[en] We present a family of one-dimensional accelerating beams in the paraxial limit whose trajectory is described, in normalized coordinates (ξ, s), by a non-parabolic curve ξ = α(s) with α(s) denoting the transverse shift of beams. When taking different values of the parameters appearing in α(s), three types of accelerating beams are observed. The first type accelerates initially along X direction and almost stops accelerating after traveling a large distance. The second type is seen to travel along a straight line with an angle with respect to the Z-axis at large distance. The beam of the last type is seen to leave initially upward and cross the Z-axis downward after traveling some distance. (paper)