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

To determine the equation of the parabola of the arch, we need to know the general equation for a parabola and its elements. The arch is facing down so the general equation written in its vertex form would be as follows:

y - k =  - a(x - h)^2

where (h,k) represents the vertex of the parabola and a represents the focus and tells you where the graph opens. If a is positive then it opens upward and if its is negative, it opens downward. 

For this given situation, the vertex is at point (0,0) or at the origin so that h and k are zero.

y=  - a(x)^2

To determine the value of a, we use the measurements given above. We make use of the maximum height and the span of the parabola. The point we will be using would be  (18/2 , 96 ).
 
y=  - a(x)^2
96=  - a(9)^2
a = -32/27

So, the equation of the parabola would be 
y=  - (32/27)(x)^2