Lesson 2: Graphs of Functions
Duration of Days: 6
Lesson Objective
Know:
The representative points of a function and how they relate to the information the graph illustrates.
Do:
Recognize and interpret the key points of a function represented on a graph.
What defines a polynomial function?
How does the degree of a polynomial influence the shape of its graph?
What is the difference between even-degree and odd-degree polynomial end behavior?
How do the leading coefficient and degree affect the direction of the graph?
How can we determine the end behavior of a polynomial without graphing every point?
How can technology help verify predictions about polynomial graphs?
Polynomial function
Degree of a polynomial
Leading coefficient
Leading term
Standard form of a polynomial
End behavior
Even-degree polynomial
Odd-degree polynomial
Turning points
Local maximum / Local minimum
Graph behavior
HSA-APR.A.1
Understand that polynomials form a system analogous to integers, namely they are closed under addition, subtraction, and multiplication.
HSF-IF.C.7
Graph functions expressed symbolically and show key features of the graph.
HSF-IF.B.4
Interpret key features of graphs and relate them to the function.
HSF-IF.C.8
Write a function defined by an expression in different but equivalent forms to reveal properties of the function.
Students explore the structure of polynomial functions, focusing on how the degree and leading coefficient influence the shape and end behavior of graphs. Students analyze polynomial equations, predict graphical behavior, and verify their reasoning using graphing technology.
The purpose of this lesson is to help students develop conceptual understanding of polynomial behavior before learning techniques for finding zeros in later sections.
DOK 2 – Skills and Concepts
Support for Developing Learners
Provide visual anchor charts showing typical polynomial shapes.
Use graphing technology (Desmos or graphing calculators) to help students visualize end behavior.
Provide structured notes or guided worksheets.
Start with lower-degree polynomials (quadratic/cubic) before generalizing.
On-Level Strategies
Students predict end behavior before graphing.
Work in small groups to compare graphs and identify patterns.
Practice matching equations with graphs.
Enrichment / Extension
Challenge students to determine possible equations from given graphs.
Explore maximum number of turning points (n - 1).
Introduce polynomial modeling with real data.
Exit Tickets and Quiz
Algebra and Trigonometry — Stewart, Redlin, Watson
Technology Resources
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Desmos Graphing Calculator
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TI-83 / TI-84 Graphing Calculators
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SmartBoard graph demonstrations