The transition curves in the modern road and railway construction are route elements equally crucial as alignment and curve (circular). In order to prevent a sudden change of the centrifugal force, the transition curve must be applied due to the impact of the motion in a sharp curve. Over the years the application of the clothoid has become widespread in many countries in the world. However, in this study, in order to eliminate the problems concerning the road dynamics, created by clothoid for vehicles at high speed, the fourth degree parabola (Bi-quadratic parabola) is examined. The possibility of using bi-quadratic parabola in defining transition curves during design of transportation facilities is analyzed in the paper. Basic properties of bi-quadratic parabolas, their fundamental mathematical expression, calculation of point coordinates, driving-dynamic characteristics of bi-quadratic parabolas are described, and the function of change in curvature and lateral impact along the bi-quadratic parabola is presented.