It is known that the distance between the image of a quadratic function and the two intersections of the x-axis is 1. If the image of this function is translated up one unit, then it has only one common point with the x-axis; if it is translated down one unit, then it passes through the origin, and the analytic expression of this quadratic function is obtained

It is known that the distance between the image of a quadratic function and the two intersections of the x-axis is 1. If the image of this function is translated up one unit, then it has only one common point with the x-axis; if it is translated down one unit, then it passes through the origin, and the analytic expression of this quadratic function is obtained

Let the analytic expression of quadratic function be y = AX2 + BX + C, ∵ y = AX2 + BX + C, and the distance between the two intersections of X axis be 1. Let the two intersections be: (x1, 0), (X2, 0), ∵ x1-x2 | = 1, ∵ x1-x2) 2 = 1, ∵ (x1 + x2) 2-4x1x2 = 1, ∵ (- BA) 2-4 × CA = 1, and ∵ let the parabola y = AX2 + B