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Algebra 2: Unit 3 Polynomial, Rational and Radical Functions

Page history last edited by Christine Koerner 4 years, 2 months ago

 

Algebra 2 Unit 3: Polynomial, Rational and Radical Functions

Unit Driving Question 

How are real situations modeled with polynomial functions?

 

Essential Questions 

  1. What information is helpful in identifying polynomial functions?

  2. What is the relationship between changes in a polynomial equation and the graph?

  3. How do polynomials, rational and radical functions represent real-world scenarios?

  4. What information would help you identify relationships between complex and real numbers?

  5. Why are solutions considered “extraneous”?

 

Launch Task 

Big Ideas for Development Lessons 

Closure & Assessment 

1 Lesson 

8 Weeks (approximately 1-2 weeks per big idea) 

1 Week 

Launch Task: Polygraph-Polynomials

 

Note:

This Unit was originally designed to be one large unit, but has been split into two parts to better facilitate ongoing interim and benchmark assessments.  

Click on the links below to see each Big Idea's Lesson Overview (includes links to teacher notes and student activities)

Part A:

  1. Big Idea 1: The behavior, properties, and characteristics of a polynomial function can be determined by its degree. 
  2. Big Idea 2: The graph and equation of a polynomial function yields information about its roots. 
  3. Big Idea 3: Basic arithmetic operations can be performed on polynomials similar to the way they are performed on numbers. 

 

Part B: 

  1. Big Idea 4: Rational functions can be described by the division of two polynomial functions. 
  2. Big Idea 5: Radical expressions can be written as expressions with rational exponents, and both are inverse operations of integral exponents. 

 

Big Idea 1: The behavior, properties, and characteristics of a polynomial function can be determined by its degree. 

OAS-M: A2.A.1.4, A2.A.2.1, A2.F.1.1 

Lessons and Additional Activities

 

Big Idea 1 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

 

Discover and describe the characteristics of a polynomial function and its graph in mathematical and real-world situations.

  • Using various tools, generalize characteristics of polynomial functions and their graphs.

    • Use graphs, tables, and equations with the aide of technology to allow students to grapple with the differences and similarities between polynomials of varying degree, including targeting rates of change.

      • Identify how the degree of the polynomial relates to end behavior, maximum number of real roots, and possible shape of the graph.

      • Explain how the leading coefficient affects the end behavior and rate of change of the graph.

      • Identify a polynomial function as having even or odd degree based on the graph or equation.

        • Extension: Categorize other functions as even, odd, or neither.

  • Predict the degree and behavior of a polynomial given a graph, or table of values.

  • Research or develop scenarios that require a polynomial function to model the situation. Use technology to model and explain.


Identify critical points and key features of the graph of a polynomial function and interpret meanings in mathematical and real-world situations.

  • Identify the following features from the table or graph of a polynomial function using words, algebraic symbols, equations and inequalities, interval notation: intercepts, symmetry, maximum and minimum points, intervals of increase and decrease, intervals of positive and negative values, domain and range.

  • Describe critical points and key features in the correct context of a mathematical and real world situation.


Make generalizations about how transformations affect key features of the graph of a polynomial function.

  • Form conjectures about the effects each transformation will have on critical points and key features.

    • Use graphs, tables and various tools to justify the conclusions.

    • Relate translations affects to real-world scenarios.

 

Big Idea 2: The graph and equation of a polynomial function yields information about its roots.

OAS-M: A2.A.1.4 A2.A.2.1 A2.N.1.1A2.N.1.2

Lessons and Additional Activities

 

Big Idea 2 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

 

Recognize and explore connections between the graph of a polynomial and its zeros.

  • Identify zeros of a polynomial function by exploring the graph using technology.

    • Verify the zeros algebraically using a variety of methods. (ie, from the graph, from the polynomial’s function rule, etc.)

      • Add, subtract, multiply radicals and complex numbers.

    • Explore and form a conjecture about multiplicities of zeros.

      • Recognize intervals of increase and decrease and positive and negative function values.

  • Build the equation of a polynomial function from its graph.


Develop, analyze and apply general characteristics of the zeros of a polynomial function based on its graph.

  • Predict possible roots of a polynomial from the equation, table and/or graph.

  • Calculate roots of polynomial functions algebraically and graphically.

    • Choose and justify which method is most efficient to determine roots based on the characteristics of the polynomial, (ie. factoring, rational root theorem, synthetic division, long division, analyzing a graph or a table, etc.)


Solve polynomial functions in the context of real-world problems.

  • Create polynomial equations, graphs or tables that model real-world problems.

    • Identify and analyze key features and interpret the meaning of each in the context of the problem.

    • Graph and predict future outcomes using polynomial models.

   

Big Idea 3: Basic arithmetic operations can be performed on polynomials similar to the way they are performed on numbers.

OAS-M: A2.F.1.5 

Lessons and Additional Activities

 

Big Idea 3 Lessons 1-2 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

 

Analyze connections between arithmetic operations on polynomials and the effect it has on the graph.

  • Discover, through technology and other means, how arithmetic operations affect the shape and key characteristics of polynomials.

  • Predict the shape and characteristics of the resulting graph after arithmetic operations are performed on two or more functions. (ie. degree change when multiplying two or more polynomials-how will this affect end behavior, zeros, max, mins, etc.)

  • Perform arithmetic operations on polynomial functions. Justify your answers graphically. Determine key characteristics.

 

Discover how polynomials and operations on polynomials are modeled in real-world scenarios.

  • Research and develop polynomial function models that involve various arithmetic operations, using current real-world data and scenarios. (ie, revenue, cost, sales, etc.)

    • Use technology to build graphs, tables, charts to display research.  

Big Idea 4: Rational functions can be described by the division of two polynomial functions. 

OAS-M: A2.A.1.3 A2.A.2.2 A2.F.1.6

Lessons and Additional Activities

 

Big Idea 4 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

 

Discover and describe the characteristics of a rational function and its graph in mathematical and real-world situations.

  • Analyze the graph of a rational function using technology. Identify critical points and key characteristics.

    • Using graphs, tables and equations, find patterns, similarities, and difference between various rational functions. Form conjectures about various characteristics. (ie holes, asymptotes, oblique asymptotes, etc.)

  • Calculate critical points, asymptotes, holes, and intervals of importance of a rational function using various algebraic methods (ie factoring, solving equations, etc.).

    • State domain restrictions. Verify using technology.

  • Build a rational function when given information and/or a graph. 

Big Idea 5: Radical expressions can be written as expressions with rational exponents, and both are inverse operations of integral exponents. 

OAS-M: A2.N.1.4 A2.A.1.5 A2.F.1.7

Lessons and Additional Activities

 

Big Idea 5 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

 

Discover and describe the characteristics of a radical function and its graph in mathematical and real-world situations.

  • Analyze, using technology, the square root and cube root function rules and graphs. Identify key characteristics. Notice similarities and differences.

    • Use tables and graphs to conjecture about the domain, orientation, translations from the parent graph, critical points, etc.

    • Compare and contrast the quadratic function and its inverse.

    • Compare and contrast the cubic function and its inverse.

      • Extension: Explore patterns with higher degree polynomials and their inverses using technology (i.e., Desmos or graphing calculators). Compare and contrast with previous findings.

  • Discover how a radical expression can be expressed using rational exponents.

    • Verify algebraically and by using technology.

  • Investigate radical equations using technology.

    • Solve radical equations algebraically. Verify with technology when and why solutions are extraneous. 

 

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