Identify key characteristics of a polynomial function including its degree, leading coefficient, relative and absolute extrema, and points of inflection.
Determine the degree of a polynomial using first, second, third, ...nth differences.
Quick Lesson Plan
We begin Unit 2 right where we left off in Unit 1–by describing the rate of change of a function. In this unit we look at higher degree polynomial functions and rational functions.
Students are given information about the value of a stock in a large sportswear corporation over a one-month period. By applying what they learned in the previous unit, they can tell that the growth is neither linear nor quadratic (it’s cubic, but they don’t know that yet). In question 2-5, students use the graph of S to explore the stock’s maximum and minimum values and to describe how fast the stock is changing. We continue to emphasize using multiple representations to verify a conclusion. For example, students can see from the concavity of a curve that the rate of change is slowing down over the interval [3,9] but they can also see this analytically by comparing the average rate of change over two consecutive subintervals between t=3 and t=9.
What do you notice about how the stock values are changing?
How would you feel about your stock on day 3? How would you feel about your stock on day 18? What do you think is important to remember about the value of a stock?
What do you notice about the slope of the graph at the high points and low points? Why do you think this happens?
Is it possible to have more than one high point?
What is the concavity of the function’s graph on the day when the stock changes the fastest?
Today’s lesson will formally introduce the words relative and absolute minimum and relative and absolute maximum. While we don’t use first derivative justifications, we do ask students to look for places on the graph where the function S changes from increasing to decreasing or decreasing to increasing. In the QuickNotes, note that we add positive and negative slopes to indicate where the function is increasing or decreasing.
Make sure that students know that an absolute maximum is also a relative maximum unless the absolute maximum occurs at an endpoint. Question 2 of the Check Your Understanding will help solidify this as students indicate the number of extrema of a quartic polynomial. We look at the function over its entire domain, so there are no endpoints to consider.
Finally, while students already know that an inflection point indicates where a function is changing at the fastest rate (greatest positive slope or least negative slope), be sure students understand that this is also the point where the concavity of a function changes. This is because concavity indicates how the rates (slopes) are changing, so a switch from concave up to concave down means the function’s rate of change was increasing and is now decreasing and thus hit a maximum rate at that point of inflection. Similarly a function that changes from concave down to concave up indicates that a function’s rate of change was decreasing and is now increasing and thus hit a minimum rate of change (again, not slowest, but most negative) at the point of inflection.
Allow your students to really wrestle with question 4 of the Check Your Understanding. We purposefully chose not to explicitly express these features of polynomial functions related to the location of extrema such as that between any two x-intercepts the function must have a relative maximum or minimum, or that a polynomial with an even degree must have at least one absolute maximum or minimum. These ideas should surface organically as students reason about the shape (and continuity–though we’re not using that word!) of a polynomial function.