# What is the equation of the parabola that has a vertex at  (2, -9)  and passes through point  (1, 4) ?

##### 1 Answer
Feb 3, 2016

$13 {\left(x - 2\right)}^{2} - 9 = y$

#### Explanation:

When we are given the vertex we can immediately write an equation vertex form, which looks like this $y = a {\left(x - h\right)}^{2} + k$. $\left(2 , - 9\right)$ is $\left(h , k\right)$, so we can plug that in to the format. I always like to put parentheses around the value I'm inputting just so I can avoid any issues with signs.

Now we have $y = a {\left(x - \left(2\right)\right)}^{2} + \left(- 9\right)$. We can't do much with this equation besides graph it, and we don't know $a , x , \mathmr{and} y$.

Or wait, we do.

We know that for one point, $x = 1$ and $y = 4$ Let's plug those numbers in and see what we've got.

We have $\left(4\right) = a {\left(\left(1\right) - 2\right)}^{2} - 9$, and let's solve for $a$. First, let's solve ${\left(1 - 2\right)}^{2}$. $1 - 2 = - 1.$Now$, - {1}^{2} = 1$. At last we have $a \cdot 1 - 9 = 4$, which can be simplified to $a - 9 = 4$. Add $9$ to both sides and we have $a = 13$. Now we have evry piece of our equation.

Our equation needs to be for a line, not a point, so we won't be needing $\left(1 , 4\right)$ anymore. We will however need $a$, so let's plug that into our old vertex form equation, shall we?

$y = \left(13\right) {\left(x - \left(2\right)\right)}^{2} + \left(- 9\right)$ or $y = 13 {\left(x - 2\right)}^{2} - 9$ is our final form.