Contextualization

Theoretical Introduction

The Exponential Function is an essential topic in Mathematics and is fundamental to understanding various dynamics of the world around us. It is defined as a function of the type f(x) = a^x, where 'a' (base) is a positive number different from 1 and 'x' is the independent variable.

The graphs of these functions have unique characteristics that are determined by the base of the exponential: If the base is greater than 1, the function is increasing, and if the base is between 0 and 1, the function is decreasing. All graphs of exponential functions intersect the y-axis at the point (0,1) and approach the x-axis but never touch it, which we call a horizontal asymptote.

Exponential functions are inverses of logarithmic functions and together, they are key pieces for solving many mathematical problems. They involve concepts such as domain, range, and growth, vital for the graphical analysis and interpretation of their characteristics.

Contextualization

The Exponential Function is applied in various contexts of our daily lives. One of the most well-known examples is the calculation of compound interest in Financial Mathematics. Through it, it is possible to understand and predict how money accumulates over time. Additionally, this function is widely used in exact and biological sciences to describe the growth of populations, the decay of radioactive substances, and much more.

Another relevant example is in modeling phenomena related to pandemics, such as the current COVID-19. Through the exponential function, epidemiologists can predict the increase in cases and better understand the progression of the disease.

Furthermore, the exponential function allows for a better understanding of a wide range of phenomena involving accelerated growth and decay, such as the popularization of technologies, the spread of information in networks, among many others.

Practical Activity - 'Unraveling Exponentials through Graphs'

Project Objective

The objective of this activity is to provide students with a practical understanding of exponential functions, their properties, graphical behavior, and real-world applications. Students will challenge themselves to create and interpret graphs of exponential functions using online resources, discuss their findings, and write a detailed report on the experience.

Detailed Project Description

Each group of 3 to 5 students will explore two different scenarios that can be modeled by exponential functions: the growth of a population of microorganisms and the appreciation of a financial investment. For each scenario, students will create a graph of the corresponding exponential function and analyze its different elements (domain, range, growth).

Required Materials

• Computer or tablet with internet access
• Graphing software (suggestion: Desmos, GeoGebra)
• Notebook or virtual document for notes

Detailed Step-by-Step Guide for Carrying Out the Activity

1. Each group should research and choose a scenario of population growth of microorganisms and a financial investment scenario that can be modeled by an exponential function. This information should be noted down, and the sources properly documented.

2. For each scenario, students should identify the key elements that will form the basis of the exponential function (growth rate, time).

3. Using graphing software, such as Desmos or GeoGebra, students should input the exponential functions that represent the chosen scenarios and create the respective graphs.

4. Next, students should analyze the generated graphs, observing the fundamental characteristics of exponential functions (domain, range, growth). They should also discuss and take notes on the differences and similarities between the generated graphs and how these relate to the selected scenarios.

5. Finally, students should prepare a detailed report on the project. The report should include the following topics: Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables and Connection with Activities

The results of the practical activities should be documented in the written report, which should be structured as follows:

• Introduction: Students should present the chosen scenarios, explain why these scenarios can be modeled by an exponential function, and what their relevance and real-world application are. They should also indicate the project's objectives.

• Development: Here, students should present in detail the theory of the exponential function, its main elements and properties, as well as the methodology used in the project. They should then explain how they applied this theory to create the graphs corresponding to the selected scenarios, which tools were used, and present the generated graphs. Finally, they should discuss the analyses they conducted on the graphs.

• Conclusions: Students should summarize the main points of the project, showing what they learned about exponential functions and how they applied this knowledge to analyze real-world situations. They should also reflect on the experience of working in a group and on the socio-emotional skills they developed.

• Bibliography: Students should list all the sources they relied on to carry out the project, including the resources where they found the scenarios to be modeled, the sources where they researched exponential functions, and the tutorials or manuals of graphing software they used.

The report should be submitted in digital format within one week from the start date of the project.