Question

How do you determine the period for `y = cos (2pix)`?

Answer 1

You can use the `2pi` in front of `x` in the argument of `cos`.
Calling this number `k` you have that:
`k=(2pi)/(period)`
So in your case you have `period=1`
Your function does an entire oscillation in 1 radian.
graph{cos(2pix) [-2.433, 2.435, -1.215, 1.217]}

Answer 2

Another way to think about it:

`cos(theta)` has a period of `2pi`
You can think of this as meaning that the values
`" from " cos(0) " to " cos(2pi)`
make up one period.

So the question is what values of `x`
would cause `cos(2pi*x)` to take on the range
`cos(0) " to " cos(2pi)`

The answer is obviously `x = 0 " to " 1`
So the period is `1`