In this activity, students use their calculator to sketch, and then draw conclusions about the end behavior of polynomials with degree ‘n’. Please note that the focus is on end behavior, not creating a ‘good window’. To help students see the function without constantly adjusting the window, have them ‘shrink’ the graph by having an ‘a’ value of 0.2 in front of each function. As for notation, teachers can use this to discuss end behavior how they choose (using limit notation or not). This is a word document so please feel free to edit to fit your purpose.
Utah State Core Standard
Desired Results
Benchmark/Enduring Understanding
End behavior of polynomial functions is important for student understanding of polynomial functions and their graphical representations. Understanding why a function goes in the direction it does as x approaches infinity shows a level of understanding that can be assessed at various levels of Bloom’s Taxonomy.
Essential Questions / Skills
§ What are characteristics of polynomial functions and how does end behavior contribute to the overall affect of these functions? / · Students will be able to sketch graphs of polynomial functions
· Students will be able to look at graphs of polynomials and be able to determine the degree of the polynomial.
Assessment Evidence
Assessment may include observations of students conversations as they move into generalizations about end behavior. Assessment may also include students creating graphs using their knowledge of polynomials and how multiplicity (see investigating multiple roots) and end behavior help determine the shape of these functions.
Instructional Activities
Launch: warm-up or discussion about domain/range and definition of ‘end behavior’
Explore: Students work in small groups to discuss and answer questions about the degree of a polynomial and how this relates to the end behavior of the function.
Summarize: Students draw conclusions and have discourse within the class to create generalizations about end behavior of polynomials.
Materials Needed
Worksheet, calculators
Investigating End Behavior
Use your calculator to graph each polynomial and complete the information below each graph. You will be using this information to draw a conclusion about the end behavior of polynomials of degree ‘n’. To create a vertical shrink to help reduce the output values, put 0.2 in front of each equation (a = 0.2).
Zeroes: ______Zeroes: ______Zeroes: ______
Degree of polynomial: __ Degree of polynomial: __ Degree of polynomial: __
End behavior: ______End behavior: ______End behavior: ______
Zeroes: ______Zeroes: ______Zeroes: ______
Degree of polynomial: __ Degree of polynomial: __ Degree of polynomial: __
End behavior: ______End behavior: ______End behavior: ______
Zeroes: ______Zeroes: ______
Degree of polynomial: __ Degree of polynomial: __
End behavior: ______End behavior: ______
How does a negative in front of the polynomial affect the end behavior?
Use your observations from the previous graphs, explain how you would predict the end behavior of a given polynomial of degree n.
Using your prediction, sketch the following graphs without using a calculator.
Zeroes: ______Zeroes: ______Zeroes: ______
Degree of polynomial: __ Degree of polynomial: __ Degree of polynomial: __
End behavior: ______End behavior: ______End behavior: ______
Zeroes: ______Zeroes: ______Zeroes: ______
Degree of polynomial: __ Degree of polynomial: __ Degree of polynomial: __
End behavior: ______End behavior: ______End behavior: ______
Compare your answers with others, then make any necessary adjustments to your conjecture at the top of this page.