This topic, named the finale of quadratic function, involves the image and properties of quadratic function, the method of undetermined coefficient, the area of graph, Pythagorean theorem, the properties of angular bisector and other knowledge points, with emphasis on the mathematical ideas of classification discussion. The first problem involves complicated classification discussion, which makes the test more difficult.

Solution: (1) A (4,0), C (0,4). Let the analytical formula of parabola be y = ax 2+bx+c, then there are detailed ideas and solutions here. /exercise/math/799755 On both sides of the right-angled trapezoid ABCO, OA and OC are on the positive semi-axis of the coordinate axis, BC//x axis, OA=OC=4, and the parabola with the straight line x= 1 as the symmetry axis passes through A,

(1) Find the resolution function of parabola;

(2) It is known that the analytical formula of the straight line L is y=x+m, and it intersects with the X axis at point G, taking point P on one side of the trapezoid ABCO.

(1) When m=0, as shown in figure 1, point P is the intersection of the parabola symmetry axis and BC. When crossing point P, make PH perpendicular to straight line L at point H, then connect OP to find the area of Δ OPH;

(2) When m=-3, take the intersection point P as the vertical line between the X axis and the straight line L, and the vertical foot as points E and F. Is there such a point P, and the triangle with the vertices of P, E, F, E and F is an isosceles triangle? If it exists, find the coordinates of point P; If it does not exist, please explain why.

The process of discussion and calculation is complicated and needs patience and careful consideration. Come on ~