write an equation of a cosine function with Amplitude=1/2, Period=6pi, and Phase Shift=pi/3?
To write the equation of a cosine function, we need to use the general form:
y = A * cos(B(x - C)) + D
Where:
A represents the amplitude.
B represents the frequency.
C represents the phase shift.
D represents the vertical shift.
Given:
Amplitude (A) = 1/2
Period = 6π
Phase Shift (C) = π/3
To find the frequency (B) from the period, we can use the formula:
Frequency = 2π / Period
Substituting the given value, we get:
B = 2π / (6π)
B = 1/3
Now let's substitute the known values into the general equation:
y = (1/2) * cos((1/3)(x - π/3)) + D
Since the amplitude is 1/2, the vertical shift (D) will also be 1/2 to center the graph. Thus, D = 1/2.
Substituting this value into the equation, we have:
y = (1/2) * cos((1/3)(x - π/3)) + 1/2
Therefore, the equation of the cosine function with an amplitude of 1/2, period of 6π, and phase shift of π/3 is:
y = (1/2) * cos((1/3)(x - π/3)) + 1/2