Write an equation for a cosine function with an amplitude of 2/3 , a period of pi, and a vertical shift of 2 units up.

To write an equation for a cosine function with the given parameters, we need to consider the general form of a cosine function:

y = A*cos(Bx - C) + D

Where:
A represents the amplitude,
B represents the frequency (and therefore, the period is given by T = 2π/B),
C represents a phase shift, and
D represents a vertical shift.

Given that the amplitude is 2/3, the period is pi, and the vertical shift is 2 units up, we can substitute these values into the general equation to find the specific equation:

A = 2/3 (amplitude)
T = pi (period)
D = 2 (vertical shift)

Amplitude:
The amplitude of a cosine function is the absolute value of A, so the amplitude value is the same. Therefore, A = 2/3.

Period:
The period is given by T = 2π/B. Since the period is pi, we can solve for B as follows:
pi = 2π/B
B = 2π/pi
B = 2

Vertical Shift:
The vertical shift is given by D, so D = 2.

Now, we have all the necessary values to write the equation:

y = (2/3)*cos(2x) + 2

Therefore, the equation for the cosine function with an amplitude of 2/3, a period of pi, and a vertical shift of 2 units up is:

y = (2/3)*cos(2x) + 2