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It is known that the parabola passes through B (0,4) C (5,9) and its vertex falls on the positive half axis of X axis
Because it passes through point B, so C = 4
And because the vertex is on the positive half axis of the X axis
So B & sup2; - 4ac = 0
That is B & sup2; = 16A
And because of C
That is: 9 = 25A + 5B + 4
A = 1, B = - 4 or a = 1 / 25, B = 4 / 5
Because it is on the positive half axis, the axis of symmetry - B / 2A > 0
a> 0 so B
Finding the analytic expression of quadratic function
The symmetric axis of the parabola y = x2 + BX + C is x = 1, intersecting the x-axis at points a and B (a is on the left side of B), and ab = 4, intersecting the y-axis at point C. find the analytic function of the parabola and the coordinates of its vertex M
Because the axis of symmetry is x = 1 and the distance between a and B is 4, the distance between a and B is 2
So a is (- 1,0), B (3,0)
We can get y = (x + 1) (x-3)
That is y = x ^ 2-2x-3
The formula is y = (x-1) ^ 2-4
The vertex is (1, - 4)
Finding the analytic expression of quadratic function
It is known that the vertex m of the quadratic function y = x2 + BX + C is on the line y = - 4x, and the image passes through the point a (- 1,0)
If you substitute the point a (- 1,0) into the quadratic function, you can get B-C = 1 (square ①), the quadratic function can be changed into y = (x + B / 2) ^ 2 + C-B ^ 2 / 4, you can get the vertex m (- B / 2, C-B ^ 2 / 4), you can get B ^ 2 + 8b-20c = 0 (square ②) by substituting the point a (- 1,0). Finally, the simultaneous equations ① and ② can be solved to get b = 2, C = 1 or B = 10, C = 9, so the analytical formula is y = x ^ 2 + 2x + 1 or y = x ^ 2 + 10x + 9
How to find the analytic expression of quadratic function?
The intersection formula of quadratic function: y = a (x-x1) (x-x2) (A & ne; 0) x1, X2 and x0d are the abscissa of the intersection of parabola and X-axis respectively. Given the abscissa of the intersection of parabola and x-axis, it is easier to use the intersection formula when solving the analytic formula of quadratic function
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