# Using the vertex form, how do you solve for the variable a, with the points (3,1) the vertex and(5,9)?

May 30, 2015

The answer depends upon what you intend by the variable $a$

If the vertex is $\left(\hat{x} , \hat{y}\right) = \left(3 , 1\right)$
and another point on the parabola is $\left(x , y\right) = \left(5 , 9\right)$

Then the vertex form can be written
$\textcolor{w h i t e}{\text{XXXXX}}$$y = m {\left(x - \hat{x}\right)}^{2} + \hat{y}$
which, with $\left(x , y\right)$ set to $\left(5 , 9\right)$, becomes
$\textcolor{w h i t e}{\text{XXXXX}}$9 = m(5-3)^2+1

$8 = 2 m$

 m =4)#

and the vertex form is

$y = 4 {\left(x - 3\right)}^{2} + 1$

Option 1: (less likely option, but possible)
The vertex form is sometimes written as
$\textcolor{w h i t e}{\text{XXXXX}} y = m {\left(x - a\right)}^{2} + b$
in which case
$\textcolor{w h i t e}{\text{XXXXX}} a = 3$

Option 2:
The generalized standard form of a parabola is usually written as
$\textcolor{w h i t e}{\text{XXXXX}} y = a {x}^{2} + b x + c$
in which case
$\textcolor{w h i t e}{\text{XXXXX}} a = 4$