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 5

Rational functions describe the quotient of two polynomial functions.

1 week
See Standards
Common Core: Major Standards:
A-APR.B.2

Understand the relationship between zeros and factors of polynomials.  Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

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).

A-CED.A.1

Create equations that describe numbers or relationships. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A-REI.A.1

Understand solving equations as a process of reasoning and explain the reasoning. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-REI.A.2

Understand solving equations as a process of reasoning and explain the reasoning.  Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A-REI.B.4b

Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

A-SSE.A.2 Algebra II

Use the structure of an expression to identify ways to rewrite it. Tasks are limited to polynomial, rational, or exponential expressions.  For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).   

Common Core: Supporting Standards:
A-APR.D.6

Rewrite rational expressions.  Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.  From the PARCC Model Content Frameworks: “This standard sets an expectation that students will divide polynomials with remainder by inspection in simple cases. For example, one can view the rational expression (x+4)/(x+3) as (x+4)/(x+3) = (x+3)+1/(x+3) = 1 + 1/(x+3) ."

Evidence of Understanding

  • describe connections between a rational equation and the features of its graph
    • describe the general shape and patterns for the graph of a rational function including domain, range, end behavior, and asymptotes
    • use the equation to justify vertical asymptotes on the graph (limits on the domain)
      • possible extension: use the degree of the numerator and denominator of the rational function to predict oblique asymptotes and use long division to determine their equations
    • use the structure of an equation to identify ways to rewrite it and determine asymptotes (mainly via factoring but possibly also applying properties of polynomials)
  • solve rational functions in the context of a situation
    • build equations that model the relationship between two variables in a rational function
    • connect equivalent mathematical representations (an equation to a situation, etc.) and justify the connection
    • use multiple mathematical representations (equation, table, graph) to uncover solutions to a problem or answer questions for a given situation
    • rewrite rational expressions in different forms using long division when necessary
      • Example:  (x+4)/(x+3) = (x+3) +1/(x+3) = 1 + 1/(x+3)
      • Example:  (x3 + 5x2 + 3x - 2)/(x + 1) = x2 + 6x + 9 + 7/(x + 1)
    • solve rational equations and verify extraneous solutions

Develop conceptual understanding:

rational function, asymptote, extraneous solution, (+)oblique asymptote

Supporting terms to communicate:

domain, range, end behavior, degree, polynomial, roots, zeros, variable, term, binomial factor
Core Resource

A core resource supports multiple days of instruction.