# How do you find the equation of a parabola when given a vertex (3, -3) point (0, 6)?

Jul 27, 2015

I found: $y = {x}^{2} - 6 x + 6$

#### Explanation:

You have
1] the general equation of your parabola:
$y = a {x}^{2} + b x + c \text{ } \left(1\right)$
2] two points of your parabola:

$x = 3 , y = - 3$ which is also the vertex. At the vertex ${x}_{v} = 3 = - \frac{b}{2 a}$;

$x = 0 , y = 6$

we can try use these information into equation (1):

$- 3 = 9 a + 3 b + c$

$6 = 0 a + 0 b + c$ so that $\textcolor{red}{c = 6}$

$3 = - \frac{b}{2 a}$ and $b = - 6 a$

So we can get, from the first (with $c = 6$ and $b = - 6 a$):
$- 3 = 9 a - 18 a + 6$ and $\textcolor{red}{a = 1}$
and so: $b = - 6 a = \textcolor{red}{- 6}$

The equation of the parabola is:
$y = {x}^{2} - 6 x + 6$
Graphically:
graph{x^2-6x+6 [-16.02, 16.02, -8.01, 8.01]}