Contextualization

The Second Degree Function, also known as a quadratic function, is one of the fundamental topics in Mathematics. It is so important that its application is seen in various areas such as Engineering, Physics, Economics, among others. The quadratic function is expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Understanding the concepts involved in the second-degree function, such as its roots, the vertex of the parabola, and concavity, is crucial to a deeper understanding of mathematical functions and even differential equations.

Second-degree functions can be used to describe the motion of an object thrown vertically upwards in a vacuum, the perfect trajectory of an arch, or even to maximize or minimize certain quantities in the economy. These functions also help us understand how change occurs and predict where and when it will happen. This is very useful when we want to predict what will happen in the future using mathematics. With this project, you will learn what second-degree functions are, their properties, and how to find their inputs and outputs, which will help build a solid foundation for more advanced mathematical studies.

In real terms, the 'input' in a function is the value you give to the function, and the 'output' is what you get after applying that function. In the case of a quadratic function, the input, also known as the independent variable, is usually represented by x, and the output value, or dependent variable, is the result of the calculation based on the values of a, b, c, and x.

Understanding the inputs and outputs of a function presents a series of benefits, including the ability to interpret and solve real-life problems. A good example of this can be seen in calculating the area of a plot of land, where the input would be the dimensions of the land and the output would be the total area. Similarly, businesses can use quadratic functions to maximize profits or minimize costs, where the inputs would be different production levels (or prices) and the output would be profitability (or cost).

To delve deeper into the subject, I recommend reading the book 'Fundamentos de Matemática Elementar - Vol. 1: Conjuntos e Funções' by Gelson Iezzi and consulting the website Só Matemática, which provides a detailed and objective explanation of second-degree functions. Additionally, the YouTube channel 'Matemática Rio com Prof. Rafael Procopio' has very didactic videos on the subject.

Practical Activity

Activity Title:

'Maximizing Profits and Minimizing Costs: An Adventure with Quadratic Functions'

Project Objective:

Explore the concept of Second Degree Functions and their practical applications in the business world.

Detailed Project Description:

Groups will be challenged to act as owners of an orange juice production company. For this, they will need to use second-degree functions to solve optimization problems, that is, maximize profit and minimize costs.

Necessary Materials:

• Computers with internet access
• Electronic spreadsheet software (Google Sheets, Excel, etc.)
• Calculators

Detailed Step-by-Step for Activity Execution:

1. Each group must create a second-degree function that represents the cost of producing orange juice, taking into account factors such as the quantity of oranges, workers' salaries, expenses with electricity, among others.

2. Next, students must create another second-degree function that represents the revenue obtained from the sale of orange juice, considering the selling price and the quantity sold.

3. The goal is to find the number of juices that should be sold to maximize profit (revenue - cost). To do this, students must derive and solve the profit function, finding the vertex of the parabola represented by it.

4. Students must use electronic spreadsheet software to graphically represent the cost, revenue, and profit functions, and visually demonstrate the answer from step 3.

5. Finally, students must create a report that includes:

   • Introduction: Explaining the concept of second-degree functions, their real-world applications, and the project's objective.
   • Development: Describing in detail how the cost and revenue functions were created and how the profit maximization value was obtained. In this section, they must include charts from the spreadsheet software, along with a detailed explanation of how these charts illustrate their findings.
   • Conclusion: Discussing the practical implications of their findings. What should the company do to maximize profits? What other aspects of the business could be optimized using second-degree functions?
   • Bibliography: list all resources (books, websites, videos, etc.) used during the project.

Project Deliverables:

Each group must deliver:

1. The cost and revenue functions, as well as the profit equation.
2. Image of the graph created in the spreadsheet software.
3. Final report containing all the sections mentioned above.

The report will be used to evaluate not only the students' understanding of second-degree functions but also their research skills, teamwork, time management, and problem-solving abilities. Additionally, the project can help them understand how mathematics is used in real situations, thus improving their appreciation for the discipline.