2.1.2—Hyperbolic functions

sinh x is an odd increasing curve through the origin; cosh x is an even U-shaped curve with minimum 1; tanh x is odd, increasing and bounded between −1 and 1.

The limits as x→±∞ explain the horizontal asymptotes of tanh. Evenness gives symmetry about the y-axis, while oddness gives rotational symmetry about the origin.

cosh(−x)=cosh x, so its graph mirrors across the y-axis. tanh x approaches 1 from below as x grows and −1 from above as x decreases.

Do not transfer the sine/cosine range or periodicity to hyperbolic functions; none of these basic hyperbolic graphs is periodic.