It is known that the square of quadratic function y = ax + BX + C and a point of intersection coordinate of X axis is (8,0), and if x = 6 is y, it has the minimum negative value of 12, the analytic expression of quadratic function is obtained

When x = 6 is y, the minimum value is negative 12, that is, the vertex coordinates are (6, - 12)
The axis of symmetry is x = 6, and the point of intersection with the X axis is (8,0)
So the other intersection is (4,0)
Let y = a (x-4) (X-8)
Substituting (6, - 12) into: - 12 = a (6-4) (6-8), a = 3
So, y = 3 (x-4) (X-8)

It is known that the square of quadratic function y = ax + BX + C passes through O (0,0), a (4,0), and when x ∈ R, there is a minimum value - 4
(1) , find the analytic expression of quadratic function
(2) When 1 ≤ x ≤ 5, find the value range of Y
Note: A is not equal to 0, R is a real number set

What is behind your question? Please complete it
1) The analytic expression of quadratic function is f = x2-4x if the minimum point is (2, - 4)
2) In [1,2], f monotonically decreases, in [2,5], f monotonically increases, the minimum value of F in [1,5] is f (2) = - 4, f (1) = - 3, f (5) = 5, so as to be [- 4,5]

The graph and properties of quadratic function y = ax ^ 2 + BX + C answer the following questions
(1) What is the relationship between the image of function y = ax ^ 2 and y = x ^ 2?
(2) What is the relationship between the function y = a (x-m) ^ 2 + K and the image of y = ax ^ 2?

(1) When a > 0, the former is 1 / a compression in the horizontal direction of the latter image (obviously, if A1 is stretching); if a > 0, the former is 1 / a compression in the horizontal direction of the latter image

Properties of square + BX + C of quadratic function y = ax

1: When a is greater than 0, the opening of the function is upward. On the left side of the symmetry axis, y decreases with the increase of X; on the right side of the symmetry axis, y increases with the increase of X
When a is less than 0, the opening of the function is downward. On the left side of the symmetry axis, y increases with the increase of X, and on the right side of the symmetry axis, y decreases with the increase of X

It is known that the square of quadratic function y = ax + BX + C and a point of intersection coordinate of X axis is (8,0), and if x = 6 is y, it has the minimum negative value of 12, the analytic expression of quadratic function is obtained