Big Idea:

Big Idea 1

The degree of a polynomial function determines its behaviors and properties.

1 week

Evidence of Understanding

  • describe connections between a polynomial equation and the features of its graph
    • correlate the equation’s degree to the maximum number of real roots and end behavior
    • distinguish a minimum or maximum point as absolute or relative using a function’s degree  
    • generalize how the equation’s lead coefficient impacts the graph’s direction and rate of change
    • classify a function as odd, even or neither from both the equation and its graph
      • analyze examples and nonexamples including polynomial, absolute value, trigonometric, rational, exponential, logarithmic, and radical functions
      • describe how the graphs of even or odd functions reflect in the coordinate plane
      • explain why the equation for an odd function cannot have a constant term
         
  • investigate and and determine specific features of a parabola and its quadratic equation
    • study various forms of a quadratic equations (factored, standard, vertex) to understand why a quadratic function has at most two real zeros
      • possible extension: show how the degree limits a quadratic to at most 2 real zeros
    • explore how the axis of symmetry relates to roots, the vertex, and other points on the parabola
      • justify why the vertex always lies on the axis of symmetry
      • use the axis of symmetry to explain why the vertex is located in the middle of the roots, and is the only point with an unrepeated output value
    • approximate the roots of a given parabola and verify them using its quadratic equation
      • use the degree to classify roots as real or imaginary
      • connect characteristics of the roots with characteristics of the quadratic equation  
    • predict and justify the end behavior of a quadratic equation using its degree and lead coefficient

Develop conceptual understanding:

polynomial, degree, roots, zeros, absolute, relative, odd, even, quadratic, cubic, quartic, axis of symmetry, vertex  

Supporting terms to communicate:

lead coefficient, exponent, reflect, minimum, maximum, end behavior, vertex form, standard form, factored form

Core Resource
A core resource supports multiple days of instruction.
  • Distinguishing Functions: Even, Odd, Neither
    This single-day resource supports students in identifying even and odd functions in both table and graph form and come up with their own definition.
    Resource:
    Distinguishing Functions: Even, Odd, Neither

    Students will look at examples of Even and Odd functions to generate their own definitions and then further refine their definitions and understanding with a matching activity. There will also be opportunity for students to re-visit and refine their understanding of different function families.

    All Resources From:
Instructional Routine: Connecting Representations
These tasks are embedded within the instructional routine called Connecting Representations.