Algebra II Unit Number 4

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1 Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees. Analysis of polynomial functions provides the foundations for the examination of more complex functions across multiple mathematical concepts and subject areas. Suggested Time Frame Algebra II Unit Number 4 3 rd and 4 th Six Weeks Suggested Duration: days Guiding Questions How are the graphs of ff(xx) = aa(xx h) 3 + kk and ff(xx) = ( 1 (xx bb h)3 + kk related to the graph of ff(xx) = xx 3? How do you sketch the graph of a polynomial function in intercept form? What are some ways to factor polynomials and how is factoring useful? What is the difference between rational and complex solutions in a polynomial equation? Vertical Alignment Expectations Sample Assessment Question COMING SOON Ongoing TEKS Mathematical Process Standards - The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: Apply 1A apply mathematics to problems arising in everyday life, society, and the workplace; Mathematical Practices 1) Make sense of problems and persevere in

2 1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; 1E create and use representations to organize, record, and communicate mathematical ideas; 1F analyze mathematical relationships to connect and communicate mathematical ideas; 1G display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. solving them. 2) Reason abstractly and quantitatively. 3) Construct viable arguments and critique the reasoning of others. 4) Model with mathematics. 5) Use appropriate tools strategically. 6) Attend to precision. 7) Look for and make use of structure. 8) Look for and express regularity in repeated reasoning. Communicate: Have the students: 1) Keep a journal 2) Participate in online forums for the classroom like Schoology or Edmoto. 3) Write a summary of their daily notes as part of the assignments. Create, Use The students can: 1) Create and use graphic organizers for things such as parent functions. This organizer can have a column for a) the name of the functions, b) the parent function equation, c) the graph, d) the domain, e) the range, f) any asymptotes. This organizer can be filled out all at once, or as you go through the lessons. They could even use this on assessments at the teacher s discretion. Analyze Display, Explain, Justify

3 The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. Portions of District Specificity/Examples are from the Austin Area Math Supervisors Clarifying the TEKS Documents found within the Region XI Mathematics resources page. Knowledge and Skills with Student Expectations District Specificity/ Examples Vocabulary Suggested Resources Resources listed and categorized to indicate suggested uses. Any additional resources must be aligned with the TEKS. 2A.7E 2A.6A 2A.7B,C,D Major Points Graphing Cubic Functions Graphing Polynomial Functions Adding and Subtracting Polynomials Factoring Polynomials Multiplying Polynomials Factoring Polynomials Dividing Polynomials Finding Rational Solutions of Polynomial Equations Finding Complex Solutions of Polynomial Equations Coefficient Factor Parameter Real Number Term Degree Transformation Monomial Binomial Trinomial Polynomial Cubic Function Root Textbook Resources HMH Algebra II Text Book Websites:

4 2.A7 Number and Algebraic Methods The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to: 7E determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping;

5 7B add, subtract, and multiply polynomials 7C determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two;

6 7D determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods; 2A.6 Cubic, Cube Root, Absolute Value and Rational Functions, Equations, and Inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to: 6A 3 analyze the effect on the graphs of f(x) = x 3 and f(x) = xx when f(x) is replaced by af(x), f(bx), f(x - c), and f(x) + d for specific positive and negative real values of a, b, c, and d; Example 1: Vertical and Horizontal Transformations of Cubic functions Use the graph of f(x) = x 3 to graph g(x) = (x 2) 3 and h(x) = x The graph of g(x) =(x 2) 3 is the graph of f(x) = shifted x 3 down 2 units. The graph of h(x) = x is the graph of f(x) = x 3 shifted right 4 units. Graph on next page: Example 2: 3 Step 1: Choose several points from the parent function f(x) = xx Step 2: Multiply the y-coordinate by 2. This stretches the parent function by the factor of 2 and reflects the result in the x-axis. Step 3: Translate the graph from step 2 to the left 5 units and down 2 units.. (Example continues on next page)

7 3 3 The graph of g(x) = - xx is the graph x + 5 of f(x) = xx down 2 units. shifted

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