Answer
See graph and explanations.
Work Step by Step
Step 1. We are given $y=\frac{x^3+2}{2x}=\frac{1}{2}x^2+\frac{1}{x}$, which can be considered as the sum of two functions.
Step 2. It is easier to graph the function by first plotting the known functions $y=\frac{1}{2}x^2$ (which is a parabola) and $f(x)=\frac{1}{x}$ where $f'(x)=-\frac{1}{x^2}\lt0$ and is thus decreasing over all its open intervals. The asymptotes are $x=0$ and $y=0$ for this function.
Step 3. Finding the first derivative $y'=x-\frac{1}{x^2}$ and $y'=0$ gives $x=1$ as a critical point (plus when $x=0$ where $y'$ is undefined). We have the $y'$ sign changes as $..(-)..(0)..(-)..(1)..(+)..$ which gives decreasing regions of $(-\infty,0),(0,1)$, decreasing regions $(1,\infty)$, and a local minimum at $x=1$.
Step 4. The new asymptotes are $x=0$ only.
Step 5. To graph the given function from $y=\frac{1}{2}x^2$ and $f(x)=\frac{1}{x}$, use the above information and add the corresponding y-coordinates at some key points and connect the points to form curves as shown in the figure.