Rational and Polynomial Functions

9 Components
Initial Task
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Representing Polynomials
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 4 End of Unit Assessment Teacher Resources ALG II Unit 4 Find aligned Regents questions with Quiz Banker
A2 U4: Rational and Polynomial Functions
Browse Components
Initial Task
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Representing Polynomials
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 4 End of Unit Assessment Teacher Resources ALG II Unit 4 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 1

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

1 week
See Standards
Common Core: Major Standards:
F-IF.B.4

Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

A-APR.B.3

Understand the relationship between zeros and factors of polynomials.​  Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Tasks include quadratic, cubic, and quartic polynomials and polynomials in which factors are not provided.  For example, find the zeros of f(x) = (x2 - 1)(x2 + 1).

Common Core: Supporting Standards:
F-IF.C.9

Analyze functions using different representations. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Common Core: Plus Standards:
N-CN.C.9(+)

Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

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