Contextualization

The quadratic function is one of the most prominent and important topics within the universe of Mathematics. It is fundamental for us to be able to understand diverse phenomena and situations of our daily life, since it is present in many areas of human knowledge, such as Physics, Engineering, Economy, among others.

The quadratic function is called so because the highest exponent of the unknown is 2. It is also called square function, since it is related to the area of square geometric figures. Its general expression is given by f(x)=ax²+bx+c, where a, b and c are real numbers and a ≠ 0.

To fully understand the quadratic function, it is necessary to know and understand concepts such as coefficients, roots or zero of the function, vertex of the parabola, sign of the function and the concavity of the parabola. All these are essential elements that help us draw the graph of a quadratic function and interpret it correctly.

And why is it so important to understand the quadratic function? Because it has practical applications in diverse fields of knowledge. In Physics, for example, it is applied in the study of projectile movements, where the graph of this function represents the trajectory of an object thrown in the air. In Engineering, it is used in the calculation of structures and in the dimensioning of materials. In Economy, the quadratic function is used to analyze the behavior of markets and to maximize the production and profit of enterprises.

To go deeper into the theoretical concepts of the quadratic function, you can consult the following sources:

• Book: "Fundamentos de Matemática Elementar: Geometria analítica, funções, sequências" by Osvaldo Dolce and José Nicolau Pompeo.
• Web page: Só Matemática (www.somatematica.com.br) - In the "Ensino Médio" section, you will find the "Funções" topic, where the quadratic function is explained in a very didactic way.
• Videos: YouTube channel "Matemática Rio com Prof. Rafael Procopio" - This channel has a series of videos about the quadratic function, with clear explanations and practical examples.

Practical Activity: "Quadratic in Action"

Project Objective

Develop a practical and interdisciplinary project involving the quadratic function, connecting mathematics, physics and art, in order to deepen the students' understanding of the topic and stimulate their creativity and collaboration.

Detailed Project Description

In this project, you will create a projectile launch model, using concepts of physics and mathematics, specifically the quadratic function. And to make the project more fun and engaging, you will also create an explanatory video, incorporating arts and creative expression into the project.

This project should be carried out by teams of 3 to 5 students and should take at least twelve hours per student participant to complete.

Necessary Materials

• Materials to build a mini-catapult or projectile launcher (could be cardboard, rubber bands, spoons, etc.)
• A small and light object to be the projectile (such as a marshmallow or a paper ball)
• A camera or smartphone to film the projectile launch
• Video editing software (can be any free one available online)
• A ruler or tape measure
• Paper and pen to write down the observations

Detailed Step-by-Step

1. Research about the parabolic movement, which is the movement described by an object thrown in the air, and understand how the quadratic function is used to represent this movement.

2. Build a simple projectile launcher and choose a light object to be the projectile. The objective is for the projectile to trace a parabolic trajectory when launched.

3. Make several attempts to launch the projectile, measuring the launch distance and the maximum height reached by the projectile. Write down all observations.

4. Use the collected data to build a quadratic equation that represents the projectile's trajectory. Remember that the quadratic equation has the form y=ax²+bx+c, where y is the height of the projectile, x is the horizontal distance, a, b and c are constants that you should determine from your data.

5. Film the projectile launch and create an explanatory video about the project. The video should include a theoretical explanation of the quadratic function and the parabolic movement, a practical demonstration of the projectile launch and a discussion of the results obtained. Be creative in editing and presenting the information!

Project Deliveries and Written Document

At the end of the project, you must deliver the following:

• The built projectile launcher model.
• The explanatory video created.
• A written report containing:
  • Introduction: Contextualization of the topic, its relevance and application in the real world, as well as the objective of this project.
  • Development: Explanation of the theory behind the quadratic function and the parabolic movement, detailed description of the activity carried out, methodology used (how you built the model, how you collected and analyzed the data, how you created the video), presentation and discussion of the results obtained.
  • Conclusion: Return to the main points, explanation of the obtained learnings and the conclusions drawn about the project.
  • Bibliography: The sources you based yourself on to work on the project (books, web pages, videos, etc.).

Remember that the written document is a complement to the practical work carried out and to the video created, so it should be well-structured and contain all necessary information to understand the project and the obtained results.

Good luck, have fun and learn a lot!