Lesson 2 | Polynomials | 11th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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.

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.
F.IF.B.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 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.

Interpreting Functions

F.IF.B.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 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.

F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Linear, Quadratic, and Exponential Models

F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

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

Describe end behavior as the value of y the function approaches as the value of $$x$$ either increases to infinity or decreases to negative infinity.
Describe how the shape of the function, domain of the function, and end behavior change based on whether the degree of the function is odd or even.
Describe the effect the leading coefficient has on the shape of the function and the end behavior of the function.
Explain that when all coefficients and terms are unchanged except the degree of the polynomial, the rate of change over a fixed interval will increase with an increasing degree.

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Anchor Problems

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

Problem 1

What do all of these functions have in common? What's different?


$${h(x)=x^2}$$	$${k(x)=x^4}$$	$${l(x)=x^6}$$	$${r(x)=x^8}$$

What do all of these functions have in common? What's different?


$${t(x)=x}$$	$${j(x)=x^3}$$	$${m(x)=x^5}$$	$${w(x)=x^7}$$

Guiding Questions

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Problem 2

Write a function and draw a sketch that meets the constraints below.

Your function tends toward negative infinity when $$x$$ is very small and tends toward positive infinity when $$x$$ is very large.
Your function has the marked points as $$x$$-intercepts.

Guiding Questions

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References

Building Polynomial Functions — Challenge #1

Target Task

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

Match the graph with the equation and describe the end behavior.

1.	a. $${y=\frac{1}{2}x^2}$$
2.	b. $${y=x^3-8}$$
3.	c. $${y=-x^4-x^3+4x+2}$$
4.	d. $${-3x^3}$$
5.	e. $${3x^5-x^3+4x+2}$$

References

EngageNY Mathematics Algebra II > Module 1 > Topic B > Lesson 15 — Exit Ticket

Algebra II > Module 1 > Topic B > Lesson 15 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

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 an “always sometimes never” statement is given, such as “the end behavior is the same for even functions.”

I Speak Math Polynomial Roots and End Behavior Card Sort — Incorporates non-real roots as well as real roots
EngageNY Mathematics Algebra II > Module 1 > Topic B > Lesson 15
Building Polynomial Functions — Use the rest of the problems from this activity
Mathematics Vision Project: Secondary Mathematics Three Module 3: Polynomial Functions — Lesson 3.3, pages 12 and 13

Lesson 1

Lesson 3

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

Building Functions

Interpreting Functions

Interpreting Functions

Linear, Quadratic, and Exponential Models

Foundational Standards

Interpreting Functions

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Target Task

References

Additional Practice

Lesson Map

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