Question

what is the function of cos that has an amplitude of 4 period of 2x and has a point at (0,2)

Answer 1

the amplitude should be 4, therefore it should be 4cosx.
period should be 2x... this doesn't quite make sense
I think the question is that the period should be 2pi

Answer 2

So how do i answer the question?

Answer 3

If you want an amplitude of A and a period of \(T\), then the equation you want is
$$
A\cos(\cfrac{2\pi}{T}t)
$$
Where \(t\) is the independent variable.

Now, to make this function go through <0,2>, use the phase, \(\theta\):
$$
y=A\cos(\cfrac{2\pi}{T}t+\theta)
$$
The objective then is to find the phase such that y passes through <0,2>:
$$
2=A\cos(\cfrac{2\pi}{T}(0)+\theta)\\
2=A\cos(\theta)\\
\implies \theta = \arccos \cfrac{2}{A}
$$

That's it!

Make sense?