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##).

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