y-axis as directrix, it looks like this: x = ay^2 + by + c and a > 0
Passing through (5,1), we have 5 = a + b + c
Passing through (13,-7), we have 13 = 49a - 7b + c
x-coordinate of Vertex must be on (0,5].
This all leads to a = (1/7)*(6-c), so 6-c > 0 or c < 6
Finally, b = (1/7)*(29-6c) and this suggests c > 2.27, approximately. I'll let you pin it down exactly.
Where does that leave us?