# What is the equation has a graph that is a parabola with a vertex at (-2, 0)?

Nov 21, 2016

A family of parabolas given by ${\left(x + h y\right)}^{2} + \left(2 + \frac{c}{2}\right) x + b y + c = 0$ . Upon setting h = 0, b =4 and c = 4, we get a member of the family as represented by ${\left(x + 2\right)}^{2} = - 4 y$. The graph for this parabola is given.

#### Explanation:

The general equation of parabolas is

(x+hy)^2+ax+by+c=0. Note the perfect square for the 2nd degree

terms.

This passes through the vertex $\left(- 2 , 0\right)$. So,

$4 - 2 a + c = 0 \to a = 2 + \frac{c}{2}$

The required system ( family ) of parabolas is given by

${\left(x + h y\right)}^{2} + \left(2 + \frac{c}{2}\right) x + b y + c = 0$.

Let us get a member of the family.

Upon setting h = 0, b = c = 4, the equation becomes

${\left(x + 2\right)}^{2} = - 4 y$. The graph is inserted.

graph{-1/4 (x+2)^2 [-10, 10, -5, 5]}