(1) Analysis: ∵ Parabola f (x) = x 2+bx+c passes through the origin and point p, where p is the moving point. Starting from the origin, it moves in the positive direction of the X axis at a speed of one unit per second.

Let the exercise time be t (t > 0)

∴f(0)=c=0,|OP|=t

f(t)=t^2+bt=0==>； t(t+b)=0== >b=-t

∴f(x)=x^2-tx

(2) Analysis: ABCD is a rectangle, A( 1, 0), b (1, 5), D(4, 0).

∴C(4,-5)

parabola

Intersect with rectangular ABCD at m

The size of ∠AMP will not change when P moves.

Let x =1= = > f( 1)= 1-t

∴M( 1, 1-t)

∫P(t，0)

∴tan∠amp=( 1-t)/( 1-t)= 1

∴∠AMP=45

(3) analysis: ∵ rectangular ABCD,

A( 1，0)、B( 1，-5)、C(4，-5)、D(4，0)

∴ The advantages of the rectangle are:

(2,- 1),

(2,-2),

(2,-3),

(2,-4),

(3,- 1),

(3,-2),

(3,-3),

(3,-4)

The parabola divides these "advantages" into two equal parts.

When the parabola passes through (2, -3), (3,-1)

-3=2^2-2t==>； t = 7/2； - 1=3^2-3t==>； t= 10/3

When the parabola passes through (2, -4), (3, -2)

-4=2^2-2t==>； t = 4； -2=3^2-3t==>； t= 1 1/3

∫7/2 > 10/3，4 > 1 1/3

∴ take 7/2 < t <; 1 1/3

That is, when t is 7/2.