This unit bundles student expectations that address an introductory exploration of the parent functions covered in Algebra II, attributes of functions including domain and range, and inverses of functions. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.
Prior to this unit, in Grade 8, students were introduced to linear functions. In Algebra I, students experienced an in-depth study of linear and quadratic functions and their characteristics. Also in Algebra I, students investigated exponential functions. In Geometry, students analyzed pattern situations and represented them using linear and quadratic functions as appropriate.
During this unit, students explore representations of the families of functions to be covered in Algebra II, f(x) = , f(x) = , f(x) = x^{3}, f(x) = , f(x) = b^{x}, f(x) = |x| and f(x) = log_{b }x, where b is 2, 10, and e, including tables, graphs, verbal descriptions, and algebraic generalizations. Students analyze key attributes of functions, including domain (represented in interval notation, inequalities, set notation), range (represented in interval notation, inequalities, set notation), intercepts, symmetries, and asymptotic behaviors. Using various representations, students describe and analyze the relationships between a function and its inverse (quadratic and square root, logarithmic and exponential, cubic and cube root), including restriction(s) on domain. Students graph and write the inverses of functions in function notation using notation such as f^{–1}(x). Students use composition of two functions, including domain restrictions, to determine if functions are inverses of one another.
After this unit, students will experience a more in-depth study of each family of functions by describing and applying the parent function, identifying the transformational effects of parameter changes on the graph of the parent function, and analyzing the characteristics of the family of functions. Representative functions will also be used to formulate and solve equations and inequalities in problem situations. Inverses will be addressed again when comparing and contrasting the quadratic and square root functions, exponential and logarithmic functions, and cubic and cube root functions. The concepts in this unit will also be applied in subsequent mathematics courses.
In Algebra II, graphing families of functions and identifying their key attributes is identified as STAAR Readiness Standard 2A.2A. Describing and analyzing inverse functions is identified as STAAR Readiness Standard 2A.2C. All STAAR Readiness Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Writing and graphing inverse functions is identified as STAAR Supporting Standard 2A.2B. Using the composition of functions to determine if functions are inverses is identified as STAAR Supporting Standard 2A.2D. Both STAAR Supporting Standard 2A.2B and 2A.2D are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses. Writing the domain and range of a function in inequality, set, and interval notation is STAAR Supporting Standard 2A.7I and part of STAAR Reporting Category 1: Number and Algebraic Methods. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.
According to the National Council of Teachers of Mathematics (NCTM):
The Algebra Standard emphasizes relationships among quantities and the ways in which quantities change relative to one another. To think algebraically, one must be able to understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; and analyze change in various contexts. In high school, students create and use tables, symbols, graphs, and verbal representations to generalize and analyze patterns, relations, and functions with increasing sophistication, and they convert flexibly among various representations. They compare and contrast situations modeled by different types of functions, and they develop an understanding of classes of functions, both linear and nonlinear, and their properties. (NCTM, 2002, p. 2-3)
In addition, Focus in High School Mathematics: Reasoning and Sense Making (2009) from the National Council of Teachers of Mathematics (NCTM), states “Functions are one of the most important mathematical tools for helping students make sense of the world around them, as well as preparing them for further study in mathematics” and lists the key elements of reasoning and sense making with functions as being “using multiple representations of functions, modeling by using families of functions” (p. 41). According to NCTM (2000), students need to learn to use a wide range of explicitly and recursively defined functions to model the world around them. Navigating through Data Analysis in Grades 9 – 12 (2003) states, “a fundamental goal of the mathematics curriculum: to develop critical thinking and sound judgment based on data” (NCTM, p. 1). According to Navigating through Algebra in Grades 9 – 12 (2002), “The Algebra Standard emphasizes relationships among quantities and the ways in which quantities change relative to one another. To think algebraically, one must be able to understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; and analyze change in various contexts” (NCTM, p. 2).
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (2003). Navigating through data analysis in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics, Inc. Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf |