Which cosine function has maximum of 2, a minimum of -2, and a period of 2pi/3?
The general form of a cosine function is:
f(x) = A cos(Bx + C) + D
where A is the amplitude, B is the frequency (or 1/period), C is the phase shift, and D is the vertical shift.
To find the specific cosine function that has a maximum of 2, a minimum of -2, and a period of 2pi/3, we need to use the following information:
- The amplitude is the distance from the maximum to the minimum, so A = (2 - (-2))/2 = 2.
- The frequency is the inverse of the period, so B = 1/(2pi/3) = 3/2pi.
- The phase shift is 0 because the function starts at the maximum.
- The vertical shift is the average of the maximum and minimum, so D = (2 + (-2))/2 = 0.
Therefore, the cosine function that satisfies these conditions is:
f(x) = 2 cos(3x/2pi)