trajectory
A trajectory is the path an object traces out through space as it moves — the invisible line you could draw connecting all the places it has been. When you toss a ball and watch its graceful arc, that arc is its trajectory. It answers a purely geometric question: what shape does the journey make in space?
Precisely, the trajectory is the curve of successive positions the object occupies, with time not shown directly — it is the route on the map, not the timetable. For a projectile moving under gravity alone (no air resistance), combining steady horizontal motion with accelerating vertical free fall gives a trajectory that is exactly a parabola: y as a function of x has the form y = (tan theta) x - (g / (2 v_0^2 cos^2 theta)) x^2, an inverted U for a launch above the horizontal. Different launch speeds and angles give different-sized parabolas.
Keep trajectory distinct from the motion graphs. A trajectory is a picture of actual space — where the object is — so a projectile's trajectory really is a parabola arcing through the air. A position-time or velocity-time graph, by contrast, plots a quantity against time and is not a map of the path. Also, a trajectory is a parabola only for the idealized no-air case; real air resistance flattens the far side of the arc and pulls the landing point in closer.
A garden hose held at an angle sends water along a curved arc that rises, peaks, and falls — a parabolic trajectory. Aim it steeper and the arc is tall and narrow; aim it flatter and the arc is long and low.
A projectile's trajectory (ignoring air) is a parabola — a curve in real space, not a graph.
A trajectory shows where, not when: two objects can share the same parabolic trajectory yet travel it at different speeds. To recover the timing you need the velocity or the position-time relationship, which the shape alone does not give.