and solving for x. In this case, 2/3x - π/4 = 0. Solving for x, we find x = π/4 * (3/2) = 3π/8. Therefore, the phase shift is 3π/8 to the right (since it is positive).
4. Vertical Shift:
The vertical shift is given by the value of D. In this case, D = 4. Thus, the function is shifted vertically upwards by 4 units.
To summarize:
- Amplitude: 1/3
- Period: 3π
- Phase Shift: 3π/8 (to the right)
- Vertical Shift: 4