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

Dec 16, 2015

Convert into vertex form to get
vertex at $\left(\frac{1}{3} , 3 \frac{2}{3}\right)$

#### Explanation:

General vertex form:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{y = m {\left(x - a\right)}^{2} + b}$
with vertex at $\left(\textcolor{g r e e n}{a , b}\right)$

Given
$\textcolor{w h i t e}{\text{XXX}} y = 3 {x}^{2} - 2 x + 4$

Extract $m$ from terms including $x$
$\textcolor{w h i t e}{\text{XXX}} y = 3 \left({x}^{2} - \frac{2}{3} x\right) + 4$

Complete the square
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{red}{3} \left({x}^{2} - \frac{2}{3} x \textcolor{b l u e}{+ {\left(\frac{1}{3}\right)}^{2}}\right) + 4 - \textcolor{red}{3} \cdot \textcolor{b l u e}{{\left(\frac{1}{3}\right)}^{2}}$

Re-write as a squared binomial and simplify the numeric expression
$\textcolor{w h i t e}{\text{XXX}} y = 3 {\left(x - \frac{1}{3}\right)}^{2} + 3 \frac{2}{3}$
which is the desired "vertex form"
graph{3x^2-2x+4 [-1.07, 2.775, 2.868, 4.788]}