Every year, my students study the general characteristics of polynomial functions. We investigate the various shapes of various functions and slowly shift parameters to watch changes in the graphs. Eventually, we deduce the roles of the constant term, leading coefficient, and degree.
It should be noted that Desmos makes this process much easier than years previous. Just set up the generic polynomial, add sliders, set specific ones to play (depending on what you want to investigate), and have students discuss in groups.