# How do you write a quadratic equation with vertex; ( 2,3 ); point: ( 4,11 )?

May 1, 2017

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

#### Explanation:

The general vertex form for a quadratic is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b r o w n}{y} = \textcolor{g r e e n}{m} {\left(\textcolor{\mathmr{and} a n \ge}{x} - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$
with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$
and a "spread factor" of $\textcolor{g r e e n}{m}$

Given the vertex: $\left(\textcolor{red}{2} , \textcolor{b l u e}{3}\right)$
this becomes
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b r o w n}{y} = \textcolor{g r e e n}{m} {\left(\textcolor{\mathmr{and} a n \ge}{x} - \textcolor{red}{2}\right)}^{2} + \textcolor{b l u e}{3}$

We are given that one solution point is $\left(\textcolor{\mathmr{and} a n \ge}{x} , \textcolor{b r o w n}{y}\right) = \left(\textcolor{\mathmr{and} a n \ge}{4} , \textcolor{b r o w n}{11}\right)$

So we have
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b r o w n}{11} = \textcolor{g r e e n}{m} {\left(\textcolor{\mathmr{and} a n \ge}{4} - \textcolor{red}{2}\right)}^{2} + \textcolor{b l u e}{3}$

Simplifying:
$\textcolor{w h i t e}{\text{XXX}} 8 = \textcolor{g r e e n}{m} {\left(2\right)}^{2}$

$\textcolor{w h i t e}{\text{XXX}} 8 = 4 \textcolor{g r e e n}{m}$

$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{m} = 2$

We can substitute this back into our vertex equation, to get
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b r o w n}{y} = \textcolor{g r e e n}{2} {\left(\textcolor{\mathmr{and} a n \ge}{x} - \textcolor{red}{2}\right)}^{2} + \textcolor{b l u e}{3}$

If your instructor prefers this in "standard form" we can expand the right side to get:
$\textcolor{w h i t e}{\text{XXX}} y = 2 {x}^{2} - 8 x + 11$

For verification purposes, here is the graph of $y = 2 {\left(x - 2\right)}^{2} + 3$ 