Algebra I, Unit 4

  • Graphical Analysis and Modeling of Quadratic Functions
  • Algebraic Analysis of Quadratic Functions

Quadratic Functions and Modeling

Overview (Big Ideas):
Graphical Analysis and Modeling of Quadratic Functions
Students will begin to manipulate expressions to include the addition, subtraction and multiplication of linear and quadratic functions with the recognition that the result is a new function. This skill will be used to build higher degree functions in Algebra II. For the purposes of Algebra I, students will focus on building quadratic functions and on recognizing the properties of quadratic functions as they emerge. As students investigate quadratic patterns, they should be able to expand upon their skills in Unit 2 to express the patterns both recursively and explicitly. Students will begin to make connections between quadratic behavior in the real world and the features of the recursive, explicit and graphical representations of a quadratic function. By this point students should be able to distinguish among linear, exponential and quadratic functions given verbal, numeric, and/or graphical representations.
Students can extend their work with modeling in Unit 1 to begin to include quadratic functions, creating equations to model a given situation and graphing these functions on coordinate axes. Students will identify and interpret key features, such as intercepts, intervals where the function is increasing or decreasing, intervals where the function is positive or negative, relative maxima and minima, symmetries, and end behavior in order to sketch graphs. Additionally, they will identify any domain restrictions within the context of the problem situation. Students will graph simple cases by hand but are encouraged to use graphing technology where appropriate.
Students will explore transformations of quadratic functions, examining the effects 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. Through the process of experimentation, students will be able to generate rules for transformation. Students should be able not only to sketch a graph of a transformed function given a symbolic representation, but also to determine the value of k given a graphical representation of a transformed function.

Algebraic Analysis of Quadratic Functions
Students will interpret the parts of quadratic expressions in terms of a problem situation, including terms, coefficients, and factors. This should lead to the investigations of the structure of equivalent expressions and student interpretations about what different forms of quadratic expressions might reveal about a problem or a graph of the function. Once students have discovered the different forms their expressions can take and what those forms reveal, they should begin to develop skills at re-writing these expressions through factoring, and completing the square. Throughout this unit, it is extremely important for instructors to create a balance between conceptual understanding and procedural fluency. Emphasis should be placed on factoring as a means to acquire the zeroes of a function and on completing the square as a means to determine the maximum or minimum of the function. It should be reinforced that these features reveal information about the graph of the function as well as the real-world situation.
Students should use the method of completing the square in order to derive the quadratic formula. They are expected to solve quadratic equations by methods of taking square roots, completing the square, factoring, or the quadratic formula. Emphasis should be placed on solving the equation using the method most appropriate for the form of equation given. At this point, students should be exposed to the existence of the complex number system and should recognize when a quadratic function would have complex solutions. Students will not solve quadratics with complex solutions until Algebra II.

Enduring Understandings:
1. For functions that map real numbers to real numbers, certain patterns of co-variation, or patterns in how two variables change together, indicate membership in a particular family of functions and determine the type of formula that the function has.
2. A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the covariation between two variables.
3. Members of a family of functions share the same type of rate of change. This characteristic rate of change determines the kinds of real-world phenomena that the functions in the family can model.
4. Quadratic functions are characterized by a linear rate of change, so the rate of change of the rate of change (second difference) is constant.
5. Functions can be represented in various ways, including through algebraic means, graphs, word descriptions, and tables.
6. Changing the way that a function is represented does not change the function, although different representations highlight different characteristics, and some may only show part of the function.
7. Some representations of a function may be more useful than others, depending on the context.
8. Links between algebraic and graphical representations of functions are especially important in studying relationships and change.
9. Reasoning about the vertex form of a quadratic allows deducing that the quadratic has a maximum or minimum value and that if the zeros of the quadratic are real, they are symmetric about the x-coordinate of the maximum or minimum point.
10. For functions that map the real numbers, composing a function with “shifting” or “scaling” functions changes the formula and graph of the function in readily predictable ways.

Source:
Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing Essential Understanding of Functions Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.


Essential Questions:
Graphical Analysis and Modeling of Quadratic Functions
  • When linear and quadratic expressions are added, subtracted, or multiplied, what is the resulting function?
  • What are the differences between the numeric patterns and graphical representations of linear, exponential and quadratic functions? How might these differences appear in a real-world problem?
  • How can quadratic models be represented recursively? How might they be represented explicitly?
  • How can quadratic models be used to represent real-world problems?
  • What do the key features of the graph of a quadratic function reveal about the problem?
  • How do parameters introduced in the context of the problem affect the symbolic, numeric and graphical representations of a quadratic function?
  • How do changes to the symbolic representation of a quadratic function affect the graphical representation of the function?


Algebraic Analysis of Quadratic Functions
  • What are the parts of the expression and what information do they reveal about the problem?
  • What do different forms of an expression reveal about the graph of a quadratic function?
  • Can the methods of completing the square or factoring be used to gain more information about the problem or the graph of the function?
  • How can an expression be rewritten to reveal the necessary information?
  • What method is most efficient for solving a quadratic equation given in specific forms?
  • When does a quadratic equation have a complex solution?


Common Misconceptions:
Graphical Analysis and Modeling of Quadratic Functions
  • When multiplying binomials, students may confuse x2 with 2x.
  • Students often confuse the effects of f(x) + k with the effects of f(x) + k.
  • Students have difficulty with vertical stretches.

Algebraic Analysis of Quadratic Functions
  • Students find it difficult to recognize equivalent expressions, especially when quadratic expressions are written as the product of two linear functions.
  • Students struggle to write quadratic expressions as a product of its factors due to weakness in number sense. In particular, students have difficulty with expressions where a > 1 and/or where c < 0.
  • Students often forget to balance the equation when completing the square.
  • Students make sign mistakes when squaring negative numbers.