Interpret the behavior of a rational function in context, specifically its horizontal asymptote.
Determine the end behavior of a rational function by comparing the dominance of the polynomials in the numerator and denominator.
Explain why the end behavior of a rational function is determined by the quotient of the leading terms in the numerator and denominator.
Quick Lesson Plan
Today students dive into the world of rational functions by looking at the concentration of anesthesia in a patient’s body. First students make sense of the shape of a rational function by sharing noticings and wonderings about the graph of C. In question 3, students practice interpreting a graph to determine when the anesthesia is effective. This is another opportunity to connect graph behavior to contextual scenarios. A large focus of the activity is thinking about what happens to the concentration of anesthesia after a long, long time, arriving at the idea of a horizontal asymptote. It is interesting to note that while the concentration does not actually reach 0 according to this model, a drug is considered out of the patient’s system if the concentration is below 0.7%.
Note that we use a numerical approach in question 2c, a contextual or experiential approach in question 4 and an analytical approach in question 5 to determine what happens to the anesthesia concentration after “a long, long time.”
The analytical approach tends to be the most challenging for students. As you are working with groups, ask them to recall basic characteristics of fractions, namely that a fraction with a (comparatively) very large denominator has a very small value and that a fraction with a (comparatively) very large numerator has a very large value.
As you are monitoring students, continue to use the language of “after a long, long time” or “after a very long time” to get at the idea of a horizontal asymptote.
Is C a polynomial function? How can you tell from the graph? How can you tell from the equation?
What do you think would happen if we evaluated C for t=72?
Will the concentration ever hit 0 exactly? Why or why not?
What happens to the value of a fraction if the denominator gets very large?
What happens to the value of a fraction if the numerator gets very large?
How can you tell whether the numerator or denominator is growing faster?
This lesson on end behavior of rational functions continues to build on what students have already explored with polynomial functions about dominating behavior. We will use limit notation in this course to describe the value that the output is approaching as x increases (and decreases) without bound. Instead of simply giving “rules” for end behavior, have students make their own conjectures about whether there will or will not be a horizontal asymptote and why. Encourage them to think about what they know about fractions and what will make a fraction small versus large.
A major difference in how end behavior is addressed in AP Precalculus compared to Precalculus is the focus on the ratio of leading terms being the end behavior model. Instead of simply comparing the degree, finding the actual end behavior model eliminates the need for different “cases” based on whether the degree of the numerator is greater than, less than, or equal to the degree of the denominator. Instead of telling students to look at the ratio of leading coefficients if the degree of the numerator and denominator is the same, we can help students see that the rational function actually behaves like the constant function y=k (where k is the ratio of leading coefficients) for large values of x. Instead of looking at many different cases, we take time establishing why the end behavior is what it is for simple power functions, and then we compare all other functions to those power functions.