If the symmetry axis of the image of quadratic function f (x) is a straight line x = - 2, the intercept of the image on the Y axis is 1, and the length of the line segment cut by the X axis is 22, find f (x)

If the symmetry axis of the image of quadratic function f (x) is a straight line x = - 2, the intercept of the image on the Y axis is 1, and the length of the line segment cut by the X axis is 22, find f (x)

Let the quadratic function be y = AX2 + BX + C (a ≠ 0), because the intercept of the image of the quadratic function on the y-axis is 1, so C = 1, and the symmetry axis is a straight line x = - 2, so − B2A = - 2 & nbsp; & nbsp; & nbsp; according to the length of the line segment cut by the x-axis of the image of the quadratic function is 22, that is, the absolute of the two differences of the equation AX2 + BX + 1 = 0

Given that the image of a quadratic function passes through P (- 2,7), the symmetry axis is a straight line x = 1, and the length of the line segment of the image cut on the X axis is 8, the analytic solution of the function is obtained

Because the axis of symmetry is a straight line x = 1, the length of the line segment cut on the x-axis of the image is 8, so the analytic solution of this function is obtained
So the function must pass the points (- 3,0) and (5,0)
Let the function be y = a (X-5) (x + 3) because it passes through the point (- 2,7)
7 = a (- 2-5) (- 2 + 3)
So a = - 1
So y = - x ^ 2 + 2x + 15

Let the quadratic function y = f (x) satisfy the following conditions: when x = 2, there is a minimum value of - 1, and the intercept of its image on the Y axis is 1

When x = 2, there is a minimum value of - 1, which means that the vertex coordinates of the function are (2, - 1)
Therefore, it can be expressed in the form of vertex
Let the function expression be y = a (X-2) & sup2; - 1, and the function passes through the point (0,1)
Substituting (0,1) into the function expression: 1 = 4a-1. A = 1 / 2
So the function expression is y = 1 / 2 (X-2) & sup2; - 1