Zunyi mathematics examination questions

1、(20 10? Zunyi) as shown in the figure, it is known that the vertex coordinate of the parabola y=ax2+bx+c(a≠0) is Q(2,-1), which intersects with the Y axis at point C (0,3), intersects with the X axis at points A and B (point A is on the right side of point B), and point P is the moving point on the parabola.

(1) Find the function relation of parabola;

(2) When △ADP is a right triangle, find the coordinates of point P;

(3) Under the conclusion of question (2), if point E is on the X axis and point F is on the parabola, is there a parallelogram with vertices A, P, E and F? If it exists, find the coordinates of point F; If it does not exist, please explain why. Test center: Quadratic function synthesis problem. Special topic: finale.

Analysis: (1) Given the vertex coordinates of the parabola, we can set the analytical formula of the parabola as the vertex, and then substitute the coordinates of point C transmitted by the function image into the above formula to get the analytical formula of the parabola;

(2) Due to the PD‖y axis, ∠ ADP ≠ 90. If △ADP is a right triangle, two cases can be considered:

(1) Take point P as the right-angled vertex, at this time AP⊥DP, and point P is located on the X axis (that is, it coincides with point B), from which the coordinates of point P can be obtained;

② With point A as the vertex of the right angle, it is easy to know that OA=OC, then ∠ OAC = 45, so OA divides ∠CAO equally, and then D and P are symmetrical about X, so we can get the analytical formula of straight line AC. Then set the abscissa of d and p, and the ordinate of d and p is expressed according to the analytical formula of parabola and straight line AC. Since the two points are symmetrical about x, the ordinate is opposite to each other.

(3) Obviously, when P and B coincide, a quadrilateral with vertices A, P, E and F cannot be formed, so only one case of (2)② meets the problem, and it can be seen from ② that P and Q coincide at this time; Suppose there is a parallelogram that meets the requirements, then according to the properties of parallelogram, we know that the vertical coordinates of P and F are inverse numbers, so we can find the vertical coordinates of point F, and substitute them into the analytical formula of parabola to find the coordinates of point F. 。

Solution: Solution: (1)∵ The vertex of the parabola is Q(2,-1).

∴ Let the analytical formula of parabola be y=a(x-2)2- 1,

Substituting c (0 0,3) into the above formula gives:

3=a(0-2)2- 1，a = 1；

∴y=(x-2)2- 1, that is, y = x2-4x+3;

(2) There are two situations:

(1) when point P 1 is a right-angled vertex, point P 1 coincides with point b;

Let y=0, x2-4x+3=0, x= 1, x = 3；;

Point a is to the right of point b,

∴b( 1,0),a(3,0)；

∴p 1( 1,0)；

② When point A is the right vertex of △APD2;