Trajectory of a projectile is the path that particle will take under the action of gravity, neglecting all other forces, such as friction from air resistance.

It is given by 
 y = xtanФ(1 - x/R)

where y = height attained
x = horizontal distance
R = total range of the projectile
Ф = Angle of projection
Consider a projectile thrown with a velocity 'v' with an angle Ф with the horizontal .
horizontal motion is uniform motion
the distance travelled by the projectile in time ' t'
x= vcosФt
t= x/v cosФ  ........ 1
vertical motion is uniformly accelerated motion
distance travelled by the particle in vertical direction in time 't' is
using (s=ut+1/2at²)

  y=vsinФt - 1/2 gt²  ...........2

from 1 and 2

y= vsinФ× x/cosФ - 1/2 g x²/v²cos²Ф

y= x tanФ - 1/2 g x²/v²cos²Ф


hence the projectile is parabolic in nature.