# How do you write the vertex form equation of the parabola y = x^2 - 4x + 3?

Oct 17, 2017

$y = {\left(x - \textcolor{red}{2}\right)}^{2} + \textcolor{b l u e}{\left(- 1\right)}$
with vertex at $\left(\textcolor{red}{2} , \textcolor{b l u e}{- 1}\right)$

#### Explanation:

The general vertex form is
$\textcolor{w h i t e}{\text{XXX}} y = {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
with vertex at (color(red)a,color(blue)b)

+----------------------------------------------------------------------------------+
| The process used (below) is often referred to as$\textcolor{w h i t e}{\text{XXXxx}}$ |
| $\textcolor{w h i t e}{\text{XXX}}$completing the square$\textcolor{w h i t e}{\text{XXXXXXXXXXXXXX..x}}$|
+----------------------------------------------------------------------------------+

In order to convert the given equation: $y = {x}^{2} - 4 x + 3$
into vertex form, we need to a term of the form ${\left(x - \textcolor{red}{a}\right)}^{2}$

Since ${\left(x - \textcolor{red}{a}\right)}^{2} = \left({x}^{2} + 2 \textcolor{red}{a} x + {\textcolor{red}{a}}^{2}\right)$
and
since the coefficient of $x$ in the given equation is $\left(- 4\right)$
then
$\textcolor{w h i t e}{\text{XXX")2color(red)a=-4color(white)("xxx")rarrcolor(white)("xxx}} \textcolor{red}{a} = - 2$
and
$\textcolor{w h i t e}{\text{XXXXXXXXXXXXXXX}} {\textcolor{red}{a}}^{2} = \textcolor{g r e e n}{4}$

That is the first term of the vertex form must be
$\textcolor{w h i t e}{\text{XXX}} {\left(x - \textcolor{red}{2}\right)}^{2}$

We need to add $\textcolor{g r e e n}{4}$ to ${x}^{2} - 4 x$ from the original equation to make ${\left(x - \textcolor{red}{2}\right)}^{2}$
...but if we are going to add $\textcolor{g r e e n}{4}$ then we will also need to subtract $\textcolor{g r e e n}{4}$ so the equation will not really change.

$y = {x}^{2} - 4 x + 3$
$\textcolor{w h i t e}{\text{XXX}}$will therefore become
$y = \underbrace{{x}^{2} - 4 x \textcolor{g r e e n}{+ 4}} + \underbrace{3 \textcolor{g r e e n}{- 4}}$
$\textcolor{w h i t e}{\text{XXX}}$re-writing as a squared binomial and simplifying the constants
$y = \underbrace{{\left(x - \textcolor{red}{2}\right)}^{2}} + \underbrace{\textcolor{b l u e}{\left(- 1\right)}}$
$\textcolor{w h i t e}{\text{XXX}}$which is the vertex form