Here is a lesson I love because of graphing calculators. My goals for my students were to be able to look at a graph of a polynomial and know its degree and find an equation given various points on the polynomial. I also wanted them to be able to look at an equation and be able to give a quick sketch of the graph with the intercepts and shape correct.
At the start of class before I mentioned anything about shapes of non parabola polynomials, I handed out graphing calculators and we all got the same window and turned the grid on. In Y=, I had them type in y=(x-2)(x+1)(x-1) (or something similar previously checked out to make sure it fits in the window). I told them not to graph it but to think: what are the x-int, y-int, and make a conjecture about the shape. THEN they could graph and see if they were right. We then had a discussion to link the intercepts with the equation.
Then they typed in y=2(x-2)(x+1)(x-1). I asked them to think about how this might affect the graph and intercepts and shape and THEN graph and confirm their reasoning. We did one more, and then I discussed the shape and degree.
Then same process with y=(x+2)(x-1)^2. They had to think and do the same analysis as above. We then did y=(x-1)(x+2)^2 and such.
Then the fun part: I typed in an equation and just showed them the graph, and they had to match it. We did that a few times. I made sure to change the constant in front sometimes and to make it "upside down" and to have various powers. Then I asked them to find an equation for a 5th degree polynomial with 2 "bounces" off the x-axis going in the same direction (both below the axis or both above). Then for the ones that finished that early: opposite bounce direction.
Then we took notes, and I believe they had a good sense of polynomial graphs.