If that and the fact that the graph contains (3, 0) and (6.25, 10), there are an infinite number of such functions. Obviously, there exist a unique linear function through those points but since you say "the function cannot be linear or quadratic", you could just try a cubic: . You could pick any two of a, b, c, d arbitrarily and use the two conditions to solve for the other two.
Or you could use a sin function: Take y= a sin(x)+ b. Then you must have a sin(3)+ b= 0 and a sin(6.25)+ b= 10. Solve those for a and b.