Given the image of quadratic function through the origin and point... Mathematical problems Given that the image of a quadratic function passes through the origin and points (- 1 / 2, - 1 / 4), and the distance between the image and another intersection of X axis and the origin is 1, then the analytic expression of the quadratic function is obtained_______ I'm previewing. Please tell me the process Thank you.

Given the image of quadratic function through the origin and point... Mathematical problems Given that the image of a quadratic function passes through the origin and points (- 1 / 2, - 1 / 4), and the distance between the image and another intersection of X axis and the origin is 1, then the analytic expression of the quadratic function is obtained_______ I'm previewing. Please tell me the process Thank you.

Because it goes through the origin, so C = 0
Let f (x) = ax ^ 2 + BX
Substitution point (- 1 / 2, - 1 / 4)
-1/4=a/4-b/2
Another point may be (1,0) or (- 1,0)
Take them in separately and count them on one side
Finally, we get f (x) = - x ^ 2 / 3 + X / 3 or F (x) = x ^ 2 + X