How do I convert the equation f(x)=x^2-2x-3 to vertex form?

##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##).