Algebra 2 (& GT)
Unit 4: Polynomial Functions
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What will my child learn?
Students will:
Perform arithmetic operations on polynomials.
A.APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials (SAT® Content  PAM.04).
Build a function that models a relationship between two quantities.
F.BF.A.1
Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations.
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Interpret functions that arise in applications in terms of a context.
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; and end behavior.*
Build new functions from existing 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.
Analyze functions using different representations.
F.IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Understand the relationship between zeros and factors of polynomials.
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.
Represent and solve equations graphically.
A.REI.D.11
Explain why the xcoordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.
Use polynomial identities to solve problems.
A.APR.C.4
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
A.APR.C.5 (+)
Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s triangle. (Algebra II GT only)
Understand the relationship between zeros and factors of polynomials.
A.APR.B.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x) (SAT® Content  PAM.05).
Students will:
Perform arithmetic operations on polynomials.
A.APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials (SAT® Content  PAM.04).
 Background Info.
 Check for Understanding: Adding and Subtracting Polynomials  Multiplying Polynomials
 Review/Rewind: Adding and Subtracting Polynomials Review  Multiplying Polynomials Review
 Enrichment Tasks: Powers of 11
Build a function that models a relationship between two quantities.
F.BF.A.1
Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations.
 Background Info.
 Check for Understanding: Modeling with OneVariable Equations and Inequalities
 Review/Rewind: Modeling with Combined Functions
 Enrichment Tasks: Lake Algae  A Sum of Functions
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
 Background Info.
 Check for Understanding: Zeros of Polynomials and their Graphs  Positive and Negative Intervals of Graphs  Intro to End Behavior
 Review/Rewind: Zeros of Polynomials and their Graphs  Positive and Negative Intervals of Polynomials  End Behavior of Polynomials
 Enrichment Tasks: Graphs of Power Functions  Running Time
Interpret functions that arise in applications in terms of a context.
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; and end behavior.*
 Background Info.
 Check for Understanding: Interpreting Features of Functions  Positive and Negative Parts of Functions
 Review/Rewind: Match Features of a Modeling Function to Its Real World Meaning  Positive and Negative Parts of a Function
 Enrichment Tasks: Telling a Story with Graphs  Average Cost
Build new functions from existing 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.
 Background Info.
 Check for Understanding: Even and Odd Functions  Shifting and Reflecting Functions
 Review/Rewind: Intro to Function Symmetry  Shifting and Reflecting Functions
 Enrichment Tasks: Building a General Quadratic Function  Transforming the Graph of a Function
Analyze functions using different representations.
F.IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
 Check for Understanding: Comparing Features of Functions
 Review/Rewind: Comparing Features of Functions Review
 Enrichment Tasks: Throwing Baseballs
Understand the relationship between zeros and factors of polynomials.
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.
 Background Info.
 Check for Understanding: Using Zeros to Graph Polynomials
 Review/Rewind: Zeros of Polynomials and their Graphs
 Enrichment Tasks: Graphing From Roots
Represent and solve equations graphically.
A.REI.D.11
Explain why the xcoordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.
 Background Info.
 Check for Understanding: Intersecting Functions  Systems of Nonlinear Equations
 Review/Rewind: Solving Systems of Equations Graphically  Interpreting Equations Graphically
 Enrichment Tasks: Population and Food Supply  Two Squares are Equal
Use polynomial identities to solve problems.
A.APR.C.4
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
 Background Info.
 Check for Understanding: Polynomial Identities
 Review/Rewind: Analyzing Polynomial Identities
 Enrichment Tasks: Trina's Triangles
A.APR.C.5 (+)
Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s triangle. (Algebra II GT only)
 Check for Understanding: Binomial Theorem
 Review/Rewind: Intro to the Binomial Theorem
 Enrichment Tasks: Powers of 11
Understand the relationship between zeros and factors of polynomials.
A.APR.B.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x) (SAT® Content  PAM.05).
 Background Info.
 Check for Understanding: Reminder Theorem of Polynomials
 Review/Rewind: Polynomial Remainder Theorem
 Enrichment Tasks: Zeros and Factorization of a Non Polynomial Function  The Missing Coefficient
What are some signs of student mastery?

Tools & Technology
Desmos is a free online graphing calculator that works on any computer or tablet without requiring any downloads. A FREE Desmos iPad app is available too! Practice your skills with Adding and Subtracting Polynomials Battleship 