Look at Einstein's theory on gravity. It is shown on a parabolic graft.
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The nature of displacement-time graph is parabolic if the acceleration is constant(uniform). When acceleration is constant, displacement is directly proportional to the square of time which results into a parabolic structure of graph.
If f(x) = x2 + 25, then to plot f(x) on a graph would give you a parabolic curve extending infinitely upward with a minimum value of 25, and it's vertex at the point (0, 25).
The synonym for the adjective parabolic would be parobolical.
In a parabolic curve it would be called an asymptote, where only one integer is exluded. If multiple integers are excluded, or you are dealing with piece-wise functions it is called a jump discontinuity.
No. If you tilt a parabola, you will still have a parabolic curve but it will no longer be a parabola.