Expand the ideas of factors, dividends, divisors, quotients, and remainders from numbers to functions.
Divide polynomials using an area model.
Explain why rewriting a rational function in equivalent ways can reveal different characteristics of the function, including slant asymptotes.
Quick Lesson Plan
In today’s activity students look at the similarities between writing rational numbers in different but equivalent ways (as improper fractions and mixed numbers) and writing rational functions in different but equivalent ways. The first 2 questions are designed to get students thinking about the different properties and meanings of division. An improper fraction can be thought of as a division problem and the mixed number can be thought of as the answer to the division problem, as well as the ratio or comparison of numerator and denominator, answering the question of “how many times greater?”. This is a helpful perspective for studying end behavior and foreshadows some topics in AP Calculus (L’hospital’s rule!). The purpose behind the number line is the emphasis on locating a number based first on its whole number component. This is exactly what students will do with rational functions as they look for an end behavior model!
On the second page of the activity, students explore the end behavior of the rational function, and discover why this rational function has a slant asymptote. Similarly to Lesson 2.6, students realize that the term 22/(x+4) becomes negligible as x increases and decreases without bound. Thus, at the ends of the graph f behaves like the linear function y=2x-3.
How did you write the improper fraction as a mixed number? What does the 3 represent? What does the 1 represent? What does the 7 represent?
If x+4 divided f(x) evenly, what would the product of the two expressions be?
Is 2x-3 an overestimate or an underestimate for how many times x+4 goes into f(x)? How do you know?
Does f have a horizontal asymptote? How do you know?
What would change if the numerator had the term 2x^3 instead of 2x^2?
It is critical that students understand the relationship between the dividend, divisor, quotient, and remainder. The margin notes are designed to get students to see this connection, first with numbers, and then with rational functions. Establishing the idea of the remainder first as a whole number is imperative to students’ understanding of polynomial long division, specifically how to write the final remainder term. When using the area model, look at the constant term and determine what number needs to be added or subtracted to make the dividend. This is the remainder. Writing this out as a multiplication problem first, just as the margin notes for question 4a do, is very helpful for getting students to identify the remainder and then give the answer to the division problem.