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

##### 1 Answer
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
$\textcolor{w h i t e}{\text{XXXXX}} a = 4$