Contextualization

Exponential functions are one of the most important topics in mathematics and are at the core of many natural and social phenomena. It is a mathematical function expressed in the form f(x) = a^x, where 'a' is a positive constant different from one. This type of function has very rapid growth - that is, the value of f(x) increases significantly as x increases - and that is why we find applications for exponential functions in many different areas of life and science.

We can see exponential functions in action in the population of bacteria, which grows exponentially under ideal conditions, or in the amount of money you can have in a compound interest account. It is also found in rates of radioactive decay, which also decrease exponentially over time. Learning about exponential functions is not only an essential step to advance in more advanced mathematics, but it is also a tool to understand the world around us.

Introduction

In this project, we will explore in depth the concept of exponential functions and how they are graphically represented. By learning to draw these graphs, we can begin to understand the characteristics of this function such as its domain, range, and growth rate. Additionally, we can observe how this growth rate translates into real-world situations.

We will start with a review of the basic concepts of exponential functions and the theory behind the graphs of these functions. To help us in this goal, we will use various educational resources including videos, websites, and math books. Next, we will work in groups to create our own graphs of exponential functions, both manually and with the help of math software.

We want you to delve into understanding how and why these graphs are formed, and how changes in the parameters of an exponential function affect its graph. This is essential knowledge for anyone wishing to delve into mathematics and its applications.

Practical Activity: 'Exploring the World of Exponential Functions'

Project Objective

Identify, understand, and apply the concepts related to exponential functions and their graphical representation.

Project Description

Student groups will carry out a series of activities involving the creation of graphs of exponential functions and the analysis of the implications of these functions in real-world scenarios.

Required Materials

1. Pencils, ruler, and graph paper for manual graph creation.
2. Internet access.
3. Computers with graphic software (e.g., GeoGebra, Desmos, or other math software chosen by the group).

Activity Steps

1. Form groups of 3 to 5 people.
2. Review the theory of exponential functions using the suggested learning resources.
3. Think of different real-world scenarios that could be represented by an exponential function. Choose one of these scenarios to work on the project.
4. Create an exponential function that represents the chosen scenario. Justify the choice of values in the function.
5. Draw the graph of the chosen exponential function manually, using paper and pencil.
6. Use math software to digitally draw the graph of the same function.
7. Analyze and compare the graphs created manually and digitally. Verify if both show the same behavior.
8. Discuss how changes in the function's parameters impact the graph and the real-world scenario it represents.
9. Finally, prepare a report (minimum of 5 pages) on the project.

Deliverables and Report Format

Students must submit a report containing:

• Introduction: In this section, provide a brief introduction to the concept of exponential function, its importance, and practical applications in the real world. Also, include the exponential function chosen for the project and the real-world scenario it represents.
• Development: Explain in detail the chosen exponential function, the parameters, and how it relates to the chosen real-world scenario. Include the manually and digitally created graphs. Compare the graphs and detail the similarities and differences. Also, discuss the consequences of changes in the function's parameters.
• Conclusions: Describe the main learnings obtained from the project. Comment on the importance of the exponential function and what you learned about the behavior of this function. Summarize the essential aspects of the exponential function that were worked on.
• Bibliography: Cite all sources consulted for the project, including books, web pages, videos, etc.

The report will be the basis for project evaluation, so it is recommended that students produce it with care and dedication.