Answer
The required function is $-2\cos\left(\dfrac{3x}{2}\right)$. The period is $\dfrac{4\pi}{3}$ and amplitude is $2$ units.
Work Step by Step
Note that the maximum value of function is $2$, so, the amplitude is $2$ units.
The function can be obtained using transformation of $\cos{x}$ as follows:
$\underline{\textbf{Step 1}}$: To make maximum value of $2$ and minimum value of $-2$, multiply the standard function by $2$.
$\underline{\textbf{Step 2}}$: From the given graph if we consider one complete wave, it starts from $x=\dfrac{-\pi}{3}$ and ends at $x=\pi$, which has length of $\pi-\dfrac{-\pi}{3}=\dfrac{4\pi}{3}$. Thus, the function repeats itself after every $\dfrac{4\pi}{3}$, so, the period is $\dfrac{4\pi}{3}$. The original wave have period of $2\pi$, so, the wave is compressed horizontally by a factor of $\dfrac{2\pi}{\frac{4\pi}{3}}=\dfrac{3}{2}$.
$\underline{\textbf{Step 3}}$: Note that the given graph shows reflection of the standard function about x-axis, so, multiply the function by $-1$.
Therefore, the function we obtained is $-2\cos\left(\dfrac{3x}{2}\right)$. [Remember that if a function is $f(x)$, it is compressed horizontally by a factor of $k$, then, the new function is $f(k\cdot x)$]
