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.
I have only been teaching for 2 years, but am already beginning to encounter the recursive nature of the profession. I have had several repeat classes in my 4 semesters of teaching, and they require the achievement of the same outcomes. This does not bother me, in general, because I am excited to see the improvement in my teaching. There is one unit, however, that has already frustrated me. Its ability to sabotage creative exploits is unrivalled throughout the mathematics curriculum; I am speaking of the unit on polynomial factoring.