In this unit, students build the foundational understandings that functions model relationships between quantities. In addition, students build an arsenal of tools to study functions throughout the course including the routine of examining functions with multiple representations. “Students should develop ways of thinking that are general and allow them to approach any type of function, work with it, and understand how it behaves, rather than see each function as a completely different animal in the bestiary” (Grade 8, High School Functions progression document, page 7). Specific features of linear, exponential, quadratic, and polynomial functions are each studied extensively in their own units in Algebra 1. Many standards that are introduced in Unit 1 will be further developed and/or utilized across these future units; look for these standards under the Unit Level Standards heading.
Contexts are important for developing conceptual understanding. Drawing from contexts, students justify that one quantity depends on another and that each element of the domain corresponds to exactly one element of the range. They can qualitatively describe aspects of functions (e.g., increasing, decreasing) and reason about a function’s domain and range leading to conjectures about additional representations. In addition, students use contexts to make sense of function notation. For example, if the function h represents the height in centimeters of a bean sprout at specified days, t, then students should be able to talk about and/or identify each of the following: h(3), 5=h(t), h(6.5)=13.2, and h(t)=n.
When functions are expressed symbolically, students input values into the equations to generate a table of output values and use these corresponding values to graph the function. After manually generating numeric and then graphic representations, students will use technologies such as a graphing calculator to generate both tables of values and graphs of functions. In doing so, they should pause to reflect upon the functional relationship between variables exhibited in these representations and why the representations make sense. By generating and comparing multiple cases, students recognize that functions can be organized by similar and different features like patterns of change, restrictions in the domain/range, and general shape. This organization of functions both serves as as an introduction to the families of functions that they will study throughout the course and equips students with strategies to study and represent functions.
Students should not be expected to generate symbolic representations until later units. Likewise, symbolic manipulation of equations to reveal key features of functions (e.g., intercepts, maximum, horizontal asymptote) is included later in specific function units.
