Introduction

The second-degree function, also known as a quadratic function, is a mathematical expression of great importance and applicability in various areas. It is represented as follows: f(x) = ax² + bx + c, where 'a' is the parameter that influences the shape (concavity) of the resulting parabola on the graph, 'b' is the parameter that influences the position relative to the y-axis, and 'c' is the parameter that defines the position of the vertex relative to the y-axis.

The graph of the second-degree function, known as a parabola, has very interesting and useful characteristics. One of the main features of parabolas is that they always have a minimum or maximum point, called the vertex. The position and shape of the parabola are defined by the coefficients a, b, and c of the quadratic function.

To plot the parabola on the Cartesian plane, you need to identify the axis of symmetry (a vertical line passing through the vertex), the zeros of the function (points where the function crosses the x-axis), and the vertex. The parabola can open upwards (a > 0) or downwards (a < 0), and is symmetric with respect to the axis of symmetry.

Contextualization

The second-degree function is a powerful tool that helps us model and solve many real-life problems. For example, the trajectory of a projectile launched into the air, optimizing the size of a box to maximize its volume, studying the profits and losses of a company, among other applications. Understanding the graph of this function is essential for solving these problems.

Furthermore, recognizing the graph of a second-degree function, its parameters, and sketching a graph of a second-degree equation is a vital skill for various professions, especially in engineering, physics, and economics. Therefore, we need a solid and intuitive understanding of the graphs of second-degree functions, as they are key to understanding many real-world phenomena.

Practical Activity

Activity Title: 'Mapping Parabolas: An Interdisciplinary Approach'

Project Objective

This project aims to develop technical and socio-emotional skills in high school students, integrating concepts from Mathematics and Physics. By the end of this project, students should be able to:

1. Mathematically model real-life situations using second-degree functions;
2. Sketch the graph of a second-degree function, recognizing its parameters;
3. Use digital tools to assist in problem-solving and graph creation;
4. Cooperate in a team, manage project time, and communicate effectively;
5. Write a scientific report.

Detailed Project Description

The project will be carried out by groups of 3 to 5 students and is estimated to last 12 hours per student. The activity will involve mathematical modeling of projectile trajectories, analysis of the resulting graphs, and application of physical concepts in interpreting the results.

Required Materials

1. Paper, pencil, and eraser for sketches and preliminary calculations;
2. Computer with internet access and programs/software for plotting graphs (Geogebra, Desmos, or similar);
3. Access to books or online sources of Mathematics and Physics for research;
4. Support materials for report creation (word processor).

Detailed Step-by-Step for Project Execution

1. Theoretical Research: The group should study the concepts of second-degree functions (parameters, graphs, applications) and projectile motion in Physics.

2. Mathematical Modeling: Based on a problem situation presented by the teacher (e.g., a basketball player shooting the ball, a rocket being launched, etc.), students should create a mathematical model using a second-degree function. This model should take into account: initial height, initial velocity, and launch angle.

3. Graph and Analysis: Using a graphing software, the group should plot and analyze the function, identifying the vertex, axes of symmetry, function zeros, and general graph characteristics.

4. Physical Interpretation: Using kinematics concepts, the group should interpret the graph in physical terms (e.g., flight time, maximum range, maximum height, etc).

5. Final Report: The group will compile all steps and results into a final report, including: Introduction, Development, Conclusions, and Bibliography.

Project Deliverables

The project consists of delivering a final report that documents all stages of the project. This report should follow the structure:

1. Introduction: Brief contextualization of the second-degree function, its application in Physics, project objective, and the problem situation to be solved.

2. Development: Detailed explanation of the project stages: theoretical research, mathematical modeling, graph and analysis, and physical interpretation. This part should include the created graphs, the formulas used, and all calculations and interpretations made by the students.

3. Conclusions: Here, students should summarize the main points of the work, highlight what they learned, the difficulties encountered, how they overcame these difficulties, and their final observations on the project.

4. Bibliography: References to the materials used during the project, such as books, websites, videos, software, among others.

This project will contribute to learning important concepts and developing essential skills, both in Mathematics and Physics, as well as socio-emotional skills, such as time management, problem-solving, cooperation, and communication.