CMS Curriculum Guides 2011-2012 Algebra II Unit Title: Polynomial Functions and Operations Unit Title: Time: Suggested Suggested Timeline: 15 days Enduring understanding (Big Idea): Factor polynomials in multiple forms, analyze polynomial functions and their graphs by identifying end behavior and roots, graph polynomial functions and inverses Essential Questions: How are factors and roots of a polynomial equation related? How are a function and its inverse function related? Common Core Standards Mathematical Textbook Practices Alignment A-SSE.1. Interpret expressions that represent a quantity in terms of its ★ context. 1, 2,3 1-1 o Interpret parts of an expression, such as terms, factors, and coefficients. Connection to 2003 Standards Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). 7 4-4, 4-5, 4-6, 4-7 A-APR.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. 1 5-3, 5-4 A-APR.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). 1 *See 5-4 Remainder Theorem A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct 4 *See 5-2 problem 2, 3 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems. CMS Curriculum Guides 2011-2012 Algebra II a rough graph of the function defined by the polynomial. A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. 2, 3 5-5, 5-6 A-APR.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.1 1 5-7 F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated ★ cases. 4 5-1, 5-2 o c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. o a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F-BF.4. Find inverse functions. o Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. o (+) Verify by composition that one function is the inverse of another. o (+) Read values of an inverse function from a graph or a table, given that the 2.06 Use cubic equations to model and solve problems. a. Solve using tables and graphs. Interpret constants and coefficients in the context of the problem. 4, 8 5-8 1 6-6, 6-7 2.01 Use the composition and inverse of functions to model and solve problems; justify results. CMS Curriculum Guides 2011-2012 Algebra II function has an inverse. (+) Produce an invertible function from a non-invertible function by restricting the domain. Prior Knowledge: Solving equations, basic quadratic factoring, graphing quadratic functions Key Vocabulary Discriminant, Greatest Common Factor, Quadratic Formula, End Behavior, Relative Maximum, Relative Minimum, Turning, Point, Synthetic Division, Composite Function, Inverse Function Resources: Pearson Algebra II Dynamic Activities (log in to pearsonsuccessnet.com and access the Interactive Digital Path): 4-4, 4-5, 4-6, 4-7, 5-2, 5-4 Problem-Based Task: o Chapter 4 Tasks : Task 3 only o Chapter 5 Tasks o Chapter 6 Tasks : Tasks 3 and 4 only PowerAlgebra.com, Teacher Online Access pack, Algebra 2 Companion – vocabulary, key concepts, got it? Honors – see Challenge problems at the end of sections, advanced level assignments, dynamic activities, enrichment worksheets, extended concept bytes o 4-5 Enrichment o 4-7 Enrichment o Chapter 5 Project o Chapter 6 Project