The parabola has the same "level'"/values of "y" at x= 2 and x= 4;


hence. the symmetry line is  x= 3, and the parabola has the vertex form


    y = .


At x= 2,  we have  3 =  = 1 + c;  hence, in this vertex form, c= 2.   


In the "general form equation"


     y = x^2 - 6x + 9 + 2 = x^2 - 6x + 11,    c = 11.    ANSWER

Substitute (x,y) = (2,3) in

     y = x^2 + bx + c
     3 = 2^2 + b(2) + c
     3 = 4 + 2b + c
    -1 = 2b + c
2b + c = -1

Substitute (x,y) = (4,3) in

     y = x^2 + bx + c
     3 = 4^2 + b(4) + c
     3 = 16 + 4b + c
   -13 = 4b + c
4b + c = -13

Solve this system:



and get b = -6, c = 11

Edwin