The Connected command sets all equations to use the usual graph style - a connected line. In addition, this graph style is made the default, so that when a variable is deleted it will revert to this graph style. The other possible setting for this option is Dot.
Compare this to the GraphStyle( command, which puts a single equation into a specified graph style.
The Connected and Dot commands don't depend on Graphing_Mode, and will always affect all functions, even in other graphing modes. The exception to this is that sequence mode's default is always the dotted-line style, even when Connected mode is set. The Connected command will still change their graphing style, it just won't change the default they revert to.