Answer:
The parabolic shape of the door is represented by  . (See attachment included below). Head must 15.652 inches away from the edge of the door.
. (See attachment included below). Head must 15.652 inches away from the edge of the door. 
Step-by-step explanation:
A parabola is represented by the following mathematical expression:

Where:
 - Horizontal component of the vertix, measured in inches.
 - Horizontal component of the vertix, measured in inches.
 - Vertical component of the vertix, measured in inches.
 - Vertical component of the vertix, measured in inches.
 - Parabola constant, dimensionless. (Where vertix is an absolute maximum when
 - Parabola constant, dimensionless. (Where vertix is an absolute maximum when  or an absolute minimum when
 or an absolute minimum when  )
)
For the design of the door, the parabola must have an absolute maximum and x-intercepts must exist. The following information is required after considering symmetry:
 (Vertix)
 (Vertix)
 (x-Intercept)
 (x-Intercept)
 (x-Intercept)
 (x-Intercept)
The following equation are constructed from the definition of a parabola:



The parabolic shape of the door is represented by  . Now, the representation of the equation is included below as attachment.
. Now, the representation of the equation is included below as attachment. 
At x = 0 inches and y = 22 inches, the distance from the edge of the door that head must observed to avoid being hit is:



If y = 22 inches, then x is:



Head must 15.652 inches away from the edge of the door.