By looking at a graph of a parabola. How would one tell what the equation for the line of symmetry is?

if you draw a line paralel with Ox/o than it will intersect the parabola in two points. You can calculate the coordonates of the point which is at the middle of the intersection points. The middles of those points will always be on the symethry line. If you can try it I think you might understand.