For this case we have:
The equation in vertex form of the parabola is given by:
[tex]y = a (x-h) ^ 2 + k\\[/tex]
The vertex is (h, k) and is given by the highest or lowest point of the parabola, in this case it is observed that it is [tex](h, k) = (- 3, -1)\\[/tex]
Thus, the equation is given by:
[tex]y = a (x - (- 3)) ^ 2 + (- 1)\\\\y = a (x + 3) ^ 2-1\\[/tex]
We look for the value of a, substituting a point of the parabola in the equation in the form of vertex, we will take the point [tex](x, y) = (0,8)\\[/tex]
Substituting we have:
[tex]8 = a (0 - (- 3)) ^ 2 + (- 1)\\\\8 = a (0 + 3) ^ 2-1\\\\8 = a (3) ^ 2-1\\[/tex]
[tex]8 = 9a-1\\\\8 + 1 = 9a\\\\9 = 9a\\[/tex]
[tex]a = \frac{9}{9}\\\\a = 1\\[/tex]
Thus, the equation of the parabola is given by:
[tex]y = (x + 3) ^ 2-1\\[/tex]
Answer:
[tex]y = (x + 3) ^ 2-1[/tex]
