Answer
$\color{blue}{f(x)=\left(\dfrac{1}{4}\right)^x}$
Work Step by Step
The graph of the function $f(x)=a^x$ contains the point $(2,\frac{1}{16})$.
This means that when $x=2$, $y = \frac{1}{16}$.
Substitute $x$ and $y$ into $f(x) =a^x$ to obtain:
$\begin{array}{ccc}
\\&f(x) &= &a^x
\\&f(2) &= &a^{2}
\\&\dfrac{1}{16} &= &a^{2}\end{array}$
Note that $\dfrac{1}{16} = \left(\dfrac{1}{4}\right)^2$. Thus, the expression above is equivalent to:
$\left(\dfrac{1}{4}\right)^2=a^2$
Use the rule "$a^m=b^m \longrightarrow a=b$" to obtain:
$\dfrac{1}{4} = a$
With $a=\frac{1}{4}$, the function whose graph is given is $\color{blue}{f(x)=\left(\dfrac{1}{4}\right)^x}$.