Answer
$y=-1\left(2^{x}\right)$
Work Step by Step
The exponential function we want has the form $y=Ab^{x}$.
$\left[\begin{array}{ll}
\text{point on} & \text{corresponding}\\
\text{the graph} & \text{equation}\\
(2,-4) & -4=Ab^{2}\\
(4,-16) & -16=Ab^{4}
\end{array}\right]$
Dividing the equations,
$\displaystyle \frac{-16}{-4}=\frac{Ab^{4}}{Ab^{2}}$
$4=b^{2}$, so
$b=2.$
Back-substituting into the first equation,
$-4=A(2)^{2}$
$-4=4A$
$A=-1$
The model is
$y=Ab^{x}=-1\left(2^{x}\right)$
