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Analysis: (1) Substitute the origin coordinates of point A into the analytical function, and use the undetermined coefficient method to find the solution of the quadratic analytical function;

(2) Calculate the distance from point P to AO according to the area formula of triangle, and then solve it in two cases: point P is above and below the X axis.

Solution: (1) is derived from {c = 0 of the known condition.

{a×(-4)2-4×(-4)+c=0

The solution is {a =- 1

{c=0

In this paper, the coordinate characteristics of points on quadratic function image are studied by undetermined coefficient method. (2) Attention should be paid to the discussion and solution of the upper and lower split points P on the X axis. So the analytic formula of this quadratic function is y =-x2-4x;

(2) The coordinate of point ∫ A is (-4,0),

∴AO=4，

Let the distance from point P to axis X be h,

Then S△AOP= 1/2×4h=8,

The solution is h=4,

(1) When the point p is above the X axis, -x2-4x=4,

The solution is x=-2,

Therefore, the coordinate of point P is (-2,4),

② When the point P is below the X axis, -x2-4x=-4,

X 1=-2+2√2，x2=-2-2√2，

Therefore, the coordinates of point P are (-2+2√2, -4) or (-2-2√2, -4).

To sum up, the coordinates of point P are: (-2,4), (-2+2 √ 2,4), (-2-2 √ 2,4).

In this paper, the coordinate characteristics of quadratic resolution function and midpoint in quadratic function image are studied by undetermined coefficient method. (2) Attention should be paid to the discussion and solution of the upper and lower split points P on the X axis.

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