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# Analyzing Polynomial Functions

Lesson 11 of 15

## Objective: SWBAT name intervals on which a polynomial is increasing or decreasing and identify minima and maxima of the functions.

## Big Idea: Polynomial functions behave differently over different values of input. We can characterize this behavior using terms like "increasing," "decreasing," "minimum," and "maximum."

*90 minutes*

#### Warm-up

*20 min*

As my students enter the classroom today, I hand each a slip of paper from the collection in Polynomial Vocabulary Matching. Each slip is either a polynomial vocabulary term or the definition of the term. When everyone has arrived, I ask students to match up the term with the definition by moving around the room and talking to each other about what is on their own paper. I tell students that when they find the person who has the matching paper, they should gather up their belongings and sit together to work today.

#### Resources

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When everyone has found their match, I ask the pair that had "increasing" to stand up and read aloud their strips of paper. Then I ask the "decreasing" pair to do the same. I write out the definitions as they share and then show an example of a polynomial graph that increases on some intervals and decreases on others. I emphasize that we talk about where a function increases or decreases using x values because this is often a source of confusion for my students.

We discuss each of the terms in this way, with me writing out the definitions on the board and providing examples where necessary [MP6].

#### Resources

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For the next segment of the lesson, my students will continue to work with the partners they found in the Warm Up. WS Analyzing Graphs of Polynomials is a collection of four polynomial graphs for students to analyze. For each graph they will determine:

- The location (x-value) of any local minima
- The location (x-value) of any local maxima
- The domain and the range
- The end behavior
- The intercepts (both x- and y-)
- The intervals where the function is increasing
- The intervals where the function is decreasing

As students examine the graphs, I will interact with them to make sure that they are actively practicing "decontextualizing" [**MP2**] and making use of structure [**MP7**] to reason clearly, and, in their own words. When my students have finished this work, we will review all four questions as preparation for the following day's quiz on graphing polynomials.

#### Resources

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#### Exit Ticket and Assignment

*15 min*

For today's **Exit Ticket**, my students individually answer Exit Ticket Polynomial Graphs, a set of four questions on analyzing polynomial graphs. I send these questions to students through the TI NSpire Navigator so that we can all see the results immediately.

The assignment for this evening is to prepare for tomorrow's quiz on graphing polynomials. This quiz will cover:

- Creating a good sketch of the graph of factorable polynomial functions
- Making a good guess about the algebraic form of a polynomial presented graphically
- Analyzing the graph of a polynomial function

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- LESSON 1: Seeing Structure in Expressions - Factoring Higher Order Polynomials
- LESSON 2: Proving Polynomial Identities
- LESSON 3: Polynomial Long Division and Solving Polynomial Equations
- LESSON 4: The Remainder Theorem
- LESSON 5: The Fundamental Theorem of Algebra and Imaginary Solutions
- LESSON 6: Arithmetic with Complex Numbers
- LESSON 7: Review of Polynomial Roots and Complex Numbers
- LESSON 8: Quiz and Intro to Graphs of Polynomials
- LESSON 9: Graphing Polynomials - End Behavior
- LESSON 10: Graphing Polynomials - Roots and the Fundamental Theorem of Algebra
- LESSON 11: Analyzing Polynomial Functions
- LESSON 12: Quiz on Graphing Polynomials and Intro to Modeling with Polynomials
- LESSON 13: Performance Task - Representing Polynomials
- LESSON 14: Review of Polynomial Theorems and Graphs
- LESSON 15: Unit Assessment: Polynomial Theorems and Graphs