# What is the equation of the parabola that has a vertex at  (-18, -12)  and passes through point  (-3,7) ?

Apr 9, 2017

$y = \frac{19}{225} {\left(x + 18\right)}^{2} - 12$

#### Explanation:

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

Since the vertex is given $P \left(- 18 , - 12\right)$, you know the value of $- b$ and $c$,

$y = a {\left(x - - 18\right)}^{2} - 12$
$y = a {\left(x + 18\right)}^{2} - 12$

The only unkown variable left is $a$, which can be solved for using $P \left(- 3 , 7\right)$ by subbing $y$ and $x$ into the equation,

$7 = a {\left(- 3 + 18\right)}^{2} - 12$
$19 = a {\left(15\right)}^{2}$
$19 = 225 a$
$a = \frac{19}{225}$

Finally, the equation of the quadratic is,

$y = \frac{19}{225} {\left(x + 18\right)}^{2} - 12$

graph{19/225(x+18)^2-12 [-58.5, 58.53, -29.26, 29.25]}

Apr 9, 2017

There are two equations that represent two parabolas that have the same vertex and pass through the same point. The two equations are:

$y = \frac{19}{225} {\left(x + 18\right)}^{2} - 12$ and $x = \frac{15}{361} {\left(y + 12\right)}^{2} - 18$

#### Explanation:

Using the vertex forms:

$y = a {\left(x - h\right)}^{2} + k$ and $x = a {\left(y - k\right)}^{2} + h$

Substitute $- 18$ for $h$ and $- 12$ for $k$ into both:

$y = a {\left(x + 18\right)}^{2} - 12$ and $x = a {\left(y + 12\right)}^{2} - 18$

Substitute $- 3$ for $x$ and 7 for $y$ into both:

$7 = a {\left(- 3 + 18\right)}^{2} - 12$ and $- 3 = a {\left(7 + 12\right)}^{2} - 18$

Solve for both values of $a$:

$19 = a {\left(- 3 + 18\right)}^{2}$ and $15 = a {\left(7 + 12\right)}^{2}$

$19 = a {\left(15\right)}^{2}$ and $15 = a {\left(19\right)}^{2}$

$a = \frac{19}{225}$ and $a = \frac{15}{361}$

The two equations are:

$y = \frac{19}{225} {\left(x + 18\right)}^{2} - 12$ and $x = \frac{15}{361} {\left(y + 12\right)}^{2} - 18$

Here is a graph of the two points and the two parabolas: