Answer
$\begin{cases}
x=\cos t\\
y=4\sin t
\end{cases}$
with $-\dfrac{\pi}{2}\leq t\leq \dfrac{\pi}{2}$
Work Step by Step
The given curve is half an ellipse centered at the origin with the elements:
$a=4$
$b=1$
The equation of the ellipse is:
$\dfrac{x^2}{1^2}+\dfrac{y^2}{4^2}=1$
$\dfrac{x^2}{1}+\dfrac{y^2}{16}=1$
Find parametric equations that define an ellipse:
$\begin{cases}
x=b\cos t\\
y=a\sin t
\end{cases}$
with $0\leq t\leq 2\pi$
Substitute $a$ and $b$ and determine the domain for the half ellipse:
$\begin{cases}
x=\cos t\\
y=4\sin t
\end{cases}$
The starting point $(0,-4)$:
$\begin{cases}
\cos t=0\\
4\sin t=-4
\end{cases}$
$\Rightarrow t=-\dfrac{\pi}{2}$
The ending point $(0,4)$:
$\begin{cases}
\cos t=0\\
4\sin t=4
\end{cases}$
$\Rightarrow t=\dfrac{\pi}{2}$
The parametric equations are:
$\begin{cases}
x=\cos t\\
y=4\sin t
\end{cases}$
with $-\dfrac{\pi}{2}\leq t\leq \dfrac{\pi}{2}$
