Answer
$\color{blue}{f(x)=\left(\dfrac{1}{2}\right)^x}$
Work Step by Step
The graph of the function $f(x)=a^x$ contains the point $(-3,8)$.
This means that when $x=-32$, $y=f(-3)=8$.
Substitute $x$ and $y$ into $f(x) =a^x$ to obtain:
$\begin{array}{ccc}
\\&f(x) &= &a^x
\\&f(-3) &= &a^{-3}
\\&8 &= &a^{-3}\end{array}$
Using the rule $a^{-m} = \dfrac{1}{a^m}$, the expression above is equivalent to:
$8=\dfrac{1}{a^3}$
Multiply $a^3$ to both sides of the equation to obtain:
$8(a^3) = \dfrac{1}{a^3} \cdot a^3
\\8a^3=1$
Divide $8$ to both sides to obtain:
$\dfrac{8a^3}{8} = \dfrac{1}{8}
\\a^3=\dfrac{1}{8}$
Note that $\dfrac{1}{8} = \left(\dfrac{1}{2}\right)^3$. Thus, the expression above is equivalent to:
$a^3 = \left(\dfrac{1}{2}\right)^3$
Use the rule "$a^m=b^m \longrightarrow a=b$" to obtain:
$a=\dfrac{1}{2}$
With $a=\frac{1}{2}$, the function whose graph is given is $\color{blue}{f(x)=\left(\dfrac{1}{2}\right)^x}$.
