Revise the rules from graph translations to sketch the derived, inverse or other related functions, complete the square to find turning points, and find expression for composite functions.

Revise the rules from graph translations to sketch the derived, inverse or other related functions, complete the square to find turning points, and find expression for composite functions.

We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold M with b1(M) ≥ 2. For a ...