As shown in figure 1, in the plane rectangular coordinate system, the parabola y=ax2+ 1 intersects with the positive semi-axis of the X axis at point F (16,0), ABCD intersects with the positive semi-axis of the Y axis at point E (0,016) with a side length of/kloc-0.

(1) Find the function expression of parabola;

(2) As shown in Figure 2, if the square ABCD moves in the plane and the straight line of BC side is always perpendicular to the X axis, then the parabola always intersects with AB side at point P and intersects with CD side at point Q (when moving, point P does not coincide with points A and B, and point Q does not coincide with points C and D). The coordinate of point A is (m, n) (m > 0).

① when PO=PF, the coordinates of point p and point q are obtained respectively;

② On the basis of ①, when the square ABCD moves left and right, please write the range of m directly;

(3) When n=7, whether there is a value of m to make point P the midpoint of AB side, and if so, find the value of m; If it does not exist, please explain why.

Solution: (1) parabola y=ax2+c passes through points e (F( 16) and F( 16,0) to obtain:

{O= 162a+c 16=c

Solve? {a =-116c =16, (3 points)

∴? Y =- 1 16x2+ 16。 (4 points)

(2)① Make the PG⊥x axis of point G pass through point P,

PO = PF，

∴OG=FG，

∫F( 16，0)，

∴OF= 16，

∴OG=？ 12，OF=？ 12× 16=8,

That is, the abscissa of point p is 8,

Point p is on a parabola,

∫m > 0，

∴y=？ - 1 16×82+ 16= 12,

That is, the ordinate of point P is 12,

∴p(8 12), (6 points)

The ordinate of point P is 12, and the side length of square ABCD is 16.

∴ The ordinate of point Q is -4,

Q point is on a parabola,

∴? -4=- 1 16x2+ 16，

∴? x 1=85，x2=-85，

∵m>0,∴？ X2=-85 (truncated) ∴? x=85，

∴? Q(85，-4)。 (8 points)

②8? 5- 16 0，

∴x2=-20 (rounded)

∴x=20，

∴Q point coordinates (20, -9),

∴ Point Q coincides with point C, which contradicts that point Q does not coincide with point C,

When n=7, there is no such value of m, that is, p is the midpoint of the AB side. (14)