Question

Choose the function whose graph is given by:

Answer 1

using a graph tool

I proceed to graph each case

case a)

$y=cos (\frac{1}{2} x)$

see the attached figure N 1

case b)

$y=cos (\frac{1}{4} x)$

see the attached figure N 2

case c)

$y=cos (4x)$

see the attached figure N 3

case d)

$y=cos (2x)$

see the attached figure N 4

therefore

the answer is the option A

$y=cos (\frac{1}{2} x)$

Answer 2

The period of the function $y=\cos (kx)$ is $T=\dfrac{2\pi }{k}$. From the graph you can see that period of given function is $4\pi$ (you can see that at x=0, y=1 and next time y=1 when x=4π ).

Then 

$4\pi =\dfrac{2\pi}{k} ,\\ 4\pi k=2\pi,\\ k=\dfrac{2\pi }{4\pi } =\dfrac{1}{2}$.

For $k=\dfrac{1}{2}$ the function $y=\cos (kx)$ becomes $y=\cos (\dfrac{1}{2} x)$.

Answer: correct choice is A.