# What is the equation of the parabola that has a vertex at  (-5, 4)  and passes through point  (6,125) ?

Apr 17, 2017

$y = {\left(x + 5\right)}^{2} + 4$

#### Explanation:

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

For the vertex $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right) = \left(\textcolor{red}{- 5} , \textcolor{b l u e}{4}\right)$ this becomes
$\textcolor{w h i t e}{\text{XXX}} \textcolor{m a \ge n t a}{y} = \textcolor{g r e e n}{m} {\left(\textcolor{c y a n}{x} - \textcolor{red}{\left(- 5\right)}\right)}^{2} + \textcolor{b l u e}{4}$
$\textcolor{w h i t e}{\text{XXXX}} = \textcolor{g r e e n}{m} {\left(x + 5\right)}^{2} + \textcolor{b l u e}{4}$

Since this equation hold for the point $\left(\textcolor{c y a n}{x} , \textcolor{m a \ge n t a}{y}\right) = \left(\textcolor{c y a n}{6} , \textcolor{m a \ge n t a}{125}\right)$
color(white)("XXX")color(magenta)(125)=color(green)m(color(cyan)6+5)^2+color(blue)(4
$\textcolor{w h i t e}{\text{XXXXX}} = \textcolor{g r e e n}{m} \cdot {11}^{2} + \textcolor{b l u e}{4}$
$\textcolor{w h i t e}{\text{XXXXX}} = 121 \textcolor{g r e e n}{m} + \textcolor{b l u e}{4}$

$\rightarrow \textcolor{w h i t e}{\text{X}} 121 = 121 \textcolor{g r e e n}{m}$
$\rightarrow \textcolor{w h i t e}{\text{X}} \textcolor{g r e e n}{m} = 1$

and the equation is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{m a \ge n t a}{y} = \textcolor{g r e e n}{1} {\left(\textcolor{c y a n}{x} + 5\right)}^{2} + 4$