(2)① According to the parabola y=(x-3)(x+ 1), calculate the coordinates of point C and point E, connect BC, let point C be CH⊥DE in H, and get CD= by Pythagorean theorem.
2
，CB=3
2
It is proved that △BCD is a right triangle. PC and DC extend respectively and intersect with X axis at points Q and R. According to the similarity of two triangles with equal corresponding angles, it is proved that △BCD∽△QOC, then
commander
OQ
=
laser record
Civil band
=
1
three
The coordinates of q (-9,0) are obtained, and the analytical formula of straight line CQ is y=-
1
three
X-3, the analytical formula of straight line BD is y=2x-6, and the equations are solved.
y=-
1
three
x-3
y=2x-6
, you can find the coordinates of point p;
(2) It is discussed in two cases: (1) When point M is on the right side of the symmetry axis, if point N is on the ray CD, such as alternating graph 1, the intersection point of MN's Y axis at point F is extended, so that the intersection point M is the MG⊥y axis of point G. First, it is proved that △MCN∽△DBE, and then MN=2CN is obtained. The ratio of the corresponding side of similar triangles. Let CN = If n points are on the ray DC, the coordinates of m points can also be obtained; (2) When point M is on the left side of the symmetry axis, because ∠ BDE < 45, we get ∠ CMN < 45. According to the complementarity of two acute angles of a right triangle, we get ∠ MCN > 45, and any point K on the left side of the parabola has ∠ KCN < 45, so point M.
Solution: (1)∵ Parabola y=(x-3)(x+ 1) intersects the X axis at two points A and B (point A is on the left of point B).
∴ When y=0, (x-3)(x+ 1)=0,
X=3 or-1,
The coordinate of point B is (3,0).
∫y =(x-3)(x+ 1)= x2-2x-3 =(x- 1)2-4，
∴ The coordinate of vertex D is (1,-4);
(2)① As shown on the right.
∵ parabola y=(x-3)(x+ 1)=x2-2x-3 intersects the y axis at point C,
The coordinate of point ∴C is (0, -3).
The symmetry axis is a straight line x= 1,
∴ The coordinate of point E is (1, 0).
Connect BC, and point C is CH⊥DE in H, so the coordinate of point H is (1, -3).
∴CH=DH= 1，
∴∠CDH=∠BCO=∠BCH=45，
∴CD=
2
，CB=3
2
, △BCD is a right triangle.
Extend PC and DC respectively, and intersect with X axis at points Q, R. 。
∠∠BDE =∠DCP =∠QCR，
∠CDB=∠CDE+∠BDE=45 +∠DCP，
∠QCO=∠RCO+∠QCR=45 +∠DCP，
∴∠CDB=∠QCO，
∴△BCD∽△QOC，
∴
commander
OQ
=
laser record
Civil band
=
1
three
∴OQ=3OC=9, which means q (-9,0).
The analytical formula of straight line CQ is y=-
1
three
x-3，
The analytical formula of straight line BD is y = 2x-6.
Through the equation
y=-
1
three
x-3
y=2x-6
, solution
x=
nine
seven
y=-
24
seven
.
∴ The coordinates of point P are (
nine
seven
,-
24
seven
);
② (Ⅰ) When point M is on the right side of the symmetry axis.
If the point n is on the ray CD, as shown in the standby diagram 1, extend the Y axis of the intersection point of MN to the point f, and make the intersection point m the MG⊥y axis to the point g. 。
∠∠CMN =∠BDE，∠CNM=∠BED=90，
∴△MCN∽△DBE，
∴
Communication network (short for Communicating Net)
All merchant ships
=
exist
Delaware
=
1
2
∴MN=2CN.
Let CN=a, then Mn = 2a.
∠∠CDE =∠DCF = 45，
∴△CNF and △MGF are isosceles right triangles,
∴NF=CN=a,CF=
2
One,
∴MF=MN+NF=3a，
∴MG=FG=
three
2
2
One,
∴CG=FG-FC=
2
2
One,
∴M(
three
2
2
a，-3+
2
2
a)。
Substitute parabola y=(x-3)(x+ 1) to get a=
seven
2
nine
∴M(
seven
three
,-
20
nine
);
If the point n is on the ray DC, as shown in Figure 2, MN intersects the Y axis at point F, and the passing point M is the MG⊥y axis at point G. 。
∠∠CMN =∠BDE，∠CNM=∠BED=90，
∴△MCN∽△DBE，
∴
Communication network (short for Communicating Net)
All merchant ships
=
exist
Delaware
=
1
2
∴MN=2CN.
Let CN=a, then Mn = 2a.
∫∠CDE = 45，
∴△CNF and △MGF are isosceles right triangles,
∴NF=CN=a,CF=
2
One,
∴MF=MN-NF=a，
∴MG=FG=
2
2
One,
∴CG=FG+FC=
three
2
2
One,
∴M(
2
2
a，-3+
three
2
2
a)。
Substituting the parabola y=(x-3)(x+ 1), we get a=5.
2
∴m(5, 12)；
(ii) When the point m is located on the left side of the symmetry axis.
∠∠CMN =∠BDE < 45，
∴∠MCN>45，
And any point k on the left side of the parabola has ∠ kcn < 45,
Point m does not exist.
To sum up, the coordinates of point M are (
seven
three
,-
20
nine
) or (5, 12)