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.
See sample here. (Sliding "a" to "0" invites an excellent conversation; same with "b" etc.)
After we work with the transition from function to graph, we go the opposite direction. (makes sense, right?)
My favorite types of problems, however, ask students to play with parameters to influence results while leaving some characteristics consistent. For example, they might be asked to write a polynomial function that has an identical Range but different y-intercept. Or an identical end behavior but a different number of x-intercepts.
We play with these choices for a while. (I have them come up with lists of characteristics that are impossible...this is a great conversation)
The student work below comes right before the exam is written. They are asked to write a personal ad for a polynomial of their choice as if it were joining an online dating service. They cannot state their degree, leading coefficient, or constant explicitly. The result is an interesting exercise in encoding and decoding sets of possibly parameters in polynomial functions.
I am a polynomial function, super fun and curvy with two turning points. I am currently on a down slope in my life, but I don't want to sound negative. I am an infinite range of y-values and infinite domain of x-values. It will never be a dull moment. I am looking for a polynomial that is more calm than me. Someone who is basic but positive and going up in life. They need to have 1 y-intercept and 1 x-intercept. I don't want anyone who will throw o curve at me. I hope to have domain and range in common--be something special we share.
I am a very negative and odd function. I have been working my way from quadrant 2 to quadrant 4. I have no curvy parts and I like to rest right at the origin. I am looking for a function to put a little more life in me. Two turning points is a must. I'm, looking for a positive influence in my life.
I am mostly laid back and enjoy to stick close to home. Ever since I was born I have never changed. I prefer to not cross paths with my enemy, but am willing to take the chance to see my friends on the other side. I don't have much of a range. I can't tell which quadrant I start and end which makes me mysterious. I am looking for a soul mate which will help me take risk in life. Someone who leaves the x-axis regularly; three times would be the perfect number. I would like someone who picks me up regularly and doesn't mind hanging out at our common y-intercept. Be curvy and outgoing, but willing to stay close to home as well.
Once students have a grasp on the abstractions, they can begin to play.