Abuilding has an entry the shape of a parabolic arch 96 ft high and 18 ft wide at the base.
find an equation for the parabola if the vertex is put at the origin of the coordinate system.

Since the parabola has its vertex at the origin [Vertex(0,0)], its equation becomes : y = ax², with y-axis as axis of symmetry.
Moreover, since this parabola opens down the coefficient a is negative:

So y = - ax². Let's calculate a:
Let A & B be the intersection of the parabola with the base & let's calculate their respective coordinates:
On the left we have A(-9, -96) & on the right B(+9, -96)
You plug any coordinates of A or B. Let's take B for instance: B(+9, -96)
-96 = a(9)² ↔ -96 = 81.a & a = -96/81 = -32/27.

 Finally the equation is : y = (-32/27)x²