The flow of an ideal fluid through a tube is a quartic function -- the flow rate varies with the radius to the 4th degree. So if you double the radius of a tube, 16 times more fluid can pass through the tube in the same interval of time.
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St. Louis Arch is an example of a quadratic graph. Umm... many arches are actually *catenaries*, visually indistinguishable from a parabola - this answer should be checked for accuracy.
A real world example of a cubic function might be the change in volume of a cube or sphere, depending on the change in the dimensions of a side or radius, respectively.
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Using your ICE table in doing equilibrium calculations of concentrations in chemistry yields a quadratic function. X = Vot +(1/2)at2 is an equation of kinematics in physics.