It is known that the quadratic function f (x) satisfies f (4-x) = f (x), the length of the line cut on the x-axis is 6, and the function image passes (3, - 8). The analytic expression of F (x) is obtained

It is known that the quadratic function f (x) satisfies f (4-x) = f (x), the length of the line cut on the x-axis is 6, and the function image passes (3, - 8). The analytic expression of F (x) is obtained

F (4-x) = f (x) shows that f (x) is symmetric with respect to x = 2
When the length of the line segment is 6 on the x-axis, we can know that the position with the length of 3 on the left and right sides of the symmetry is the zero point
That is - 1 and 5 are zeros
So f (x) can be expressed as a (x + 1) (X-5)
Substituting (3, - 8), we get a * 4 * (- 2) = - 8 and a = 1
So f (x) = (x + 1) (X-5) = x & # 178; - 4x-5

It is known that the quadratic function f (x) satisfies f (4-x) = f (x), the length of the line cut on the X axis is 6, and the function image passes (3, - 8). The analytic expression of the function f (x) is obtained

The quadratic function f (x) satisfies f (4-x) = f (x)
F (x) is symmetric with respect to x = 2,
The length of the line cut on the X axis is 6,
Then the intersection a and B of parabola and X axis are symmetric with respect to x = 2,
∴A(-1,0),B(5,0)
Let f (x) = a (x + 1) (X-5)
Image over (3, - 8) substitution
a*(3+1)(3-5)=-8
So a = 1
∴f(x)=(x+1)(x-5)
That is, f (x) = x ^ 2-4x-5

Given that the image of quadratic function passes through points (0,3), vertex coordinates, (- 4,18), find the analytic expression of the quadratic function, and the coordinates of the intersection of the image and X axis
After point (0.3), I didn't write it wrong.

Let the analytic formula of quadratic function be y = ax + BX + C
Let (0,3) and (- 4,18) be substituted into the equations
Then solve a, B and C respectively, and then answer it
The coordinates of the image and the x-axis. There is a formula in the book. It is: x = - B / 2a, and then the coordinates
OK?