Instructional Unit

Graphing Polynomials

Background: My choice of mathematical content came from my teaching of Algebra II in the past few years.  One particular chapter from the textbook that we used in that class was devoted to the study of polynomials.  Some of the topics covered in the chapter were the addition, subtraction, and multiplication of polynomials.  My focus in this instructional unit is on other topics covered in that chapter: the solving of roots, end behavior, and an introduction to relative extrema.

Objectives Addressed in these Lessons (from Georgia Performance Standards for Accelerated Mathematics 2):

á          Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change, and end behavior.  (MA2A1b)

á          Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros.  (MA2A3b)

á          Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior. (MA2A3d)

á          Find real and complex roots of higher degree polynomial equations using the factor theorem, remainder theorem, rational root theorem, and fundamental theorem of algebra, incorporating complex and radical conjugates. (MA2A5a)

á          Solve polynomial, exponential, and logarithmic equations analytically, graphically, and using appropriate technology.  (MA2A5b)

Lessons from the Instructional Unit (and Suggested Class Time.  I believe these times would be based on 50-minute class periods.  It might be the case that lessons here be presented in a different order.)

Lesson #1: End Behavior  (2 classes)

Lesson #2:  DescartesŐ Rule of Signs (2 classes)

Lesson #3: Introduction to Fundamental Theorem of Algebra (2 classes)

Lesson #4: Investigation on Roots of Polynomials (2 classes)

Lesson #5: Introduction to Relative Extrema (2 classes)