Is there any reason to preffer using
\begin{axis}
\addplot[blue, ultra thick, domain=-2.25:3.5,samples=200] {(1/2)*(\x)^3-(\x)^2-2*(\x)+4};
\end{axis}
over
\draw[blue, ultra thick, domain=-2.25:3.5,samples=200] plot (\x, {(1/2)*(\x)^3-(\x)^2-2*(\x)+4});
?