Introduction

Mathematics is constantly around us, from basic calculations to the most complex ones that help us understand the world. Among these applications, the study of functions stands out for its wide applicability in various areas, being the basis for understanding physical, economic, biological phenomena, among others.

Functions

Functions are, essentially, a mathematical relationship that describes how one quantity depends on one or more other quantities. In high school, we deal with functions of one variable, where one variable depends on the value of another. The graphs of these functions are visual representations that help us understand how this dependency occurs. They are a powerful tool in the interpretation and analysis of real-life situations.

Function Graphs

Function graphs allow us to visualize the behavior of a function, facilitating the understanding of important concepts such as growth and decay, maximum and minimum points, continuity, among others. The study of function graphs is fundamental for the application of functions in various areas of knowledge.

Contextualization

In the modern world, function graphs are present in our daily lives, although we may not always realize it. They can be used, for example, to understand the variation in prices of products and services, the population of a city over time, the speed of a moving object, the growth of a plant, among many other examples.

Therefore, understanding function graphs is essential not only for the study of Mathematics itself but also for the understanding and analysis of various real-world phenomena. This project aims to provide you with a practical experience in the analysis and construction of function graphs, in order to consolidate this important concept.

Practical Activity

Activity Title: "From Home to Graph: Analyzing the World with Functions"

Project Objective:

The activity aims to provide you with a practical experience in the analysis and construction of function graphs, based on data collected in daily life, enhancing the understanding of the concepts involved, as well as developing the ability to apply the theory learned in the classroom to real situations.

Detailed Project Description:

Each group will choose a phenomenon of their choice, in which they believe a function can be applied and data can be collected (for example; temperature variation during the day; energy consumption variation at home; tide height variation; product prices variation over time; etc.). The groups will collect data, plot the graph of the function that best represents the data, and present an analysis of these graphs in relation to the observed phenomenon.

Required Materials:

• Paper and pen for data notation.
• Calculator.
• Computer with internet access to use the Desmos tool.

Detailed Step-by-Step for Activity Execution:

1. Group Formation and Phenomenon Choice: Form groups of 3 to 5 members and choose a daily life phenomenon that you believe can be explained by a function.

2. Data Collection: Choose a time period (depending on the phenomenon, it can be a day, a week, a month, etc.) and collect data related to this phenomenon.

3. Data Analysis: Analyze the collected data and identify the function that best fits the observed phenomenon (it can be a linear, quadratic, exponential function, etc).

4. Graphical Representation: Use the Desmos tool to plot the graph of the function obtained from the data.

5. Graph Analysis: Analyze the obtained graph, identifying its characteristics (maximum, minimum, inflection point, growth, decay, etc) and how these characteristics relate to the observed phenomenon.

6. Report Elaboration: Describe the entire work process, from the choice of the phenomenon, through data collection and analysis, choice of the function, graph plotting, and graph analysis. This report should include, but not be limited to, an introduction, development, conclusions, and bibliography.

Project Deliverables:

At the end of the project, each group must deliver:

1. Written Report: In the report, students must explain the entire process carried out, from the choice of the phenomenon, data collection and analysis, function choice, graph plotting, and graph analysis. The report should be organized into the following topics:
   • Introduction: In this section, students explain the chosen phenomenon, why they believe it can be represented by a function, and the relevance of studying this phenomenon.
   • Development: Students must describe step by step the work done, explaining how they collected the data, how they analyzed this data, which function they chose and why, how they made the graph, and what are the characteristics of this graph.
   • Conclusions: Students should comment on what they learned from the work, the relationship between the graph and the observed phenomenon, and how the study of the function helped to understand it. They should also reflect on group work, what worked, what could be improved, etc.
   • Bibliography: They must cite all sources used to carry out the work, such as books, websites, videos, etc.

After the whole process, students should be able to identify, in practice, the relevance of functions and graphs in interpreting and understanding the world around them, a skill that is essential for the formation of a critical citizen capable of interpreting data and information.