Functions Derived From Data

Topics

Representations of functions as data sets and graphs, finding symbolic form through pattern building, function behaviors, and modeling

Overview

Scientists who study real-world relationships often gather numerical data as an integral part of the scientific process. The business person will do the same thing before making a decision that has monetary ramifications. The activities in this problem focus on identifying the type of function a data set might represent, finding the symbolic representation of a function through pattern building, and looking at behaviors of functions through a variety of real-world contexts.

The "Hook"

Given a data set, can I decide what “type” it is by looking at the data (numbers)? Might a graph help me decide what kind of function can best represent the data relationship? If I know what type it is, how do I find the symbolic representation of the data? Will the geometric behaviors of the relationship help me find a model (the symbolic representation)? Why is the symbolic representation of a data set important? That is, why might I need it? How could I use it?

The Investigation

The concept of function, when presented as a “topic” from a list of other topics to be “covered,” has inherent difficulties. The notation for functions can be similar to equation notation, but different because you do not “solve” functions. Rather, you represent functions in numerical and graphical forms so you can analyze their behaviors. Knowing how to identify the behaviors of functions (increasing/decreasing, max/min, domain/range) will offer solutions to the related equations, help solve related inequalities, lead to factoring and other operations with polynomials, and provide tools for modeling 2-variable data sets.

Activity 1 presents 14 sets of real-world data and asks students:

Activity 2 introduces students to f(x) notation and contrasts it with the product of variables. 

Activity 3 focuses on the parameters in a function equation.

Activity 4 analyzes the domain and range of a variety of functions. 

Activity 5 continues the investigation of the properties of functions.

Activity 6 focuses on the real-world interpretation of the zeros of a function. 

Activity 7 provides several opportunities for mathematical modeling.

Teaching Tips

• All activities REQUIRE a graphing calculator.
• Encourage students to look for patterns as they answer questions. There are opportunities to generalize based on patterns established in the questions. The ability to generalize may be more important than finding answers.  
• The student activities (except the modeling activities) are instructional in nature and do not require prerequisites other than the classroom activities.
• Activities should be assigned to small groups, but individual student use is OK too.
• Consider using the first student activity as a part of homework; that is, give it to students at the end of a class period and collect it at the beginning of the next class session – BEFORE you discuss functions in class. The 2^nd-6^th student activities should also be used in this fashion BEFORE you discuss the pertinent function behaviors in class.
• The modeling activities may be assigned with a one-week deadline. Students are expected to work on them just as working adults may do – with an established deadline and access to resources and experts (maybe parents?) in order to complete the activities.
• The data sets in the first activity are available in TI-83/84 Plus graphing calculator programs. When executed, they transfer the data to the list editor.
• An additional teaching activity is suggested here.
• See sample solutions for the activities.

Citation

Adapted from Foundations for College Mathematics (2nd ed.) & ancillary workbook, 2008, by E. D. Laughbaum.