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1 Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes STRAND 3: QUADRATIC FUNCTIONS TOPIC 3.3: APPLICATIONS Topic Notes Mathematical focus Participants put quadratic functions to use in maximization problems, learn how to extend standard textbook max/min problems to bring out the rich mathematics, and discover a real world use for complex numbers. Topic overview This topic includes 3 tasks: Task 3.3.1: Maximizing Revenue and Profit Task 3.3.2: Making Application Problems Richer Task 3.3.3: “Shocking” Numbers These tasks bring together the knowledge the participants have developed about quadratic functions. In Task 3.3.1, participants analyze ways to find a quadratic equation using matrices and finite differences. The big idea here is to examine various methods for defining a quadratic function given a pattern of a set of data. In Task 3.3.2, participants get a chance to develop their own maximization problems. Task 3.3.3 provides an example of the use of complex numbers in electrical engineering, addressing the age-old question, “Does anybody actually use this stuff?” TExES standards focus TExES Standard I.002 Number concepts. The teacher understands the complex number system and its structure, operations, algorithms, and representations. The beginning teacher: (A) Demonstrates how some problems that have no solution in the real number system have solutions in the complex number system. (E) Describes complex number operations (e.g., addition, multiplication, roots) using symbolic and geometric representations. TExES Standard II.004 Patterns and algebra. The teacher uses patterns to model and solve problems and formulate conjectures. The beginning teacher: (A) Recognizes and extends patterns and relationships in data presented in tables, sequences, or graphs. TExES Standard II.005 Patterns and algebra. The teacher understands attributes of functions, relations, and their graphs. The beginning teacher: (B) Identifies the mathematical domain and range of functions and relations and determines reasonable domains for given situations. (C) Understands that a function represents a dependence of one quantity on another and can be represented in a variety of ways (e.g., concrete models, tables, graphs, diagrams, verbal descriptions, symbols). December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 2 Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes TExES Standard II.006 Patterns and algebra. The teacher understands linear and quadratic functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems. The beginning teacher: (G) Models and solves problems involving linear and quadratic equations and inequalities using a variety of methods, including technology. TExES Standard V.018 Mathematical processes and perspectives. The teacher understands mathematical reasoning and problem solving. The beginning teacher: (E) Understands the problem-solving process (i.e., recognizing that a mathematical problem can be solved in a variety of ways, selecting an appropriate strategy, evaluating the reasonableness of a solution). TExES Standard V.019 Mathematical processes and perspectives. The teacher understands mathematical connections both within and outside of mathematics and how to communicate mathematical ideas and concepts. (A) Recognizes and uses multiple representations of a mathematical concept (e.g., a point and its coordinates, the area of a circle as a quadratic function of the radius, probability as the ratio of two areas, area of a plane region as a definite integral). (B) Understands how mathematics is used to model and solve problems in other disciplines (e.g., art, music, science, social science, business). (C) Translates mathematical ideas between verbal and symbolic forms. (D) Communicates mathematical ideas using a variety of representations (e.g., numeric, verbal, graphical, pictorial, symbolic, concrete). (E) Understands the use of visual media, such as graphs, tables, diagrams, and animations, to communicate mathematical information. TEKS/TAKS focus TEKS 2A.1 Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to: (B) collect and organize data, make and interpret scatter plots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. TEKS 2A.2 Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: (A) use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations; and (B) use complex numbers to describe the solutions of quadratic equations. TEKS 2A.3 Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to: December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 3 Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes (B) use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and (C) interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts. TEKS 2A.6 Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student is expected to: (A) determine the reasonable domain and range values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities; and (B) relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions. TEKS 2A.8 Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (A) analyze situations involving quadratic functions and formulate quadratic equations or inequalities to solve problems. High School TAKS Objective 1: The student will describe functional relationships in a variety of ways. High School TAKS Objective 2: The student will demonstrate an understanding of the properties and attributes of functions. High School TAKS Objective 5: The student will demonstrate an understanding of quadratic and other nonlinear functions. Materials Task 3.3.1: Maximizing Revenue and Profit Graphing calculator Chart paper X Task 3.3.2: Making Application Problems Richer X X Task 3.3.3: “Shocking” Numbers X Procedure These three tasks stand alone. Task 3.3.1 provides a nice review of matrix algebra. If there is time for only one task from this topic, then Task 3.3.2 is the most crucial, as it gives participants experience solving and enriching maximization problems. December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 4 Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes Summary The big ideas here are to examine various methods for defining a quadratic function given a pattern of a set of data. Participants use matrices, regression equations and finite differences to find quadratic equations. Assessment What follows is an assessment that participants should be able to complete upon finishing task 3.3.1. December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 5 Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes Reflect and Apply Consider the data below. x 0 y 1.0 1 3.01 2 5.04 3 7.09 50 126.0 1. Use the first three points to find a quadratic function that fits the data. Check your function using the 5th data point. 2. Find the formula for a linear function that passes through the second point and the third point in the table. 3. Find the value of the linear functions at x = 3 and compare that value to the value of the quadratic at x = 3 . 4. Compare the values of the linear and quadratic functions at x = 50 . 5. Use a graphing calculator to graph both functions on the same axes to estimate your answers. Approximately what x-values would you have to use so that the resulting function values for the quadratic function and the linear function would not differ from each other by more than 0.5? December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 6 Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes Reflect and Apply Solutions: 2 1. y = 0.01x 2 + 2x + 1 , 0.01( 50 ) + 2 ( 50 ) + 1 = 126 2. y = 2.03x + 0.98 . 3. Linear function: x = 3, y = 7.07 Quadratic function: x = 3, y = 7.09 4. Linear function: x = 50, y = 102.48 Quadratic function: x = 50, y = 126 5. Participants may examine the table or graph the functions to determine that this difference occurs for 0 ! x < 8.6 . Task 3.3.2 is best suited to the teacher journal, whereas Tasks 3.3.1 and 3.3.3 would work well with the student journal. Teacher use only Task 3.3.1 Task 3.3.2 Task 3.3.3 Modify for students * Ready for students * * December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.