What is the vertex, axis of symmetry, the maximum or minimum value, and the range of parabola #f(x)=x^2 -2x -15#?

You can factorise: #=(x+3)(x-5)#

Explanation:

This gives you the zero-points #x=-3andx=5#
Halfway between these lies the axis of symmetry :
#x=(-3+5)//2->x=+1#
The vertex is on this axis, so putting in #x=1#:
#f(1)=1^2-2.1-15=-16#
So the vertex #=(1,-16)#
Since the coefficient of #x^2# is positive, this is a minumum
There is no maximum, so the range is #-16<=f(x)< oo#
Since there are no roots or fractions involved the domain of #x# is unlimited.
graph{x^2-2x-15 [-41.1, 41.1, -20.55, 20.52]}