How do you find the vertex of #g(x) = x^2 - 9x + 2#?

1 Answer
Feb 9, 2016

Often the easiest way to find the vertex for a given parabolic equation is to convert it into vertex form.

Explanation:

The vertex form of a parabolic equation is
#color(white)("XXX")y=color(orange)(m)(x-color(red)(a))^2+color(blue)(b)#
which has its vertex at #(color(red)(a),color(blue)(b))#

The following process is commonly called completing the square

Given
#color(white)("XXX")g(x)=x^2-9x+2#
we can assume #color(orange)(m)=1# since that is the implied coefficient of #x^2#

To get the #(x-color(red)(a))^2 = x^2+2color(red)(a)x+color(red)(a)^2# component
we need to re-write the expression so it contains a squared binomial).
For the given expression the first two terms:
#color(white)("XXX")x^2-9x# must equal #x^2+2color(red)(a)x#
which implies
#color(white)("XXX")color(red)(a) = -9/2#
and the third term of the expanded binomial must be:
#color(white)("XXX")color(red)(a)^2= (9/2)^2 = 81/4#

We want:
#color(white)("XXX")g(x)=x^2-9x+(9/2)^2#
but instead of the #(9/2)^2# we have #2#

The solution?
Add in the #(9/2)^2# and then subtract it back off again.
#color(white)("XXX")g(x)=x^2-9xcolor(green)(+(9/2)^2)+2color(green)(-(9/2)^2)#

which can then be written as
#color(white)("XXX")g(x)=color(orange)(1)(x-color(red)(9/2))^2+(color(blue)(-73/4))#

Comparing this to the general vertex form,
we see the vertex is at #(9/2,-73/4) = (4 1/2, -18 1/4)#

We can compare this result with the graph of the given function to see that our result is reasonable
graph{x^2-9x+2 [-3.82, 10.23, -20.15, -13.127]}