# What is the vertex form of y= (3-x)(3x-1)+11 ?

May 2, 2016

$y = - 3 {\left(x - \frac{5}{3}\right)}^{2} + \frac{49}{3}$

#### Explanation:

The vertex form of a quadratic equation is $y = a {\left(x - h\right)}^{2} + k$. In this form, we can see that the vertex is $\left(h , k\right)$.

To put the equation in vertex form, first we'll expand the equation, and then use a process called completing the square.

$y = \left(3 - x\right) \left(3 x - 1\right) + 11$

$\implies y = - 3 {x}^{2} + 9 x + x - 3 + 11$

$\implies y = - 3 {x}^{2} + 10 x + 8$

$\implies y = - 3 \left({x}^{2} - \frac{10}{3} x\right) + 8$

$\implies y = - 3 \left({x}^{2} - \frac{10}{3} x + {\left(\frac{5}{3}\right)}^{2} - {\left(\frac{5}{3}\right)}^{2}\right) + 8$

$\implies y = - 3 \left({x}^{2} - \frac{10}{3} x + \frac{25}{9}\right) + \left(- 3\right) \left(- \frac{25}{9}\right) + 8$

$\implies y = - 3 {\left(x - \frac{5}{3}\right)}^{2} + \frac{49}{3}$

So, the vertex form is $y = - 3 {\left(x - \frac{5}{3}\right)}^{2} + \frac{49}{3}$ and the vertex is $\left(\frac{5}{3} , \frac{49}{3}\right)$