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

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
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{\mathmr{and} a n \ge}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
which has its vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

The following process is commonly called completing the square

Given
$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = {x}^{2} - 9 x + 2$
we can assume $\textcolor{\mathmr{and} a n \ge}{m} = 1$ since that is the implied coefficient of ${x}^{2}$

To get the ${\left(x - \textcolor{red}{a}\right)}^{2} = {x}^{2} + 2 \textcolor{red}{a} x + {\textcolor{red}{a}}^{2}$ component
we need to re-write the expression so it contains a squared binomial).
For the given expression the first two terms:
$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - 9 x$ must equal ${x}^{2} + 2 \textcolor{red}{a} x$
which implies
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} = - \frac{9}{2}$
and the third term of the expanded binomial must be:
$\textcolor{w h i t e}{\text{XXX}} {\textcolor{red}{a}}^{2} = {\left(\frac{9}{2}\right)}^{2} = \frac{81}{4}$

We want:
$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = {x}^{2} - 9 x + {\left(\frac{9}{2}\right)}^{2}$
but instead of the ${\left(\frac{9}{2}\right)}^{2}$ we have $2$

The solution?
Add in the ${\left(\frac{9}{2}\right)}^{2}$ and then subtract it back off again.
$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = {x}^{2} - 9 x \textcolor{g r e e n}{+ {\left(\frac{9}{2}\right)}^{2}} + 2 \textcolor{g r e e n}{- {\left(\frac{9}{2}\right)}^{2}}$

which can then be written as
$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = \textcolor{\mathmr{and} a n \ge}{1} {\left(x - \textcolor{red}{\frac{9}{2}}\right)}^{2} + \left(\textcolor{b l u e}{- \frac{73}{4}}\right)$

Comparing this to the general vertex form,
we see the vertex is at $\left(\frac{9}{2} , - \frac{73}{4}\right) = \left(4 \frac{1}{2} , - 18 \frac{1}{4}\right)$

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]}