What is the equation of the parabola that has a vertex at  (-4, 2)  and passes through point  (-7,-34) ?

Dec 24, 2015

To solve this you need to use the vertex form of the equation of a parabola which is $y = a {\left(x - h\right)}^{2} + k$, where $\left(h , k\right)$ are the coordinates of the vertex.

Explanation:

The first step is to define your variables
$h = - 4$
$k = 2$

And we know one set of points on the graph, so
$x = - 7$
$y = - 34$

Next solve the formula for $a$
$y = a {\left(x - h\right)}^{2} + k$
$- 34 = a {\left(- 7 + 4\right)}^{2} + 2$
$- 34 = a {\left(- 3\right)}^{2} + 2$
$- 34 = 9 a + 2$
$- 36 = 9 a$
$- 4 = a$

To create a general formula for the parabola you would put in the values for $a , h$, and $k$ and then simplify.
$y = a {\left(x - h\right)}^{2} + k$
$y = - 4 {\left(x + 4\right)}^{2} + 2$
$y = - 4 \left({x}^{2} + 8 x + 16\right) + 2$
$y = - 4 {x}^{2} - 32 x - 64 + 2$

So the equation of a parabola that has a vertex at $\left(- 4 , 2\right)$ and passes through point $\left(- 7 , - 34\right)$ is:
$y = - 4 {x}^{2} - 32 x - 62$