**Grade 11: Zero and Asymptotes**

In this unit we introduce the students to the qualitative behavior of polynomial and rational function-graphs with an emphasis on overall structure, working towards the effect of a parameter. I present these examples in different ways that depends on the students overall technical experience. If this is high I can go quickly through the first three examples and get on to the parametric equations. That’s where the mathematical thinking comes to the fore.

If I gave a Grade 11 class the equation

what sort of experience would they need in order to come up with the graph at the right? That’s what we play with in the first three examples.

In the next two examples I follow up with parameterized families like this:

I have plotted the two endpoint graphs at the right. Can the students figure out what the gradual transition from c = –1 to c = 1 might look like? For example, what happens around c = 0?

[A good question for Desmos.]

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