# End Behavior and Multiplicity

First day of the spring semester today, so we started polynomials in precal. We just did a few things – vocab, y-intercepts, the fact that solving f(x) = 0 is the same as finding the x-intercepts, and end behavior.

Today was an improvement over the past. Often, I throw a chart up on the board: odd degree, positive coefficient, you get this end behavior. Odd degree, negative coefficient, you get that. And so on.

That’s clearly not ideal – I want them to figure things out, instead of me telling them. So, today, I had them all pull out Desmos and graph a polynomial, any polynomial. Then they got into groups based on what their end behavior was – do you go up on both ends? Find everyone who is like you!

I told them to figure out what their leading coefficients had in common with their group. That may have been giving too much away, by telling them to look at the leading coefficient. Next time, I may just go with, “What do your functions have in common?” Fewer hints!

I think where I dropped the ball on this was talking about why. Ok, so we have a negative coefficient and an even degree. We see from the graph that it goes down on both ends, but why? As x becomes some huge number, what do you expect the y-values to do? Why would that happen? That conversation would have been good, and it didn’t happen at all. Maybe next time I see them…

On the other hand, we did get some tension going for next time. I gave them some examples to work through (find x-intercepts, y-intercepts, end behavior; now graph). The first few were in standard form, and the last few were in factored form, but every x-intercept had a multiplicity of 1. When they finished, I changed one of the x-intercepts to a multiplicity of 2 and asked them how things would change. Fun confusion:

“The degree is 4, so it’s going to be a W.” “Ok, so how many x-intercepts?” “4!” “But it only has 3 factors….”

One student got it perfectly, and I pointed out the key x-intercept and said, “Hey, this is great. Why did you do that?” She stared at me with huge eyes. “I don’t know…”

Even if they aren’t hungry for the answer next time, it was at least fun and planted the idea that there’s more to the story.