# What is the vertex form of  7y = -3x^2 + 2x − 13?

Aug 28, 2017

$y = \left(\textcolor{g r e e n}{- \frac{3}{7}}\right) {\left(x - \textcolor{red}{\frac{1}{3}}\right)}^{2} + \left(\textcolor{b l u e}{- \frac{38}{21}}\right)$

#### Explanation:

The general vertex form is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
for a parabola with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

Given $7 y = - 3 {x}^{2} + 2 x - 13$

Dividing both sides by $7$
$\textcolor{w h i t e}{\text{XXX}} y = - \frac{3}{7} {x}^{2} + \frac{2}{7} x - \frac{13}{7}$

Extracting the "inverse stretch" coefficient, $\textcolor{g r e e n}{m}$, from the first 2 terms:
$\textcolor{w h i t e}{\text{XXX}} y = \left(\textcolor{g r e e n}{- \frac{3}{7}}\right) \left({x}^{2} - \frac{2}{3} x\right) - \frac{13}{7}$

Completing the square
$\textcolor{w h i t e}{\text{XXX}} y = \left(\textcolor{g r e e n}{- \frac{3}{7}}\right) \left({x}^{2} - \frac{2}{3} x \textcolor{m a \ge n t a}{+ {\left(\frac{1}{3}\right)}^{2}}\right) - \frac{13}{7} \textcolor{m a \ge n t a}{- \left(\textcolor{g r e e n}{- \frac{3}{7}}\right) \cdot {\left(\frac{1}{3}\right)}^{2}}$

Simplifying
$\textcolor{w h i t e}{\text{XXX}} y = \left(\textcolor{g r e e n}{- \frac{3}{7}}\right) {\left(x - \textcolor{red}{\frac{1}{3}}\right)}^{2} + \left(\textcolor{b l u e}{- \frac{38}{21}}\right)$
which is the vertex form with vertex at $\left(\textcolor{red}{\frac{1}{3}} , \textcolor{b l u e}{- \frac{38}{21}}\right)$

For verification purposes here is the graph of the original equation and the calculated vertex point: 