Question

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

Answer 1

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]}