# What is the vertex form of y=-3x^2 -2x+5?

Mar 28, 2017

$\textcolor{g r e e n}{y = - 3 {\left(x + \frac{1}{3}\right)}^{2} + 5 \frac{1}{3}}$

#### Explanation:

Out target is the quadratic vertex form: $y = \textcolor{m a \ge n t a}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$

We are given
$\textcolor{w h i t e}{\text{XXX}} y = - 3 {x}^{2} - 2 x + 5$

Extracting the $\textcolor{m a \ge n t a}{m}$
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{m a \ge n t a}{- 3} \left({x}^{2} + \frac{2}{3} x\right) + 5$

Completing the square
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{m a \ge n t a}{- 3} \left({x}^{2} + \frac{2}{3} x \textcolor{b r o w n}{+ {\left(\frac{1}{3}\right)}^{2}}\right) + 5 \textcolor{b r o w n}{- \left(\textcolor{m a \ge n t a}{- 3}\right) \left({\left(\frac{1}{3}\right)}^{2}\right)}$

$\textcolor{w h i t e}{\text{XXX}} y = {\textcolor{m a \ge n t a}{x + \frac{1}{3}}}^{2} + 5 + \frac{1}{3}$

color(white)("XXX")y=color(magenta)(-3)(x-color(red)(""(-1/3)))^2+color(blue)(5 1/3)

This is technically a more correct version than I give in the "Answer" since it clearly identifies the vertex as being at $\left(\textcolor{red}{- \frac{1}{3}} , \textcolor{b l u e}{5 \frac{1}{3}}\right)$
but I thought the "Answer" version was easier to read.
(Check with your instructor for the preferred form).