Find the equation of the parabola that passe through these three points?

Note by symmetry that (0,-4) is the vertex.

So y = ax^2 + bx - 4

and since the vertex is on the y axis, b=0 as well

So y = ax^2 - 4

substitute in the third point to find a

a*3^2 - 4 = 3root3 - 4

a = root(3)/3

So the parabola is

y = root(3)x^2/3 - 4

x-coordinate of (0, - 4) is midway between the x-coordinates of the remaining two points. Hence, it is the vertex. Hence, its equation is of the form,

x^2 = a(y + 4)

as (- 3, 3√3 - 4) lies on the parabola,

(- 3)^2 = a (3√3 - 4 + 4) => a = √3

=> equation of the parabola is

x^2 = √3(y + 4)

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