# Representing Polynomials Graphically

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties:

- Recognizing the connection between the zeros of polynomials when suitable factorizations are available and graphs of the functions are defined by polynomials.
- Recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing
*f*(*x*) by*f*(*x*+ k),*f*(*x*) + k, -*f*(*x*),*f*(-*x*)..

## Introduction

The lesson unit is structured in the following way:

- Before the lesson, students attempt the assessment task individually. You then review their work and formulate questions that will help them improve their solutions.
- During the lesson, students work collaboratively in pairs or threes, matching functions to their graphs and creating new examples. Throughout their work students justify and explain their decisions to peers. During a whole-class discussion, students explain their reasoning.
- Finally, students improve their solutions to the initial task and complete a second, similar task.

## Materials required

- Each student will need a mini-whiteboard, pen, and eraser, and a copy of
*Cubic Graphs and Their Equations*and*Cubic Graphs and Their Equations (revisited).* - Each small group of students will need cut-up cards
*Cubic Graphs*,*Cubic Functions*and*Statements to Discuss: True or False?,*a large sheet of poster paper, and a glue stick. You may want to enlarge the cards and/or copy them onto transparencies to be used on an overhead projector to support the whole-class discussion.

## Time needed

20 minutes before the lesson, a 90-minute lesson (or two 50-minute lessons), and 20 minutes in a follow-up lesson. Timings are approximate and will depend on the needs of your students.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*

- H.A-APR: Understand the relationship between zeros and factors of polynomials
- H.A-SSE: Interpret the structure of expressions
- H.F-BF: Build new functions from existing functions
- H.F-IF: Analyze functions using different representations