find an equation of the parabola having vertex (1, -3) and containing (-2, 0)?

Answer 1

There is not a unique solution to the problem. There is one vertical parabola, one horizontal parabola and an infinite number of parabolas that are tilted.

Vertical parabola with vertex (h,k).

y - k = a(x - h)²
y + 3 = a(x - 1)²

It contains the points (-2,0). Plug the point in and solve for a.

0 + 3 = a(-2 - 1)²
3 = 9a
a = 1/3

y + 3 = (1/3)(x - 1)²
_____________

Horizontal parabola with vertex (h,k).

x - h = a(y - k)²
x - 1 = a(y + 3)²

It contains the points (-2,0). Plug the point in and solve for a.

-2 - 1 = a(0 + 3)²
-3 = 9a
a = -1/3

x - 1 = (-1/3)(y + 3)²
________

In addition there are an infinite number of tilted parabolas.

Answer 2

y = a(x - h)^2 + k

where vertex is (h,k)

since vertex is given:

y = a(x -1)^2 -3

Substitute (-2,0):

0 = a(-2 - 1)^2 - 3

a = 1/3

Therefore:

y = (1/3)(x - 1)^2 - 3