For the quadratic function y = AX2 + BX + C, if y is an integer when x takes any integer, then we call the image of the function an integral point parabola (for example: y = x2 + 2x + 2). (1) please write an analytic expression of the integral point parabola whose absolute value of the quadratic coefficient is less than 1______ (need not prove) (2) please explore: is there an integral parabola with the absolute value of the quadratic coefficient less than 12? If it exists, please write down the analytical formula of one parabola; if not, please explain the reason

For the quadratic function y = AX2 + BX + C, if y is an integer when x takes any integer, then we call the image of the function an integral point parabola (for example: y = x2 + 2x + 2). (1) please write an analytic expression of the integral point parabola whose absolute value of the quadratic coefficient is less than 1______ (need not prove) (2) please explore: is there an integral parabola with the absolute value of the quadratic coefficient less than 12? If it exists, please write down the analytical formula of one parabola; if not, please explain the reason

(1) For example: y = 12x2 + 12x, y = − 12x2 − 12x, etc. (as long as write a conditional function analytic formula) (2) suppose there is a conditional parabola, then for the parabola y = AX2 + BX + C, when x = 0, y = C, when x = 1, y = a + B + C, from the definition of the integral point parabola, we know that C is an integer, a + B + C is an integer, and a + B must be an integer 2A must be an integer, so a should be an integral multiple of 12, | a | ≥ 12; there is no integral point parabola with the absolute value of quadratic coefficient less than 12

The image intersection points a (m, 5) and B (3, n) of the first-order function y = 2x + 3 and the second-order function y = ax & sup2; + BX + C, and when x = 3, the parabola is the most 9

When x = 3, the function has an extreme value of 9, which means that the vertex coordinates are (3,9)
Because the abscissa of point B is 3, the vertex is point B
We can use the vertex formula, let the function expression be:
y=a(x-3)²+9
Substituting a (m, 5) into a function expression, 2m + 3 = 5, M = 1
The coordinates of point a are (1,5), which are substituted into quadratic function expression
4a+9=5,a=-1
y=-(x-3)²+9
Y = - X & sup2; + 6x is the function expression

Parabola y = - x ^ 2 + 4x + 12, vertex coordinates?

(2,16)

Given that the parabola y = - 1 / 3x & # 178; + BX + C passes through points (3, - 1) and (0, - 4), then the value of B + C is

Substituting the points (3, - 1) and (0, - 4) into the result:
-3+3b+c=-1
c=-4
The solution is: B = 2, C = - 4
Then: B + C = - 2