# What is the vertex form of y=-3x^2+4x -3?

Jan 13, 2018

To complete the square of $- 3 {x}^{2} + 4 x - 3$:
Take out the $- 3$
$y = - 3 \left({x}^{2} - \frac{4}{3} x\right) - 3$
Within the brackets, divide the second term by 2 and write it like this without getting rid of the second term:
$y = - 3 \left({x}^{2} - \frac{4}{3} x + {\left(\frac{2}{3}\right)}^{2} - {\left(\frac{2}{3}\right)}^{2}\right) - 3$
These terms cancel each other out so adding them to the equation isn't a problem.

Then within the brackets take the first term, the third term, and the sign preceding the second term, and arrange it like this:
$y = - 3 \left({\left(x - \frac{2}{3}\right)}^{2} - {\left(\frac{2}{3}\right)}^{2}\right) - 3$
Then simplify:
$y = - 3 \left({\left(x - \frac{2}{3}\right)}^{2} - \frac{4}{9}\right) - 3$
$y = - 3 {\left(x - \frac{2}{3}\right)}^{2} + \frac{4}{3} - 3$
$y = - 3 {\left(x - \frac{2}{3}\right)}^{2} - \frac{5}{3}$

You can conclude from this that the vertex is $\left(\frac{2}{3} , - \frac{5}{3}\right)$

Jan 13, 2018

$y = - 3 {\left(x - \frac{2}{3}\right)}^{2} - \frac{5}{3}$

#### Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is a multiplier}$

$\text{to obtain this form use the method of "color(blue)"completing the square}$

• " the coefficient of the "x^2" term must be 1"

$\Rightarrow y = - 3 \left({x}^{2} - \frac{4}{3} x + 1\right)$

• " add/subtract "(1/2"coefficient of x-term")^2" to"
${x}^{2} - \frac{4}{3} x$

$y = - 3 \left({x}^{2} + 2 \left(- \frac{2}{3}\right) x \textcolor{red}{+ \frac{4}{9}} \textcolor{red}{- \frac{4}{9}} + 1\right)$

$\textcolor{w h i t e}{y} = - 3 {\left(x - \frac{2}{3}\right)}^{2} - 3 \left(- \frac{4}{9} + 1\right)$

$\textcolor{w h i t e}{y} = - 3 {\left(x - \frac{2}{3}\right)}^{2} - \frac{5}{3} \leftarrow \textcolor{red}{\text{in vertex form}}$