Solution: (1) ∫ Point B (0,1) is on the image, ∴.........................(2 points).

∴ k =1.................. (3 points)

(2) From (1), the parabola is known as:

∴ Vertex A is (2,0) .............. (4 points)

∴OA=2,OB= 1

If C(m, n) is used as CD⊥x axis is in D, then CD=n, OD=m, and ∴AD=m-2.

From the known ∠ BAC = 90..................(5 points)

∴∠ CAD+∠ Bao = 90, while ∠ Bao+∠ OBA = 90 ∴∠ OBA = ∠ CAD.

∴Rt△OAB∽Rt△DCA

∴ (or tan ∠ oba = tan ∠ CAD)...(6 points)

∴n=2(m-2)；

And the point C(m, n) on it, ⅷ

∴, namely

∴m=2 or m =10; When m=2, n = 0；; When m= 10, n =16; .................... (7 points)

∴ The coordinates of the qualified point C are (2,0) or (10, 16)...(8 points).

(3)① According to the meaning of the question, point C (2,0) does not meet the requirements, and point C is (10, 16).

now

........................... (9 points)

The other point P is on the symmetry axis x=2 of the function image, ∴P(2,t), AP=

∴ = ....................... (10)

∵

∴, ∴1< t < 21...................... when t≥0 and S=t (11).

When t < 0 ∴, S=-t, ∴-21< t.

The value range of ∴t is 1 < t < 2 1 or-21< t.

(2) t = 0, 1,17 ......................................................... (14 points)