There is an image of a quadratic function. Three students have described some of its characteristics respectively: A: the axis of symmetry is a straight line x = 4; B: the abscissa of the two intersections with the X axis is an integer; C: the ordinate of the intersection with the Y axis is also an integer, and the triangular area product with the three intersections as the vertex is 24. Please determine an analytic formula of quadratic function that satisfies all the above characteristics

There is an image of a quadratic function. Three students have described some of its characteristics respectively: A: the axis of symmetry is a straight line x = 4; B: the abscissa of the two intersections with the X axis is an integer; C: the ordinate of the intersection with the Y axis is also an integer, and the triangular area product with the three intersections as the vertex is 24. Please determine an analytic formula of quadratic function that satisfies all the above characteristics

Let the coordinates of the intersection of the image of quadratic function and X axis be (x1,0), (x2,0), and the analytic formula be y = a (x-x1) (x-x2). From a we know that X1 + x2 = 8, from B we know that X1 and X2 are all integers. Let x1 = 2, X2 = 6, y = a (X-2) (X-6) = a (x2