Contextualization

Quadratic functions are an important topic in mathematics that every high school student must master. They are everywhere in the world around us, from the shape of a football field to the path of a basketball's arc, the height of a rocket, and the profit of a company.

A quadratic function is a second-degree polynomial equation in a single variable with the general form: f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a symmetric curve that opens either upward or downward.

Understanding quadratic functions is crucial in many science, technology, engineering, and mathematical fields. For instance, in physics, they can be used to calculate the trajectory of a projectile. In economics, they are used to model profit and loss. In computer science, they are used in algorithms and optimization problems.

Importance

Understanding and applying quadratic functions can help you make sense of the world around you. They are used in a wide range of real-world applications, such as determining the maximum or minimum points in business and economics, modeling the spread of diseases, predicting the path of a satellite, and even in the design of bridges and buildings.

In addition, learning about quadratic functions can also enhance your problem-solving skills. It teaches you to break down complex problems into smaller, more manageable parts, to analyze and interpret data, and to make predictions. These skills are not only valuable in math but also in many other areas of life and work.

Resources

To help you with this project, here are some reliable resources that you can use:

1. Khan Academy: Quadratic functions and equations
2. Math is Fun: Quadratic Functions
3. PurpleMath: Quadratic Functions
4. Desmos Graphing Calculator
5. Book: Algebra 1, Common Core Edition by John A. Carter

These resources provide a wealth of information, examples, and practice problems to help you understand and apply the concepts of quadratic functions.

Practical Activity

Activity Title: Quadratic Function in Real Life

Objective of the project:

The main objective of this project is to create a real-world scenario that can be modeled by a quadratic function. The scenario should be from one of the following domains: sports, physics, economics, or architecture. The students must formulate the quadratic function that represents their chosen scenario and interpret the meaning of the function's parameters in the context of their scenario.

Detailed description of the project:

Students will work in groups of 3 to 5 to develop their project. The project involves four main steps:

1. Choose a Scenario: Each group will select a real-world scenario from the domains mentioned above (sports, physics, economics, or architecture). The scenario should be something the group members are interested in and can easily collect data or make reasonable assumptions about.

2. Model the Scenario: Based on the chosen scenario, the group will create a quadratic function that represents the situation. The group will decide the values of a, b, and c in the general form of the quadratic function (f(x) = ax^2 + bx + c) and explain why these values were chosen.

3. Graph the Function: The group will use a graphing tool, such as Desmos, to graph their quadratic function. They should also identify and explain the vertex, axis of symmetry, and y-intercept of the parabola.

4. Interpret the Function: The group will interpret the meaning of the function's parameters (a, b, and c) in the context of their scenario. They will also use the graph to make predictions or draw conclusions about their scenario.

Necessary materials:

• Computers with internet access
• Graphing tool (e.g., Desmos)
• Notebook for brainstorming and calculations

Detailed step-by-step for carrying out the activity:

1. Form a Group: Form a group of 3 to 5 students. Make sure everyone in the group is interested in the chosen domain and has access to the necessary materials.

2. Choose a Scenario: Discuss and choose a real-world scenario from the domains given (sports, physics, economics, or architecture).

3. Model the Scenario: Based on your chosen scenario, create a quadratic function that represents the situation. Explain why you chose the values of a, b, and c.

4. Graph the Function: Use a graphing tool to plot your function. Identify and explain the vertex, axis of symmetry, and y-intercept of the parabola.

5. Interpret the Function: Interpret the meaning of the function's parameters in the context of your scenario. Use the graph to make predictions or draw conclusions about your scenario.

6. Report Writing: Each group member will contribute to writing a report that details the process of the project. The report should include an introduction, development, conclusions, and a bibliography.

   • In the Introduction, the students will contextualize the chosen scenario, its relevance, and real-world application, as well as the objective of this project.

   • In the Development section, the students will detail the theory behind quadratic functions, explain the chosen scenario and the model created, present the methodology used, and discuss the obtained results.

   • In the Conclusions, the students will revisit the main points of the project, discuss what they learned from it, and draw conclusions about the application of quadratic functions in real-life situations.

   • In the Bibliography, the students will list the resources they used for the project, including books, websites, and videos.

7. Presentation: Each group will present their project to the class. The presentation should be clear, engaging, and informative.

The project will take approximately one week to complete, with an estimated workload of three to five hours per student. At the end of the project, students should have a deeper understanding of quadratic functions and their real-world applications. They should also have improved their teamwork, problem-solving, and communication skills.