(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)