Introduction

Functions are dependency relationships between two variables that allow us to describe real-world phenomena in an abstract way. This concept is fundamental in many areas of mathematics, and its understanding is crucial for students' academic and professional progress.

A function is a mathematical relationship between two variables, typically represented as x and y, in which each value of x corresponds to a single value of y. This one-to-one relationship is what characterizes a function. Functions can be represented in various ways: numerically, through value tables; algebraically, through a mathematical expression; and graphically, through a diagram on the Cartesian plane.

Let's delve deeper into quadratic functions, also known as quadratic functions, which are a specific form of function that presents a parabolic graph. These functions are expressed in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero.

Contextualization

Functions are present in various aspects of our real world. From physics to economics, through areas such as engineering, medicine, architecture, and even in the most common daily life, such as calculating the distance traveled by a car as a function of time, or the variation in the price of a product as a function of demand.

Quadratic functions are very present in physics, especially in phenomena involving motion, as they represent the law of motion of bodies in free fall. In the financial world, the principle of economies of scale, which indicates that the cost per unit of a product tends to decrease as production increases, can also be represented through a quadratic function.

The following resources will be very useful for deepening our knowledge about functions and their applications in our daily lives. Use them to prepare for the practical activity we will develop as a group:

• Khan Academy - Functions
• Só Matemática - 2nd Degree Function
• YouTube - Matemática Rio with Prof. Rafael Procopio

Practical Activity

Title: "Quadratic Functions in Real Life"

Project Objective:

This project was designed to apply and deepen your understanding of quadratic functions, exploring their applications in the real world. Additionally, we aim to strengthen your teamwork, time management, and communication skills. The project will be developed by groups of 3 to 5 students and is estimated to last twelve hours.

Detailed Project Description:

Each group must choose a daily life phenomenon that can be modeled by a quadratic function. Examples include: the trajectory of an object in free fall, production cost forecasting, animal population behavior, among others.

After choosing the theme, the group must conduct a more detailed research on the subject, and then formulate a quadratic function that represents this phenomenon. The groups should then build a graphical model of this function and explain, step by step, the logic used to arrive at this result.

The conclusion should bring a reflection on the importance of quadratic functions in the real world, and how this knowledge can be applied in different spheres of life.

Necessary Materials:

• Books and/or online material on quadratic functions.
• Computer or drawing notebooks to create the graph.
• Paper and pen for drafts.

Detailed Step-by-Step for Activity Execution:

1. Form groups of 3 to 5 students.
2. Choose a daily life phenomenon that can be represented by a quadratic function.
3. Research and collect information about this phenomenon.
4. Formulate the quadratic function that corresponds to the chosen phenomenon.
5. Draw the graph that represents this function.
6. Write the process and logic used to arrive at the function and the graph.
7. Write a conclusion reflecting on the importance of quadratic functions in real life and how this knowledge can be applied in various areas.
8. All this information must be understood and recorded in a final report.

Project Deliverables and Connection with Suggested Activities:

The deliverables of this project are:

• Detailed research on the chosen phenomenon and its relation to quadratic functions.
• Formulation of the corresponding quadratic function and its graph.
• Detailed report of the process, including the formulation of the function, the construction of the graph, and the logic used.
• Conclusion on the importance of quadratic functions in the real world.

This report should be organized into the following topics:

1. Introduction: contextualizing the chosen phenomenon and its relation to quadratic functions.
2. Development: detailing the process of formulating the function, constructing the graph, and the logic used.
3. Conclusion: reflection on the project and the acquired learning.
4. Bibliography: referencing the sources used during the research.

Each part of the report relates directly to the practical activities carried out throughout the project. The introduction fits with the choice of the phenomenon, the development section with the formulation of the function and the construction of the graph, and the conclusion with the final reflection on the importance of quadratic functions.