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)