Contextualization

The study of quadratic functions is fundamental in the Mathematics curriculum of High School. These functions, also known as quadratic functions, are represented by a second-degree equation, and their graphs have the shape of a parabola. The importance of studying quadratic functions lies in their wide application in various areas of knowledge, from physics to economics.

The graph of a quadratic function on the Cartesian plane is a powerful visual tool that allows us to understand the relationship between variables, find solutions to the quadratic equation, identify the vertex of the parabola, the direction in which the parabola opens, among other characteristics.

Understanding the quadratic function and its graph helps students to comprehend more advanced concepts in mathematics and physics. For example, in the launch of an object under the force of gravity, the equation that describes the height of the object over time is a quadratic function. In economics, the total cost function, which relates production cost and quantity produced, can also be represented by a quadratic function.

Theoretical Introduction

The general equation of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a, b$, and $c$ are constants and $x$ is the independent variable. If the coefficient $a$ is positive, the parabola opens upwards. If it is negative, the parabola opens downwards. The graphical representation of these functions, the parabolas, has several important properties that can be directly extracted from the equation.

The vertex of the parabola is the point that represents the maximum or minimum value of the function, and its coordinates are given by $(-\frac{b}{2a}, -\frac{D}{4a})$. The parabola is symmetrical with respect to the vertical axis passing through the vertex. The roots or zeros of the function, which are the values of $x$ where the function crosses the x-axis, can be found using the Bhaskara's formula.

The concavity of the parabola (whether it opens upwards or downwards) is determined by the sign of the coefficient $a$. If $a$ is positive, the function has a minimum value at the vertex of the parabola. If $a$ is negative, the function has a maximum value at the vertex.

Practical Activity

Activity Title: "Building and Analyzing Quadratic Functions Graphs"

Project Objective

The objective of this project is to provide students with a practical and visual understanding of quadratic functions, through the creation, analysis, and interpretation of quadratic function graphs. Additionally, it is intended that students develop collaboration and teamwork skills.

Detailed Project Description

Students will be divided into groups of 3 to 5. Each group will receive three different quadratic function equations. For each equation, the group must perform the following activities:

1. Identify the coefficients a, b, and c and discuss how these numbers influence the shape and position of the parabola.
2. Use the Bhaskara's formula to find the roots of the function.
3. Calculate the vertex of the parabola.
4. Draw the graph of the function, marking the roots, the vertex, and the axis of symmetry.
5. Discuss the concavity of the parabola and what it represents.
6. Propose a real-world problem that can be modeled by the function and discuss how the function's characteristics help solve the problem.

Required Materials

• Graph paper
• Pencils and erasers
• Calculator
• Computer with internet access (optional, to use software like GeoGebra to assist in graph construction)

Step-by-Step Guide for the Activity

1. Students should first identify the coefficients a, b, and c of each equation.
2. They should then calculate the roots of the function using the Bhaskara's formula.
3. Next, they should calculate the vertex of the parabola using the vertex formula.
4. Having identified the roots and the vertex, students should then draw the graph of the function on the graph paper.
5. Once the graph is complete, students should discuss the concavity of the parabola (whether it opens upwards or downwards) and what that means in terms of the function.
6. Finally, students should propose a real-world problem that can be modeled by the function and discuss how the function's characteristics (roots, vertex, concavity) help solve the problem.

Project Deliverables

At the end of the project, each group must produce a written report that includes:

• Introduction: Description of the quadratic function and its relevance. The objective of this project should be included, as well as how it applies to real situations.
• Development: Detailing of the theory used to solve the project and a detailed description of each activity performed and the results obtained. The equations used, the roots, the vertex, the concavity of the parabola, and the proposed problem should be clearly explained.
• Conclusions: Reflections on what was learned in this project, including the technical and socio-emotional skills developed.
• Bibliography: References to all sources of information used throughout the project.

Students should ensure that the report is clear and well-organized, demonstrating a solid understanding of the topic and the skills developed.