Contextualization

The second degree function is a fundamental concept in mathematics and has several practical applications. A second degree function, also known as a quadratic function, can be represented by the equation f(x) = ax² + bx + c, where a, b, and c are real numbers and 'a' is not equal to zero.

The main characteristic of a quadratic function is its graph, which is a curve called a parabola. The parabola can be facing upwards or downwards, depending on the value of 'a'. Furthermore, the second degree function has a minimum or maximum value, called the vertex of the parabola, which can be calculated using the formula -b/2a.

The discussion about the second degree function leads us to key concepts such as coefficients, function roots (zeros), concavity of the parabola (upwards or downwards), and the vertex of the parabola. All of these are vital components for understanding and manipulating this function.

The study of the second degree function has practical applications in various areas such as Physics, Economics, Engineering, etc. For example, in Physics, projectile motion equations are second degree functions. In Economics, quadratic functions are used to model profit maximization or minimization situations.

Therefore, besides being a key topic in the mathematics curriculum, the second degree function has indispensable real-world applicability, making its study relevant and engaging.

For the initial study of the topic, I recommend the book 'Fundamentos de Matemática Elementar: Álgebra - volumes 1 e 2' by Gelson Iezzi and Carlos Murakami. Additionally, the website 'Só Matemática' has excellent resources, including a section dedicated to the second degree function. For a more in-depth study, I suggest the book 'Calculus - volume 1' by James Stewart.

Practical Activity - Second Degree Function Project: From Theory to Practice

Project Objective

This project aims to apply the knowledge acquired about the second degree function in a real situation, developing a solution to a practical problem using the concept of inputs and outputs.

Activity Title: 'Simulating the Flight of a Projectile'

Project Description

In many games, the physics of motion is one of the most important elements. One of the most common movements is the parabolic trajectory of a projectile. In this project, students will work together to simulate the physics behind this movement, which is represented by a second degree function.

The student groups should develop a program in a programming language of their choice (Python, JavaScript, etc.) that simulates the launch of a projectile on a soccer field, aiming to hit a specific target. They should model the trajectory of the projectile using a second degree function and experiment with different input values (initial velocity, angle, etc.) to see how the trajectory of the projectile changes.

The program should also be able to generate a graph of the parabola corresponding to the trajectory of the projectile. With this, students will be able to visualize the effect of different inputs on the function output.

Required Materials

1. Computers and internet access
2. Programming software (Python, JavaScript, etc.)
3. Material for analysis and discussion of results

Step-by-Step for Activity Execution

1. Study and Research: Start the project with a broad research on second degree functions and their applications in real contexts, focusing on the parabolic motion of projectiles. Take notes and prepare to present your findings to the group.

2. Project Planning: Gather the group and discuss the objectives and strategies for the project. Divide tasks among members and define a schedule for task completion.

3. Program Development: Begin coding the program, ensuring that all characteristics of a second degree function are present in the code. Experiment with different input values and observe the results.

4. Results Analysis: Analyze the results obtained, identify trends and peculiarities of the function. Discuss the conclusions with the group and prepare for a presentation.

5. Report Preparation: Write a detailed report on the project, explaining the objective, methodology, results, and conclusions of the project.

Project Delivery

At the end of the project, groups must deliver the following:

1. The developed computer program, ready to be executed.
2. A written report detailing the research, planning, programming, and results analysis process. This report should be structured with the following topics: Introduction, Development, Conclusions, and Bibliography.

The report should follow the proposed format, where in the Introduction the choice of the activity, its relevance, and the project's objective are discussed. In the Development section, the studied theoretical concepts, a detailed description of the project, the methodology, and the results obtained should be detailed. In the Conclusion, students should reflect on what they learned, how they developed throughout the project, and what conclusions were drawn. The Bibliography should list the references consulted throughout the project.

This project will not only help students understand the theoretical concepts of second degree functions but also the importance of teamwork, time management, problem-solving, and creative thinking.