It is known that a, B and C are positive integers, and the quadratic function y = AXX + BX + C. when x is greater than or equal to - 2 and less than or equal to 1, y is larger If y is greater than or equal to - 1 and less than or equal to 7, find the analytic expression of quadratic function

It is known that a, B and C are positive integers, and the quadratic function y = AXX + BX + C. when x is greater than or equal to - 2 and less than or equal to 1, y is larger If y is greater than or equal to - 1 and less than or equal to 7, find the analytic expression of quadratic function

1) When - B / 2A ≤ - 2, even if B ≥ 4a,
Ymin=4a-2b+c=-1
Ymax=a+b+c=7
From the above two formulas, we can get: 6A + 3C = 13, because a, B and C are all positive integers. Substituting a = 1 or 2, we can see that C is not an integer. Therefore, there is no solution under this condition
2) When - 2 ≤ - B / 2A ≤ - 1 / 2, i.e. a ≤ B ≤ 4a,
In this case, the symmetry axis of quadratic function x = - B / 2a is on the left side of x = - 1 / 2, and the parabola opening is upward, so
Ymin = (4ac-b Λ 2) / 4A = - 1, that is, B Λ 2 = 4A (c + 1) ≥ 8A
Ymax = a + B + C = 7, that is, C = 7-a-b
It can be seen from the above two formulas: B Λ 2 = 4A (8-a-b), when a = 1, B is not an integer; when a = 2, the positive integer B = 4, then C = 1; when a = 3, then B is not an integer; when a = 4, B Λ 2 > 32, b > 5, a + b > 9 does not hold
So a = 2, B = 4, C = 1
3) When - 1 / 2 ≤ - B / 2A