Introduction

Exponential functions are a powerful mathematical tool that has applications in almost every field of science and engineering. They are characterized by having the variable in the exponent, unlike polynomial functions, which have the variable as the base.

The simplest form of an exponential function is f(x) = a^x, where a is a positive real number and different from 1. The number a is called the base of the exponential function. The exponential function has three important properties that will be discussed throughout our project: monotony, value of a, and growth/decay.

Our goal will be to understand how to calculate the inputs (x) and outputs (y) of these functions, which are key concepts to understand how exponential functions work.

Contextualization

Exponential functions play a crucial role in many aspects of everyday life and science. They describe things that grow or decay at a constant rate, such as compound interest, bacterial populations, radioactive decay, among others. Understanding exponential functions is essential to comprehend these and many other phenomena.

Throughout this project, we will explore how the input (x) and output (y) of an exponential function are related. By examining this relationship, we will not only be learning pure mathematics but also gaining tools to understand and model the world around us.

Practical Activity:

Project: The Power of Exponentials in Real Life

Objective:

The objective of this project is to investigate and understand the role of the exponential function in different fields of life and science, and how to represent it mathematically. Students will learn to calculate the inputs (x) and outputs (y) of exponential functions, applying this knowledge to real-world problems.

Detailed Description:

The project should be done in groups of 3 to 5 students and is expected to last more than 12 hours per student.

Students will research and choose a phenomenon that can be described by an exponential function. This choice should be well-founded and appropriate for the project level.

Next, students will mathematically model the chosen phenomenon using exponential functions. To do this, they must identify and calculate the inputs and outputs of the functions.

Finally, students will present a detailed report with their findings and conclusions, as well as give an oral presentation of the project to the class.

Required Materials:

• Computer with internet access.
• Spreadsheet program (Excel or Google Sheets).
• Calculator.
• Books and articles on exponential functions and the chosen phenomenon.
• Presentation software (PowerPoint, Google Slides, or similar).

Step by Step:

1. Choosing the Phenomenon: Research and choose a phenomenon that can be described by an exponential function. Examples may include bacterial population growth, compound interest, radioactive decay, among others. The choice should be well-founded and explained in the report.

2. Mathematical Modeling: Create a mathematical model for the chosen phenomenon using exponential functions. Define the variables, identify the function equation, and calculate the inputs and outputs of the function.

3. Analysis of Results: Analyze the results obtained in the mathematical modeling. How are the input and output of the function interrelated? How do they change over time? How is the phenomenon affected by different inputs and outputs?

4. Documentation and Presentation: Document the entire project in a written report and prepare an oral presentation for the class.

Project Deliverables:

• Written Report: Students must prepare a report, containing:
  1. Introduction - Describe the chosen phenomenon, its relevance, and explain why it can be modeled by an exponential function. It should also include the project's objective.
  2. Development - Detail the mathematical modeling, explain how you identified and calculated the inputs and outputs of the function. Discuss the results obtained and how they relate to the phenomenon.
  3. Conclusion - Discuss your conclusions about the project. What have you learned about exponential functions? What have you discovered about how the input and output of an exponential function are related and how they influence the chosen phenomenon?
  4. Bibliography - List the resources that were used to complete the project.
• Oral Presentation: Students must give a presentation to the class, explaining the chosen phenomenon, the mathematical modeling, and the results. The presentation should be clear, concise, and engaging, encouraging discussion and group learning.

At the end of the project, we expect students to have improved their technical skills in Mathematics and to be able to relate theoretical content to real-world applications. Additionally, teamwork and project presentation will help improve socio-emotional skills such as communication, time management, problem-solving, and creative thinking.