Question

How do you determine the amplitude, period and vertical translation of `y+2=4cos(x/2)`?

Answer 1

Amplitude`=4`
Period`=4pi`
Vertical Translation`=-2`

Explanation:

I would write your function isolating `y` as:
`y=4cos(x/2)-2`
From this you can "see":
1] The amplitude of your cosine is `4` (the number in front of `cos`);
2] The period is `4pi`; by using the `1/2` inside the argument of `cos` you get: `period=(2pi)/color(red)(1/2)=4pi`;
3] The vertical translation is given by `-2` telling you that your `cos` oscillates about the horizontal line passing through `y=-2` (instead of oscillating about the `x` axis!).

Graphically:
graph{4cos(x/2)-2 [-16.02, 16.02, -8.01, 8.01]}

As you can see you have a cosine oscillating between `2` and `-6` about the horizontal line at `y=-2` and one complete oscillation now takes `4pi` to be completed.