# What is the vertex form of y=9x^2 - 21x + 10 ?

Aug 11, 2017

$y = 9 {\left(x - \frac{7}{6}\right)}^{2} + \left(- \frac{9}{4}\right)$ with vertex at $\left(x , y\right) = \left(\frac{7}{6} , - \frac{9}{4}\right)$

#### Explanation:

General vertex form is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
where
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m}$ is a measure of the parabolic "spread";
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a}$ is the $x$ coordinate of the vertex; and
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{b}$ is the $y$ coordinate of the vertex.

Given
$\textcolor{w h i t e}{\text{XXX}} y = 9 {x}^{2} - 21 x + 10$

Extract the spread factor $\textcolor{g r e e n}{m}$
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{9} \left({x}^{2} - \frac{7}{3} x\right) + 10$

Complete the square for the first term and subtract a corresponding amount from the second
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{9} \left({x}^{2} - \frac{7}{3} x \textcolor{m a \ge n t a}{+ {\left(\frac{7}{6}\right)}^{2}}\right) + 10 \textcolor{m a \ge n t a}{- 9 \cdot {\left(\frac{7}{6}\right)}^{2}}$

Rewrite as a squared binomial and simplify the constant
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{9} {\left(x - \textcolor{red}{\frac{7}{6}}\right)}^{2} + \textcolor{b l u e}{\left(- \frac{9}{4}\right)}$

For verification purposes, here is the graph of this function (with grid lines at $\frac{1}{12}$ units; note: $\frac{7}{6} = 1 \frac{2}{12}$ and $- \frac{9}{4} = - 2 \frac{3}{12}$)