Contextualization

Introduction

Maximums and minimums are a fundamental concept in mathematics, and are especially important in the subdiscipline of optimization. In summary, the maximum and minimum points of a function are, respectively, where the function reaches its highest and lowest value in a given interval. Both are examples of endpoint points.

In the quadratic function, one can determine the maximum or minimum point through the vertex formula of the parabola, which is given by (-B/2A, F(-B/2A)). This concept is of utmost importance as it allows us to understand the nature of graphs and predict behaviors.

The intervals of increase and decrease of a function are important to understand the direction in which the function is moving. A function increases when it moves upwards as it progresses along the x-axis, and decreases when it moves downwards.

Contextualization

Understanding maximums and minimums has numerous practical applications. In the physical sciences, these points can be used to determine the maximum or minimum heights reached by an object in free fall. In economics, they are used to maximize profit or minimize a company's cost.

In the financial world, the concepts of maximum and minimum can be used to analyze and predict trends in the stock market. Maximum points can indicate a good time to sell a stock, while minimum points may suggest a buying opportunity.

Atividade Prática

Activity Title: "Unveiling Maximums and Minimums"

Project Objective

This project was designed so that students can explore and understand in a practical and fun way the theme of maximums and minimums in quadratic functions. The central question of the activity is to identify, analyze, and calculate maximum and minimum points in everyday situations.

Detailed Project Description

Students will be divided into groups of 3 to 5 people, and each group will receive a different hypothetical scenario to analyze. Each scenario will represent a real-life situation that can be optimized through the study of maximum and minimum points. Examples of scenarios include: the trajectory of a rocket in space, maximizing a company's profit, predicting trends in the stock market, among others.

Students should analyze the scenario, plot the quadratic function that best fits the situation, identify where the maximum and minimum points are, and calculate their values. In addition, they should discuss the relevance of these points to the proposed situation.

Necessary Materials

1. Paper and pens
2. Calculator
3. Computer with internet access for research
4. Mathematical graphing software (e.g. Desmos, GeoGebra)

Detailed Step-by-Step for Carrying Out the Activity

1. The teacher divides the class into groups of 3 to 5 students and presents each group with the scenario they should analyze.
2. Students discuss the proposed scenario in groups, research the subject on the internet, and sketch a quadratic function that fits the situation.
3. With the quadratic function defined, students must identify and calculate the maximum and minimum points of the function. To do this, they should use the vertex formula of the parabola.
4. After identifying the maximum and minimum points, students should discuss among themselves the relevance of these points to the proposed scenario.
5. Each group must make a presentation to the class, where they should present the quadratic function they defined, explain how they calculated the minimum and maximum points, and discuss the relevance of these points to the proposed scenario.
6. In addition to the presentation, each group must produce a written report containing all the information from the presentation, as well as a more detailed discussion and reflections on the activity.

Project Deliverables

1. Group presentation to the class.
2. Written report containing all project details, which should include:

• Introduction: Contextualize the proposed scenario and the relevance of studying maximums and minimums for the situation.
• Development: Describe the sketched quadratic function and explain how they calculated the maximum and minimum points. They should also discuss the relevance of these points to the proposed scenario.
• Conclusions: Draw conclusions from the study, reflect on the learning from the activity, and how it can be applied in other contexts.
• Bibliography: Indicate all sources of information used for the project's development.

This project will enable students to develop technical skills such as research, calculation, and problem-solving, and socio-emotional skills such as time management, communication, creative thinking, proactivity, among others. In addition, they will have the opportunity to see the practical application of theoretical concepts studied in the classroom.