(1) According to the coordinates of the known points, the analytical expression of the known function can be determined by the undetermined coefficient method;

(2) Firstly, the analytical expression of translation function is determined according to translation, then the coordinates of point P are determined, and then the coordinates of point C are calculated, so that the analytical expression of straight line AC is determined by undetermined coefficient method, and then the value range of m is determined.

(3) Find the midpoint of AB. The intersection of a straight line passing through this point and perpendicular to AB at x= 1 should be the critical point of vertex P. Vertex P continues to move upward, without Q point, but there are two points P downward.

Solution: solution: (1) substitute A(0, -6) and B(-2, 0) into Y = (1/2) x 2+BX+C,

Get:

{? 6=c

{0=2? 2b+c，

Solution:

{b=？ 2

{c=？ 6,

∴y=( 1/2)x^2-2x-6，

∴ Vertex coordinate is (2,-8);

(2) shift the parabola obtained in (1) to the left by 1 unit length, and then shift it upward by m (m(m>0) unit lengths to obtain a new parabola y1=12 (x-2+1)

∴P( 1,-8+m)，

C (6 6,0) can be easily obtained in the parabola y = (1/2) x 2-2x-6.

∴ The straight line AC is y2=x-6,

When x= 1, y2=-5,

∴-5