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

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Jul 29, 2015

#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 completing the square.

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#).

Graph 1

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