Contextualization

The Second Degree Function is one of the fundamental pillars of modern Mathematics and reveals itself in multiple elements of our daily lives. From the parabolic motion of a basketball to the launch of space rockets, the Second Degree Function is the mathematical tool used to model these phenomena.

The graphical and tabular representation of this function provides ways to visualize the function and understand the underlying relationships. The graph helps us understand how the function behaves across x values, while the table gives us a clear picture of the behavior of y for specific x values.

Introduction

The Second Degree Function, also known as a quadratic function, is a polynomial function of degree 2. It has the general form y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Those who venture to observe its graph on the Cartesian plane identify a parabola, which can have its concavity facing upwards or downwards, depending on the value of a.

The table of values is an ordered way to represent the points belonging to the function and is often the easiest way to start sketching its graph. Understanding the relationship between the graphical representation and the tabular representation is an essential step in mastering the second-degree function, as it brings a more comprehensive view and provides additional tools for solving complex problems.

In this sense, the focus of this project is to explore the second-degree function in its foundational aspects, the table, and the graph, as well as to apply knowledge in the development of real models represented by this function.

Practical Activity

Activity Title: Parabolando - From Table to Graph

Project Objective

The objective of this project is to work on the understanding and practice of the Second Degree Function concept through the development of a real model, represented by this function, using tables and graphs.

Project Description

The groups of students, composed of 3 to 5 participants, will choose a real phenomenon that can be modeled by a second-degree function (such as throwing a ball, free fall, trajectory of a rocket, etc.). They should create a table of values and a graph from a second-degree function formula that describes the chosen phenomenon.

Necessary Materials

1. Paper for data annotation, calculations, and table creation
2. Pencils, pens, ruler
3. Computer with spreadsheet software (Microsoft Excel, Google Sheets, etc.) and graphic tools
4. Books or other resources for research

Detailed Step-by-Step for Activity Execution

1. Group Formation: Form groups of 3 to 5 students.

2. Phenomenon Selection: Each group must choose a phenomenon that can be represented by a second-degree function. The choice needs to be approved by the teacher.

3. Phenomenon Research: Research and understand the chosen phenomenon, collecting relevant information that explains the process and can be described by the second-degree function.

4. Function Creation: Develop a second-degree function that models the chosen phenomenon.

5. Development of the Table of Values: Create a table of values with appropriate levels for the function. The table should have at least 10 pairs of values for x and y.

6. Graph Development: Use the spreadsheet software to enter the values from the table and create the corresponding graph.

7. Data Interpretation: Analyze and interpret the data from the table and the graph, identifying the patterns and characteristics of the second-degree function and the phenomenon.

8. Report Preparation: Prepare a report containing Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables Description

1. Second Degree Function: Document containing the description of the chosen phenomenon and the second-degree function representing it.

2. Table of Values: Document containing a table of at least 10 pairs of x and y, representing the function.

3. Function Graph: Computer-generated graph generated from the table of values, representing the function.

4. Final Report: Document with the following sections:

   • Introduction: The student must contextualize the chosen phenomenon, its relevance and application in the real world, as well as the objective of this project.

   • Development: The student must explain the second-degree function that models the phenomenon, explain how the table of values and the graph were developed, and present the results obtained.

   • Conclusions: The student must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.

   • Bibliography: Bibliographic and web references used in the work.

Students will have one month to complete the project. The project will be evaluated based on the quality of the research, the presentation of the function, the table of values, and the graph, the content and organization of the report, teamwork, and meeting deadlines.