When graphing higher order polynomials, one can encounter a quadratic with no real roots as a factor, which results in a "floating quadratic" turn on the graph. But where do you put it? While working with a student recently, we decided to toss the equation into a graphing program to find out. The equation was y = (x+2)(2x-5)(x^2+1), in case you care. My student guessed that it would be around -1, since that's how the other factors worked, but when plotted, we actually got what looked like a cubic twist at about (0,-1).

We couldn't quite figure out why, so we just moved on. The answer came to me this morning at 6am. The quadratic term can be thought of as a non-constant vertical stretch, that has a minimum effect at the x coordinate of the vertex of the quadratic. So what we saw was a tug upward at x=0. And remember, you can quickly find the x-coordinate by partial factoring. Not generally part of the curriculum, but maybe a nice bonus question.