Course: Algebra 1 Unit #5: Quadratic Functions

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1 Course: Algebra 1 Unit #5: Overarching Question: What patterns of change are modeled by quadratic functions as seen in real-world situations, and the tables, graphs, and function rules that represent these situations? Previous Unit: Exponential Functions This Unit: Next Unit: Solving Quadratic Equations Questions to Focus Assessment and Instruction: 1. What are the similarities and differences between quadratic, exponential, and linear functions? 2. How do changes in the values of the parameters in a quadratic function change the behavior of the graph? 3. What is the relationship between the number of real roots and the graph of a quadratic equation? Why does this relationship exist? 4. How can you translate among the vertex, standard, and factored forms of quadratic equations? What is useful about each form? 5. What are the characteristics of quadratic change as shown in a graph, function rule, table, or real-world situation? Intellectual Processes (Standards for Mathematical Practice): Make sense of problems and persevere in solving them: Build new mathematical knowledge of quadratic patterns of change through problem solving Look for and make use of structure: Use representations to model and interpret physical, social, and mathematical situations involving quadratic patterns of change Attend to precision: Students communicate about quadratic change in explaining their reasoning precisely to others. This document is the property of MAISA. Page 1 of 7

2 Key Concepts: Quadratic formula Completing the square Roots Zeroes Patterns of change in quadratic functions: common second differences Multiple representations Vertex Form y = a(x - h) 2 + k, Factored(Root) Form y = a(x - p)(x - q) Standard(Polynomial) Form y= ax 2 + bx + c Transformations of quadratic graphs based on changes of a, b, and c Key features of quadratic graphs: vertex, axis of symmetry, minimum, maximum, x- intercept(s), y-intercept Modeling quadratic functions (key applications: projectile motion and profit/loss business models) Compare and contrast quadratic functions with other function families Unit Abstract An important nonlinear function category is quadratics. Understanding characteristics of quadratic functions and connections between various representations should be developed in this unit. In the table form of a quadratic function, the change in the rate of change distinguishes it from a linear relationship. In particular, looking at the second rates of change or differences is where a constant value occurs. The symmetry of the function values can be found in the table. The graphical form shows common characteristics of quadratic functions including maximum or minimum values, symmetric shapes (parabolas), location of the y-intercept, and the ability to determine roots of the function. Quadratic functions can be written in a variety of formats: polynomial form f (x) = ax 2 + bx + c, factored form f (x) = a (x - p) (x - q), and vertex form f (x) = a (x - h) 2 + k. The impact of changing the parameters a, b, c, h, k, p, and q should be explored and understood. Connections should be made between each explicit form and its graph and table. Real-world situations that can be modeled by quadratic functions include projectile motion, television dish antennas, revenue and profit models in business, and the shape of suspension bridge cables. Common Core State Standards Seeing Structure in Expressions (A-SSE) Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. 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. Creating Equations (A-CED) Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpreting Functions (F-IF) This document is the property of MAISA. Page 2 of 7

3 Interpret functions that arise in applications in terms of the context 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. 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the functionh(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions (F-BF) Build a function that models a relationship between two quantities 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Build new functions from existing functions 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. Instructional Resources NCTM Illuminations ( Building Connections: This lesson focuses on having students make connections among different classes of polynomial functions by exploring the graphs of the functions. The This document is the property of MAISA. Page 3 of 7

4 questions in the activity sheets allow students to make connections between the x- intercepts of the graph of a polynomial and the polynomial's factors. This activity is designed for students who already have a strong understanding of linear functions, some knowledge of quadratic functions, and what is meant by a polynomial function. Egg Launch Contest: Students will represent quadratic functions as a table, with a graph, and with an equation. They will compare data and move between representations. Texas Instruments ( Bridge on the River Quad (TI-Nspire): In the first part of this activity, students graph a quadratic function that models the shape of a bridge trestle. They then solve the related quadratic equation by completing the square, recording each step as they complete it. This list of steps is then generalized to deduce the quadratic formula. In the second part of the activity, students store the formula in their handhelds, compare its results with those of the nsolve command, and use it to solve several other quadratic equations. Car Stopping Distances (TI-84): This activity uses the transformation graphing application on the TI-84 calculator to discover the equation for the stopping distance of a car on dry pavement. Geometer s Sketchpad This applet displays parent functions for a variety of functions. In particular, choose the quadratic tab. Sliders are set up in sketchpad format that allow manipulation of the parameters a, b, and c in the standard form of a quadratic equation y = ax 2 + bx + c PBS Mathline ( Getting Out of Line: This lesson emphasizes patterns, discovery, and vocabulary by focusing on the basic connections between the graphs, tables, and symbolic representations for lines, parabolas, inverse models, and exponential models. Students investigate patterns and various models using the graphing calculator. It is designed for first-year high school students and would be an ideal lesson following a unit on lines. Toothpicks and Transformations: The lesson begins with a review of transformations of quadratic functions, followed by an activity where students explore a mathematical pattern that emerges as they build a geometric design with toothpicks. Finally, the class uses graphing calculators to develop another model and to verify that all of the models factored form, vertex form, and general form are equivalent. Seeing Math This applet allows the exploration of quadratic functions, by changing one of the parameters, and seeing the impact on the parent function. It allows you to duplicate and display multiple parabolas and their equations using a color-coded format. There is also a This document is the property of MAISA. Page 4 of 7

5 section that allows for transformations of the parabolas. Three different forms of quadratic functions can be explored: Polynomial or Standard form: y = ax 2 + bx + c ; Vertex Form: y = a(x! b)2 + c ; Root or Factored Form: y = a(x! b)(x! c) Mathforum Math Tools is a project of The Math Drexel, funded in part by the National Science Foundation. The goal is to create a community digital library that supports the use and development of software for mathematics education. enlvm: National Library of Virtual Manipulatives ( Transformations of families of functions including quadratic, absolute value, cubic, and linear shared&cid=emready@trfns&cf=activity Lesson plans to go with the enlvm lesson above. shared/emready@trfns/info/lessonplan.html NCTM Reasoning and Sense Making Task Bank ( Horseshoes in Flight: Students analyze the structure of algebraic expressions and a graph to determine what information each expression readily contributes about the flight of a horseshoe. This task is particularly relevant to students who are studying (or have studied) various quadratic expressions (or functions). The task also illustrates a step in the mathematical modeling process that involves interpreting mathematical results in a realworld context. Task/Horseshoes.pdf Other Computer Applets Ten quadratic topics explored in a variety of ways: Graph Transformations, Adding Functions The Polynomial Form, Products of Linear Functions, Quadratic Growth, Equivalent Quadratic Expressions, Comparison and Operation, Solving Equations, Graphic Design, Motion at Changing Speed, Economic Decisions This applet uses vertex form of a quadratic function y = a(x! h) 2 + k to have students manipulate values of a, h, k using sliders to fulfill specific criteria. There are 3 questions provided, but you could add more. There are two activities on this page. One explores parabolas by examining each variable in the vertex-form equation. The second explores quadratic relationships through two concrete examples of quadratic data. This document is the property of MAISA. Page 5 of 7

6 Move sliders to adjust the coefficients in the quadratic equation, both the Standard Form and the Vertex Form. The goal is to hit the target parabola. Assessments Students write quadratic equations to match the graphs. Find an equation of each parabola from the graph and put in the form: y = a(x-h) 2 +k Use knowledge of vertices, minimum and maximum, x-intercepts, y-intercept, translations, reflections to make an accurate sketch. Resources on algebra topics, explanation and practice Professional Resources NCTM ( Focus in High School Mathematics: Reasoning and Sense Making: This publication elevates reasoning and sense making to a primary focus of secondary mathematics teaching. It shifts the teachers role from acting as the main source of information to fostering students reasoning to make sense of the mathematics. Focus in High School Mathematics: Reasoning and Sense Making in Algebra: Reasoning about and making sense of algebra are essential to students future success. This book examines the five key elements (meaningful use of symbols, mindful manipulation, reasoned solving, connection algebra with geometry, and linking expressions and functions) identified in Focus in High School Mathematics: Reasoning and Sense Making in more detail and elaborates on the associated reasoning habits. Articles from National Council of Teachers of Mathematics ( Articles available as free downloads to NCTM members, or for a fee to non-members Bergthold, T. (2005). Curve Stitching: Linking Linear and. Mathematics Teacher, 98(5), Retrieved February 18, 2011 from Eraslan, A. and Aspinwall, L. (2007). Connecting Research to Teaching: : Students Graphic and Analytic Representations. Mathematics Teacher, 101(3), 223. Retrieved February 18, 2011 from This document is the property of MAISA. Page 6 of 7

7 Edwards, T.G. and Ozgun-Koca, A. (2009). Technology Tips: Creating a Mathematics B Movie: The Effect of nb on the Graph of a Quadratic. Mathematics Teacher, 103(3), 214. Retrieved February 18, 2011 from This document is the property of MAISA. Page 7 of 7

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