Determine the end behavior of a polynomial from its degree and leading coefficient.
Explain why the end behavior of a polynomial function is determined by its leading term.
Use limit notation to describe the end behavior of a polynomial function.
Quick Lesson Plan
In today’s lesson students look at the end behavior of a polynomial function. You will notice that the approach we use is a different one. While generally students explore end behavior from an inductive approach (here are a bunch of functions and their graphs, can you find connections between the degree, leading coefficient, and end behavior?) in AP Precalculus we are taking a deep dive into the why. While it is good for students to know that a polynomial with an even degree has the same end behavior on the left and right side and that a polynomial with an odd degree has opposite end behavior on the left and right side, the question we are asking is why? And why does a negative leading coefficient reverse the left and right end behavior?
The activity starts with a simple exercise of determining if an expression has a positive or negative value. We purposefully use expressions that they can’t easily evaluate in their heads to get at the underlying concept. Why does multiplying by any negative number make a positive expression negative? Why is any negative number raised to an even integer, positive? These are basic principles, of course, but they are critical to understanding polynomial end behavior.
In question 4 students explore an often ignored aspect of end behavior. Why do we just look at the degree of a polynomial and its leading coefficient (which together make up the leading term) rather than all the terms of a polynomial? By comparing the graphs of g and h students see that while the two functions behave very differently in the short run, in the long run they have the exact same behavior. This leads students to conclude in question 5 that the leading term dominates, or overpowers, all other terms when x is very, very large (right end behavior) or very, very small (left end behavior).
How are these questions related to what you did in question 1?
Would your answers (to 2a and 2b) be different if the constant term was 1000 instead of 5? Why or why not?
Would your answers (to 3a and 3b) be different if the second term was -900x^3? Why or why not?
Why are the y-values getting bigger and bigger? Doesn’t subtracting the cube of a number make it smaller?
Are g and h equivalent functions? Why or why not?
What is the same about g and h? What is different?
There are three ideas you will want to be sure to discuss in your debrief and most if not all of these aren’t things that would be generally covered in a Precalculus class.
First, limit notation for end behavior is introduced. For those of you who teach Calculus, this is old hat, but remember that students are only using the notation to describe the behavior, they are not evaluating limits analytically in any way. If you have never taught limits before, be sure that you are saying the limit statements correctly, and have your students practice saying them out loud as well. “The limit as x approaches infinity of f(x) is infinity” or “The limit as x approaches negative infinity of f(x) is negative infinity.” Continue to pair this formal language with more student-friendly language. “As x increases without bound, the outputs of f are getting closer and closer to infinity” or “As x gets closer and closer to infinity (on the right side of the graph), f(x) is increasing/decreasing without bound.” Note that the course framework for AP Precalculus has chosen to use the notation =∞ rather than saying that the limit does not exist because the function is unbounded.
Students often drop the function name in the limit notation. Be sure to always encourage correct notation and clear communication!
Second, we will use the language of “dominating behavior” to describe end behavior. This is very helpful language for both polynomial functions and rational functions and is a great way to describe why the end behavior is determined in certain ways. The degree and the leading coefficient determine the end behavior because the values of the leading term dominate all lower degree terms.
Thirdly, we introduce the idea of an end behavior model in the margin notes for question 4. This becomes especially helpful for when we introduce rational functions later on in this unit. Instead of learning different rules for end behavior based on different cases, being able to determine the end behavior model helps students connect any new function to one they already know. At this point, students should confidently be able to determine the end behavior of any power function, and all end behavior models for polynomial and rational functions are power functions.