Answer
$y=-4\cdot \cos(4t)$
Work Step by Step
The object starts at the vertical position $y=-a$, and moves up.
By definition, the frequency is $\omega=\frac{2\pi}{T}$, hence the function is: $y=-a\cdot \cos(\omega t)=-a\cdot \cos\left(\frac{2\pi}{T}\cdot t\right)$.
Here, we have
$y=-a\cdot \cos\left(\frac{2\pi}{T} \cdot t\right)=-4\cdot \cos\left(\frac{2\pi}{\frac{\pi}{2}}\cdot t\right)=-4\cdot \cos(4t)$.