Question

How do you find the amplitude, period and phase shift for `y = 2 cos (x/2 - Pi/6)`?

Answer 1

`y = 2cos (x/2 - pi/6)`

Explanation:

Amplitude: (-2, 2)
Period of cos x is `2pi` --> period of `x/2` is `4pi.`
Phase shift: `-pi/6`

Answer 2

There are several steps.

Explanation:

The form of this equation is

`y = Acos(Bx+C)" "` or `" "y=Acos(Bx-C)`

(Some textbooks use the first, others use the second.)

The Amplitude is `abs(A)`

Period can be found by `(2pi)/B`

Phase Shift is found by solving:
`Bx+C = 0" "` (or `Bx-C = 0" "` depending on textbook.)

For `y = 2cos(x/2-pi/6)`, note that we can write this as:

`y = 2cos(1/2x-pi/6)" "` (so it is clear that `B = 1/2`)

Amplitude: `" " 2`

Period: `" "4pi`

Found by simplifying `(2pi)/(1/2) = (2pi)/1 * 2/1 = 4pi`

Phase Shift: `" " pi/3`

Found by solving: `x/2-pi/6 = 0` so `x/2 = pi/6` and `x = (2pi)/6 = pi/3`