Analysis: (1) Substitute the three-point coordinates of A (-4,0), B (0 0,4) and C (2,0) into Y = AX 2+BX+C (A ≠ 0) by the method of undetermined coefficient, and establish a set of equations to solve. (2) Let M be the vertical line of the X axis, the vertical ruler D and the coordinate of the point M be (m, n), so that the length of MD and OD can be expressed by an algebraic expression containing M, and the areas of △AMD, trapezoidal MDOB and △AOB are calculated respectively, then the sum of the areas of △AMD and trapezoidal MDOB minus △AOB is the area of △AMB, from which the areas of S and AOB can be obtained. According to the nature of the function, the maximum value of s can be obtained. (3) To solve this problem, we need to make full use of the properties of parallelogram. Let P(x, 1/2x2+x-4), as shown in figure 1. When OB is an edge, we can know PQ∨OB according to the properties of parallelogram, then Q(x, -x). ② As shown in Figure 2, when OB is diagonal, the abscissas of P and Q are opposite (if the abscissa of P is X, the abscissa of Q is -X), that is, Q (-X, X). If the absolute value of the ordinate difference between p and o is equal to the absolute value of the ordinate difference between q and b, then 1/2 x2+x-4 =-4- is obtained. Just find the value of X. Solution: (1) Let the analytical formula of parabola be y = ax 2+bx+c (a ≠ 0), then a set of equations 16a-4b+c = 0, c =-4, 4a+2b+c = 0 are obtained. The analytical formula of ∴ parabola is y = 1/2x 2+x-4. (2) If the passing point M is the MD⊥x axis of point D and the coordinates of point M are (m, n), then AD=m+4, MD =-n, n = 1. ∴S=S△AMD+S trapezoid DMBQ-s △ ABO =1/2 (m+4) (-n)+1/2 (-n+4) (-m)-1/2× 4. (1) When OB is an edge, if we know PQ∨OB according to the properties of parallelogram, then Q(x, -x). From PQ=OB, we get |-X. ② When OB is diagonal, the abscissas of P and Q are opposite (if the abscissa of P is X, the abscissa of Q is -X), that is, Q (-X, X). If the absolute value of the ordinate difference between p and o is equal to the absolute value of the ordinate difference between q and b, 1/2x 2+x-4 =-4-X. 4). Therefore, there are three coordinates of point Q that satisfy the meaning of the question, namely (-4,4), (-2+2 √ 5,2-2 √ 5) and (-2-2 √ 5,2+2 √ 5).