Context

Second-degree functions, also known as quadratic functions, are one of the most fundamental concepts in mathematics. These functions are continuously used in various areas of science and engineering to model various natural phenomena or practical situations. The importance of studying second-degree functions in mathematics lies in their versatility and wide application. They are commonly used to describe an object in motion, an object in free fall, a planetary orbit, the optimization of a company's production cost, among many other situations.

Growing in our understanding of second-degree functions is essential for a better understanding of the world around us, especially when we look at areas such as physics, economics, engineering, among others. By applying these concepts to practical problems, it is easier to understand why they are so important.

Project Introduction

This project aims to familiarize you with second-degree functions and their practical applications, while developing teamwork and problem-solving skills. This work will involve modeling practical situations using second-degree functions, investigating related theoretical concepts, and presenting your findings in a clear and concise manner.

The second-degree equation, which is the backbone of the quadratic function, is written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a second-degree function is a curve called a parabola. The appearance and position of the parabola depend on the values of a, b, and c in the equation.

This project will be divided into three main parts: Introduction and Contextualization, Practical Activity Suggestions, and Conclusion and Correction Criteria. Each of these parts will provide an integral direction for the completion of this project.

Practical Activity

Activity Title: Free Fall and Projectile Motion Simulation

Project Objective:

Apply the knowledge of second-degree functions in the modeling and analysis of two common movements in physics: free fall and projectile motion.

Detailed Project Description:

Part 1 – Free Fall: A ball is thrown directly upwards with an initial velocity of v_0 meters per second (m/s) from the top of a building of height h meters. The goal of this part is to model the movement of the ball using a second-degree function.

Part 2 – Projectile Motion: A projectile is launched from the ground with an initial velocity of v_0 m/s at an angle θ degrees from the horizontal. The goal of this part is to model the trajectory of the projectile using a second-degree function.

Note: This is a theoretical activity that involves the application of mathematical concepts in physics. There is no need to conduct physical experiments.

Required Materials:

1. Mathematics and physics textbooks covering second-degree functions, free fall, and projectile motion.
2. Internet access for online references.
3. Graphing software to visualize the generated second-degree functions (e.g., Desmos, GeoGebra, Microsoft Excel).

Detailed Project Steps:

1. Form groups of 3 to 5 students. The project duration will be one month, and each student should expect to spend between five and ten hours during this period.

2. Start with the first part of the project: the free fall simulation. Research and review the equation of free fall motion.

3. Identify the parts of the motion equation that make it a second-degree function. Use fictitious parameters for the initial velocity and building height.

4. Graph this second-degree function using graphing software.

5. Now, move on to the second part of the project: projectile motion. Research and review the equation of projectile motion.

6. Once again, identify the parts of the motion equation that make it a second-degree function. Use fictitious parameters for the initial velocity and launch angle.

7. Graph this second-degree function using graphing software.

8. Compare and discuss the shapes of the graphs generated in the first and second parts of the project. What do they have in common? How are they different?

9. Finally, write a detailed project report. The report should follow the specified format: Introduction, Development, Conclusions, and Bibliography. The Introduction should contextualize the problem and describe the project's objectives. The Development should explain the theory behind the project, describe the methodology used, present the generated graphs, and discuss the results obtained. The Conclusion should summarize the project, the learnings obtained, and the implications of the results. The Bibliography should list all sources consulted throughout the project.

Project Submission:

The submission should be in the form of a written report, containing the sections of Introduction, Development, Conclusions, and Bibliography. This report should include a detailed discussion of the graphs and equations used, as well as an explanation of the theoretical concepts involved in second-degree functions, free fall, and projectile motion. The report needs to be clear and concise, highlighting the group's effort in understanding and applying the concepts of second-degree functions.

Students should make sure to include in the report all produced graphs, as well as a complete discussion of their results, culminating in a strong conclusion with the main findings of the study. Additionally, it is essential for students to present their sources of information in the Bibliography section.