Contextualization

The second degree function, also known as the quadratic function, is an extremely powerful tool used in various areas, from physics to finance. Understanding the inputs and outputs of this function is essential for solving numerous practical problems.

The study of second degree functions is part of a set of mathematical knowledge called Algebra. Algebra is an extremely useful tool because it allows us to represent real-world situations in an abstract way, and manipulate these representations to find solutions to practical problems.

The Second Degree Function is characterized by the expression f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0. The inputs of the function are the values that we can assign to x, while the outputs are the resulting values for each input, obtained by calculating f(x). Second degree functions have a characteristic that is the shape of a parabola when graphically represented.

Now, you might be wondering: where will I use this in real life? The answer is: in many places! Second degree functions are present in many everyday situations. For example, when an object is thrown into the air, its height over time can be described by a second degree function. Similarly, the second degree function can also represent a company's profit as a function of its production, where the goal is to maximize this profit.

Understanding how to identify and manipulate these functions is an extremely valuable skill. So, get ready to embark on an adventure through the world of second degree functions! The knowledge you will acquire in this project will be a powerful tool in your arsenal to solve problems, not only in mathematics, but in various real-life situations.

I strongly recommend reading the chapter 'Second Degree Function' from the book 'Fundamentals of Elementary Mathematics: Volume 1 - Sets and Functions' (IEZZI, Gelson; HAZZAN, Samuel; MURAKAMI, Carlos). In addition, the website 'Só Matemática' contains excellent explanations on the topic in an easy and accessible language (Link to Só Matemática - Second Degree Function). For a deeper dive, the 'Matemática Rio' channel on YouTube has a series of explanatory videos on the topic (Link to Matemática Rio - Second Degree Function).

Practical Activity

Title: The Parable of the Ice Cream Business

Project Objective

The objective of this project is to apply the concept of the second degree function in practice, analyzing the hypothetical scenario of an ice cream sales business. Students will perform calculations and create graphics to understand how the variation in the price of ice cream affects the business's profit, maximizing it through the application of the second degree function.

Project Description

Each group of 3 to 5 students will be responsible for managing a hypothetical ice cream sales business. They will start by determining a second degree function that will represent the relationship between the price of ice cream (input) and the profit obtained (output).

For this simulation, we will consider that the number of ice creams sold per day is inversely proportional to its price, meaning that the higher the price, the fewer ice creams are sold. However, the total revenue (the price of the ice cream multiplied by the number of ice creams sold) does not follow a linear relationship with the price, reaching a maximum point before starting to decrease - this is the behavior that will be modeled by the second degree function.

Required Materials

1. Paper and pencil for initial drafts.
2. Computer with Internet access.
3. Graph creation tool (e.g., Microsoft Excel, Google Sheets).
4. Calculator.

Step-by-Step for Activity Execution

1. Research and Theory Study: Students should start by reading about second degree functions, focusing on how to identify inputs and outputs and the parabolic shape these functions generate. The research should also include how to find the vertex of a parabola, as this is the maximum profit point in the proposed situation.

2. Function Creation: Students should assume reasonable values for the fixed and variable costs of the ice cream business, and based on that, create a second degree equation that represents the relationship between the price of ice cream and the generated profit. This equation should be well documented and explained.

3. Function Graph: Using the graph creation tool, students should create a graph that represents the function, clearly demonstrating the parabolic relationship.

4. Function Analysis: Students should determine the price of ice cream that maximizes profit, identifying the vertex of the generated parabola.

5. Report: Each group of students should prepare a project report, which should include:

   • Introduction: Explanation of the second degree function and its practical application in the proposed situation.
   • Development: Presentation and discussion of the created function, as well as the resulting graph and the identification of the maximum profit point. Students should detail the methodology used, discuss results, and justify their choices.
   • Conclusion: Students should highlight the conclusions drawn from the project, the learnings, and the difficulties encountered.
   • Bibliography: References to the materials and sources used during the project.

Project Deliverables

Students should deliver:

1. The generated graph, with the second degree function clearly visible and the identified maximum profit point.
2. The written report as detailed above.

This practical activity will allow students to become familiar with the inputs and outputs of a second degree function, and they will understand how the parabolic shape of the function can be used in real life to optimize profit in a business scenario. Additionally, the written report will help develop the skill of documenting and presenting their findings in an organized and clear manner.