This topic is the finale of quadratic function. Based on translation transformation, it is difficult to examine the knowledge points such as quadratic function, linear function, trigonometric function (or similar), equilateral triangle and the nature of angular bisector. There are unknowns in the resolution function, which increases the test difficulty. The second question, the key to solving the problem is to understand the meaning of "the sum of the distances from points B and C to the straight line OP is the largest, and AP=BP". The third question, there are four points p that meet the conditions, so don't miss the solution.

Solution: (1) When m= 1/2, parabola C 1: y = (x+ 1/2) 2, so the vertex d of parabola C2 is on parabola C 1. Detailed answers/exercises/math look here.

Figure, parabola c 1: y = (x+m) 2 (m is a constant, m >；; 0), the parabola y =-x 2 is translated so that its vertex d is on the image on the right side of the parabola C 1 on the y axis, and the parabola C2 is obtained. The parabola intersects the X axis at points A and B (point A is to the left of point B), the Y axis intersects with point C, and the abscissa of point D is a. 。

(1) as shown in figure 1, if m= 1/2.

A when OC=2, find the analytical formula of parabola C2;

Is there a in B, so that there is a point P on BC line, so that the sum of the distances from point B and point C to line OP is the largest and Ap=BP? If it exists, find the value of a; If it does not exist, please explain the reason;

(2) As shown in Figure 2, when OB=2, the radical number 3-m (0

This question is really difficult. I believe you will understand after reading the answer. If you don't understand, you can keep asking me.