Projeto: Connecting Graphs and Tables of Quadratic Functions

Contextualization

The quadratic function is one of the essential tools used in mathematics. This function has been studied since high school and remains an important tool in college and beyond, in fields such as engineering, physics, economics, and many others. This function is defined by an expression in general form:

[ ax^2 + bx + c = 0 ]

Where (a), (b), and (c) are constants and (a) cannot be equal to zero.

The appearance of the function on the graph is a parabola that can be opened up or down. The direction of the opening is determined by the coefficient (a). If (a) is positive, the parabola opens up. If (a) is negative, the parabola opens down.

Quadratic functions have many applications in real life. Many natural and scientific phenomena can be described using quadratic functions, from the path taken by an object thrown into the air to the analysis of gains and losses in a company. Therefore, understanding this function is extremely important for those who want to work in areas that involve mathematics.

Practical Activity

Title: Connecting Graphs and Tables of Quadratic Functions

Project Objective

The objective of this activity is to develop students' skills to create tables from quadratic functions, plot the corresponding graph, and understand the graphical interpretation of the quadratic function.

Detailed Project Description

Students will be divided into groups of 3 to 5 students. Each group will receive a different quadratic function and must:

1. Create a Value Table for the function.
2. Plot the function on the Cartesian plane.
3. Analyze the graph and the table, identifying the relevant points of the function (Vertex, Roots).
4. Explain the relationship between the graphical and tabular forms of the function.

Materials Required

1. Pencil and paper or software to create tables (e.g., Excel, Google Sheets).
2. Environment for plotting graphs (e.g., graph paper, Desmos, GeoGebra).
3. The Quadratic Function given by the teacher.

Step-by-Step Activity

1. Form student groups (3 to 5 in each group).
2. Distribute the quadratic functions among the groups.
3. Each group must create a value table for the quadratic function, choosing at least 5 different values for x, and calculate the respective values of y.
4. Using the values from the table, each group must plot the function on the Cartesian plane.
5. Based on the graph and the table, students must identify and mark the notable points of the function (vertex and roots, if any).
6. Each group will discuss and describe the relationship between the table and the graph of the function.
7. Finally, each group will write a report documenting each step of the process and their findings.

Project Deliverables

At the end of the activity, each group must produce:

1. The Value Table of the function.
2. The Graph of the function plotted.
3. A Completion Report:
   • Introduction: Description of the project and its importance for understanding the quadratic function.
   • Development: Explain the quadratic function, the creation of the value table, the plotting of the function, and the analysis of the notable points. In addition, describe the relationship between the table and the graph.
   • Conclusion: Review the main points, explain what was learned, and the conclusions drawn about the project.
   • Bibliography: Indicate the research sources used to develop the activity.

Remember, the quality of the report will have significant weight in the final evaluation. The purpose of this report is to document the students' learning and conclusions during the practical activity. Therefore, it is important that students are clear, detailed, and reflective, making connections with the content studied.