# What is the vertex form of y=6x^2 - 27x - 15 ?

Jun 11, 2017

$y = 6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{363}{8}$

#### Explanation:

For a more detailed example of method see:

https://socratic.org/s/aFtwtRb4

$y = 6 {x}^{2} - 27 x - 15$

$y = 6 {\left(x - \frac{27}{2 \times 6}\right)}^{2} + k - 15$

$y = 6 {\left(x - \frac{9}{4}\right)}^{2} + k - 15$

..............................................................
'Gets rid' of the introduced error.

Set $\text{ } 6 {\left(- \frac{9}{4}\right)}^{2} + k = 0$

$6 \times \frac{81}{16} + k = 0$

$k = - \frac{243}{8}$
.........................................................

$y = 6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{243}{8} - 15$

$y = 6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{363}{8}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check by expanding brackets:

$y = 6 \left({x}^{2} - \frac{18}{4} x + \frac{81}{16}\right) - \frac{363}{8}$

$y = 6 {x}^{2} - 27 x - 15$

Jun 11, 2017

$y = 6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{363}{8}$

#### Explanation:

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where ( h , k ) are the coordinates of the vertex and a is a constant.

$\text{for a parabola in standard form } y = a {x}^{2} + b x + c$

${x}_{\textcolor{red}{\text{vertex}}} = - \frac{b}{2 a}$

$y = 6 {x}^{2} - 27 x - 15 \text{ is in this form}$

$\text{with " a=6,b=-27" and } c = - 15$

$\Rightarrow {x}_{\textcolor{red}{\text{vertex}}} = - \frac{- 27}{12} = \frac{27}{12} = \frac{9}{4}$

$\text{substitute this value into the function for y-coordinate}$

$\Rightarrow {y}_{\textcolor{red}{\text{vertex}}} = \left(6 \times \frac{81}{16}\right) - \left(27 \times \frac{9}{4}\right) - 15$

$\textcolor{w h i t e}{\times \times \times x} = \frac{243}{8} - \frac{486}{8} - \frac{120}{8}$

$\textcolor{w h i t e}{\times \times \times x} = - \frac{363}{8}$

$\Rightarrow \left(h , k\right) = \left(\frac{9}{4} , - \frac{363}{8}\right)$

$\Rightarrow y = 6 {\left(x - \frac{9}{4}\right)}^{2} - \frac{363}{8} \leftarrow \textcolor{red}{\text{ in vertex form}}$