Question:
What is the sum of all numbers $a$ for which the graph of $y=x^2+a$ and the graph of $y=ax$ intersect one time?

Answer:
If these two graphs intersect then the point of intersection occur when  \[x^2+a=ax,\] or  \[x^2-ax+a=0.\] This quadratic has one solution exactly when the discriminant is equal to zero: \[(-a)^2-4\cdot1\cdot a=0.\] This simplifies to  \[a(a-4)=0.\]

There are exactly two values of $a$ for which the line and parabola intersect one time, namely $a=0$ and $a=4$.  The sum of these values is  \[0+4=\boxed{4}.\]