How does one find the equation y=ax²+bx+c when you are only provided with the coordinates of the turning point, (-6,-9)? note: a is +1 or -1

Start with vertex form and eliminate parentheses.
.. y = a(x -h)^2 +k . . . . . . where (h, k) is the vertex
The value of "a" will be greater than zero if the parabola opens upward, as shown.

Using this form, your equation is
.. y = 1*(x +6)^2 -9
.. y = x^2 +12x +36 -9
.. y = x^2 +12x +27 . . . . . . the desired form