Derive a cost production function give prod f, only K In this question, only K is included and L is excluded how would I go about deriving it?

Total cost= Fixed costs + Average costs.

Since the variable input costs r per unit, the variable costs is r times the number of units rQ, hence $VC= rK^\alpha$. 

Thus $C(Q)= c_0 + rK^\alpha$

Is the above solution correct? 
IS there a more mathematical way of doing it, one that involves grpahs etc.
Here you have to express $K$ in terms of $Q$, since the cost depends on the number of units of capital employed and not on the number of products produced. You have
\begin{equation}
Q = K^a \Leftrightarrow K = Q^{\frac{1}{a}}
\end{equation}
so your cost function will be
\begin{equation}
C(Q) = c_0 + rK(Q) = c_0 + rQ^{\frac{1}{a}}
\end{equation}