Lesson 4 | Polynomials | 11th Grade Mathematics

Objective

Sketch polynomial functions using sign charts and analysis of the factored form of the polynomial function.

Common Core Standards

Core Standards

The core standards covered in this lesson

A.APR.B.3 — 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.

Arithmetic with Polynomials and Rational Expressions

A.APR.B.3 — 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.

Foundational Standards

The foundational standards covered in this lesson

F.IF.B.4

Interpreting Functions

F.IF.B.4 — 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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Given the equation of a polynomial equation in factored form, identify whether the function is positive or negative between the roots.
Identify the general shape of the function based on the factors of the polynomial written in factored form.
Sketch polynomial functions by analyzing the interval between roots.
Write possible polynomial equations based on a sketch of a polynomial function and/or information about a particular polynomial function.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Brainstorming: Graphing a Polynomial Using a Sign Chart may be helpful to provide background knowledge on how to develop a sign chart for graphing functions.

Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Sketch a polynomial function with the following marked roots and that meets the conditions below.

Positive when $${x<-3}$$
Negative when $${-3<x<0 }$$
Positive when $${ 0<x<2}$$
Positive when $${x>2}$$

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 2

Below is a polynomial in factored form.

$${y=-(x+1)(x+2)^2(3x)^2}$$

Chart the roots on the number line below, and identify whether the function is positive or negative at the ends and between each root.

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Below is a polynomial function with roots at $${x=1}$$, $${x=-2}$$,and $${x=0}$$.

Write a possible polynomial equation that could model this function. Explain your reasoning.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students create a polynomial function based on constraints provided.
Include problems where an incomplete set of constraints is given and students need to modify the constraints to match a given equation.
Include an open graph of a polynomial for which students need to identify a possible equation based on constraints.

Analyze Math Graphing Polynomials

Lesson 3

Lesson 5

Lesson Map

Topic A: Polynomial Features and Graphs

Classify polynomials through identification of degree and leading coefficient. Graph a polynomial function from a table of values; prove degree using successive differences.

A.APR.A.1 F.IF.C.7.C F.IF.C.9

Identify features of polynomial functions including end behavior, intervals where the function is positive or negative, and domain and range of function.

F.BF.B.3 F.IF.B.4 F.IF.B.6 F.LE.A.3

Match and compare equations and graphs of polynomials, and identify transformations.

F.BF.B.3 F.IF.B.4 F.IF.C.7.C

Sketch polynomial functions using sign charts and analysis of the factored form of the polynomial function.

A.APR.B.3

Topic B: Operations with Polynomials

Multiply polynomials, identifying factors, degrees, and number of real roots. Identify features of a polynomial in standard form.

A.APR.A.1 F.BF.B.3

Add and subtract polynomials. Identify degree, leading coefficient, and end behavior of result.

A.APR.A.1 F.BF.A.1.B

Divide polynomials by binomials to determine linear factors.

A.APR.A.1 A.APR.B.2 A.APR.B.3

Determine if a binomial is a factor of a polynomial using the remainder theorem.

A.APR.B.2

Identify and factor with difference of two squares in quadratic and quartic polynomials. Describe identity of difference of two squares. Describe the zeros that represent the resultant factors.

A.APR.B.3 A.SSE.A.2

Factor difference and sums of cubes. Explain sums and differences of cubes as polynomial identities.

A.APR.C.4

Factor polynomials by grouping in quartic, cubic, and quadratic functions.

F.IF.C.8.A

Topic C: Polynomial Extensions

Use polynomial identities to determine Pythagorean triples.

A.APR.C.4

Identify the solution(s) to systems of polynomial functions.

A.REI.D.11

2 days

Write polynomial functions from solutions of that polynomial function.

A.APR.B.3

Polynomials

Objective

Common Core Standards

Core Standards

Arithmetic with Polynomials and Rational Expressions

Foundational Standards

Interpreting Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy