f(x)=1/3sin(2/3x-π/4)+4

Find the amplitude, period, phase shift, and vertical shift of the function.
Amplitude:1/3
Period:??????
Phase Shift:???????
Vertical Shift:+4
Please Help ?

To find the period and phase shift of the given function, we need to understand the general form of a trigonometric function with the standard form:

f(x) = a*sin(b(x - c)) + d

Where:
a is the amplitude of the graph,
b determines the period of the graph,
c represents the phase shift, and
d is the vertical shift of the graph.

In the given function f(x) = (1/3)sin( (2/3)x - π/4 ) + 4:

Amplitude: The amplitude is the coefficient of the trigonometric function, which in this case is 1/3. So, the amplitude is 1/3.

Vertical Shift: The vertical shift is the constant term added or subtracted at the end of the function, which in this case is +4. So, the vertical shift is +4.

Now, let's find the period and phase shift:

Period: The period is determined by the coefficient of x in the function (denoted by b). In this case, the coefficient of x is (2/3). To find the period, we use the formula: Period = (2π) / |b|. Therefore, the period is (2π) / |2/3| = 3π.

Phase Shift: The phase shift is determined by the value inside the brackets (denoted by c). In this case, the value inside the brackets is -(π/4). To find the phase shift, we equate the expression inside the brackets to 0 and solve for x: (2/3)x - π/4 = 0. Solving for x, we get x = (π/4)(3/2) = 3π/8. Therefore, the phase shift is 3π/8.

In summary:
Amplitude: 1/3
Period: 3π
Phase Shift: 3π/8
Vertical Shift: +4

Thus, the amplitude, period, phase shift, and vertical shift of the given function f(x) = (1/3)sin( (2/3)x - π/4 ) + 4 are as stated above.