Contextualization

The study of Quadratic Functions is one of the most important in the high school Mathematics curriculum. These functions, expressed in the form y=ax²+bx+c, where a ≠ 0, present a series of interesting properties and applications in everyday life and in various areas of science.

The graph of these functions is a parabola, which can be directed upwards (when a > 0) or downwards (when a < 0). In addition, this graph has a minimum or maximum point, called the vertex of the parabola, which also has a specific expression in terms of a, b, and c.

Below, you will find some links that contain explanations, examples, and practices on quadratic functions:

1. Brazil School Math
2. Just Math
3. World Education - UOL

The real world is full of parabolas: a trajectory of a soccer ball when a player takes a shot, the arrangement of parabolic antennas, car headlights, among others. In addition, quadratic functions are often used to model and solve problems involving optimization, such as determining the maximum quantity that can be produced with a given cost or the shortest distance between two points.

In this project, you will have the opportunity to explore quadratic functions in detail. You will produce graphs and tables, performing analysis and comparisons. You will also have the chance to apply these functions to solve concrete problems, making learning more meaningful and interesting.

Practical Activity: Quadratic Function - Graph vs Table

Project Objective

The objective of this project is to explore and understand quadratic functions using two main representations: graphs and tables. Students will be challenged to compare different quadratic functions through their respective graphical representations and tables and apply this knowledge to solve real-world problems.

Detailed Project Description

Groups of students will work together to:

1. Generate different quadratic functions, validate them, and represent them in graphs and tables.
2. Analyze the relationships and differences between the various quadratic functions.
3. Make a comparative analysis between the representation of quadratic functions in graphs and tables.

Required Materials

• Paper and pen for drawings and notes.
• Computer with internet access and graphics creation software (Excel, Google Sheets, or other free math software such as Geogebra).
• Mathematics and physics textbooks for reference.

Detailed Step by Step

Part 1: Generating Quadratic Functions

1. Each group must generate at least four different formulas for quadratic functions.
2. Using these functions, students will normalize the table with the values of "x" ranging from -10 to 10.
3. With the values in the table, students will create the graph of each function.

Part 2: Analysis of Quadratic Functions

4. Analyze the generated parabola: direction (opening up or down), vertex, zeros of the function, axis of symmetry.
5. Compare the generated functions: how do the variations in the coefficients change the graph?

Part 3: Application of Quadratic Functions

6. Formulate a real-world problem that can be solved by one of the generated quadratic functions. Describe the problem, solve it with the function, and discuss the results found. Example: projectile launch, production optimization in a factory, etc.

Part 4: Writing the Paper

7. After completing the Practical Part, students should prepare a report following the topics: Introduction, Development, Conclusions, and References used.

Project Submission

At the end of the project, each group must submit:

• Graphs of the quadratic functions that each group generated.
• Tables containing the values of the quadratic functions they generated.
• A short video (up to 5 minutes) explaining the differences they found in the functions and how the graphs change with the variation of the coefficients.
• A written essay on the topic, explaining the project, the findings, and the lessons learned.
• Bibliography: students must list all the sources of research and learning that helped in the development of the project.

Each of these components contributes to a better understanding of the topic and allows students to take responsibility for their own learning process. In addition, working in groups facilitates the learning process, since students have the opportunity to learn from each other.