# How do you write an equation in standard form of the parabola that has vertex (-8,-3) and passes through the point (4,717)?

Feb 20, 2018

$y = 5 {x}^{2} + 80 x + 317$

#### Explanation:

The formula for a parabola can be written as $y = a {\left(x - h\right)}^{2} + k$, where $\left(h , k\right)$ is the vertex of the parabola.

Here, $h = - 8$ and $k = - 3$.

We can input the above into the equation:

$y = a {\left(x - \left(- 8\right)\right)}^{2} + \left(- 3\right)$, which simplifies to:

$y = a {\left(x + 8\right)}^{2} - 3$

We know that one of the coordinates is $\left(4 , 717\right)$. So when $x = 4 , y = 717$. We need to find $a$, and so we can input:

$717 = a {\left(4 + 8\right)}^{2} - 3$

$717 = a \left({12}^{2}\right) - 3$

$717 = 144 a - 3$

$144 a = 720$

$a = \frac{720}{144}$

$a = 5$

Since $a = 5$, we can input this into $y = a {\left(x + 8\right)}^{2} - 3$.

$y = 5 {\left(x + 8\right)}^{2} - 3$

$y = 5 \left({x}^{2} + 16 x + 64\right) - 3$

$y = 5 {x}^{2} + 80 x + 320 - 3$

$y = 5 {x}^{2} + 80 x + 317$, the formula for the parabola.

We can graph it:

graph{5x^2+80x+317 [-29.24, 35.7, -11.13, 21.33]}

Notice the vertex is at $\left(- 8 , - 3\right)$.

So the above is proved.