How to find the equation of a Parabola with vertex (0,-9) and passing through (6,-8)?

Jul 23, 2015

$y = {x}^{2} / 36 - 9$

Explanation:

The general vertex form for a parabola is
$\textcolor{w h i t e}{\text{XXXX}}$$y = m {\left(x - a\right)}^{2} + b$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$where the vertex is at $\left(a , b\right)$

Given that the vertex of the desired parabola is at $\left(0 , - 9\right)$
this becomes:
$\textcolor{w h i t e}{\text{XXXX}}$$y = m {\left(x - 0\right)}^{2} - 9$

and since $\left(x , y\right) = \left(6 , - 8\right)$ is a solution point on this parabola:
$\textcolor{w h i t e}{\text{XXXX}}$$- 8 = m {\left(6 - 0\right)}^{2} - 9$

$\textcolor{w h i t e}{\text{XXXX}}$$1 = 36 m$

$\textcolor{w h i t e}{\text{XXXX}}$$m = \frac{1}{36}$

Therefore
$\textcolor{w h i t e}{\text{XXXX}}$$y = \frac{1}{36} {\left(x - 0\right)}^{2} - 9$
or
$\textcolor{w h i t e}{\text{XXXX}}$$y = {x}^{2} / 36 - 9$