Question

How do you graph `y = 1/2cos( 4x )`?

Answer 1

See explanation, graph{(1/2)cos (4x) [-10, 10, -5, 5]}

Explanation:

You have: `y=1/2 cos(4x)`

Well, the easiest way is to start from the known function `cos(x)`
which can be drawn as such:
graph{cos x [-10, 10, -5, 5]}

The cosine function is 1 at `x=0`.
The cosine function is 0 at `x=pi/2`.

That is, our function will be 0 when the inner term of the cosine function reaches `pi/2`.
But we have `(4x)` inside our cosine.
So this means that our cosine function reaches 0
when `4x=pi/2`
or after rearranging, when
`x=pi/8` (and `-pi/8` and so on).

The following is the graph of `cos(4x)`:
graph{cos (4x) [-10, 10, -5, 5]}
The factor "4" actually compresses the cosine wave along the x-axis.
(Note: if the factor were between 0 and 1, say, for example, 0.5, then `cos(0.5x)` would expand the cosine wave along the x-axis.)

Finally, we have an external multiplicative factor of `1/2`, which compresses the "height" of our cosine wave (along the y-axis this time) in half.
graph{(1/2)cos (4x) [-10, 10, -5, 5]}

That's it. Hope this helps.