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

Nov 19, 2015

You can complete the square or use this trick ...

#### Explanation:

First, here is the vertex form of a parabola (quadratic):

$y = g {\left(x - h\right)}^{2} + k$

We can find h and k very quickly using this trick and recalling that the general formula for a quadratic is $y = a {x}^{2} + b x + c$:

$h = - \frac{b}{2 a} = \frac{- 2}{2 \times 3} = - \frac{1}{3}$

$k = y \left(h\right) = 3 {\left(- \frac{1}{3}\right)}^{2} + 2 \left(- \frac{1}{3}\right) + 4 = \frac{11}{3}$

Now, going back to vertex form, plug in h and k:

$y = g {\left(x + \frac{1}{3}\right)}^{2} + \frac{11}{3}$

Last, simply determine what is g by plugging in a known coordinate from the original equation like $\left(0 , 4\right)$:

$4 = g {\left(0 + \frac{1}{3}\right)}^{2} + \frac{11}{3} = \left(\frac{1}{9}\right) g + \frac{11}{3}$

Solving for g:

$g = 3$

So, here is the vertex form:

$y = 3 {\left(x + \frac{1}{3}\right)}^{2} + \frac{11}{3}$

hope that helped