##color(red)( f(x) = (x-1)^2-4)##

The vertex form of a quadratic is given by ##y = a(x h)^2 + k## where (##h k##) is the vertex.

The ##a## in the vertex form is the same ##a## as in ##y = ax^2 + bx + c##.

Your equation is

##f(x) = x^2-2x-3##

We convert to the vertex form by .

Step 1. Move the constant to the other side.

##f(x)+3 = x^2-2x##

Step 2. Square the coefficient of ##x## and divide by 4.

##(-2)^2/4 = 1##

Step 3. Add this value to each side

##f(x)+3+1 = x^2-2x+1##

Step 4. Express the right hand side as a square.

##f(x)+4 = (x-1)^2##

Step 5. Isolate ##f(x)##.

##f(x) = (x-1)^2-4##

The equation is now in vertex form.

##y = a(x h)^2 + k## where (##h k##) is the vertex.

##h = 1## and ##k = -4## so the vertex is at (##1-4##).