Introduction

Relevance of the Theme

The Second Degree Function is one of the fundamental pillars of mathematics and its applications permeate various areas of knowledge, from physics to economics. Mastering this concept allows for the solution of practical problems, the manipulation of mathematical models, and the abstract understanding of the nature of numbers. The graph and table of this function are powerful tools that enable visualizing and understanding its behavior, characteristics, and properties.

Contextualization

Inserted in the broader context of the High School Mathematics curriculum, the study of Second Degree Function: Graph and Table is strategically located as a central topic that interconnects multiple units of learning. It serves as a basis for the study of more complex functions, such as exponential and logarithmic functions, and as preparation for advanced concepts in calculus and mathematical analysis. This section also connects with the study of analytic geometry and statistics, enriching the student's understanding of these disciplines and stimulating an integrated view of mathematical knowledge.

Theoretical Development

Components

• General Form of the Second Degree Function: The general form of a second degree function is f(x) = ax² + bx + c, with a, b, and c being constants and a ≠ 0. This essential format defines the function and its main characteristics, such as concavity, intercepts with the axes, and the vertex of the parabola.

• Discriminant (Δ): The discriminant (Δ) is an essential tool for the analysis of the second degree function and is directly related to its zeros, that is, the points where the function intersects the x-axis. If Δ > 0, the function has two distinct real zeros; if Δ = 0, the function has two real and equal zeros (i.e., a single point of intersection with the x-axis); if Δ < 0, the function has no real zeros (parabola does not intersect the x-axis).

• Coefficients (a, b, and c): The coefficients a, b, and c of the general form of the second degree function are determinants of its behavior and appearance in the graph. The coefficient a defines the concavity of the parabola (a > 0 parabola facing upwards, a < 0 parabola facing downwards), the coefficient b is related to the position of the vertex, and the coefficient c is the ordinate of the point where the parabola intersects the y-axis.

• Vertex of the Parabola: The vertex of the parabola f(x) = ax² + bx + c has coordinates x = -b/2a and y = f(x), and represents the point of maximum or minimum (depending on the concavity of the parabola) of the function. It is a point of extremum on the graph.

• Roots of the Second Degree Function: The roots of a second degree function correspond to the values of x for which f(x) = 0. These values can be determined from the Discriminant (Δ) and are of great importance for the construction of the table and the graph of the function.

Key Terms

• Second Degree Function: Also known as a quadratic function, it is a function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The main elements of this function are the coefficient a (determines the concavity of the parabola), the coefficient b (determines the position of the vertex), and the coefficient c (determines the ordinate of the point where the parabola intersects the y-axis).

• Function Graph: The graph of a second degree function is a parabola. Its shape and position in the plane are determined by the coefficients a, b, and c of the function. The analysis of the graph allows inferences about the zeros of the function, the concavity of the parabola, and the properties of the function.

• Function Table: The table of the second degree function is constructed by substituting values of x into the function. The ordered pairs (x, f(x)) obtained are the coordinates of the points that belong to the graph of the function.

Examples and Cases

1. Example 1: Let f(x) = x² - 4x + 3 be a function. This is a second degree function with a = 1, b = -4, and c = 3. We can determine the vertex and roots of the function, as well as the type of parabola (facing upwards, in this case) from these coefficients. The discriminant is Δ = b² - 4ac = (-4)² - 4(1)(3) = 16 - 12 = 4. Since Δ > 0, we know that the function has two distinct real roots.

2. Example 2: Let g(x) = -2x² + 4x - 1 be a function. In this case, the function is a parabola facing downwards, as a = -2. The vertex of the parabola can be calculated through x = -b/2a = -4/(-4) = 1. Substituting x = 1 into the function, we get y = g(1) = -2(1)² + 4(1) - 1 = 1. Therefore, the vertex of the parabola is (1,1).

Detailed Summary

Key Points

• Understanding the Notation: The second degree function, also known as a quadratic function, is represented by f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Each of these coefficients plays a fundamental role in the configuration and behavior of the function, which is depicted in the form of a parabola.

• Exploring the Discriminant: The discriminant (Δ), obtained from the formula Δ = b² - 4ac, is an essential tool in analyzing the characteristics of the parabola. From it, we can classify the function according to the number of distinct real roots: Δ > 0 indicates two roots, Δ = 0 indicates one root, and Δ < 0 indicates that the function has no real roots.

• Vertex of the Parabola and its Coordinates: The vertex of a parabola, or point of maximum or minimum, is vital to understanding the behavior of the function. For a second degree function of the form f(x) = ax² + bx + c, the coordinates of the vertex are x = -b/2a and y = f(x).

• Defining the Roots: The roots, also known as zeros, of a second degree function are the points where the function intersects the x-axis. The quantity and values of these points are determined by the discriminant (Δ).

• Constructing the Graph and Table: The function table is obtained by substituting values of x into the function and recording the ordered pairs (x, f(x)). These pairs constitute the points that belong to the graph of the function, being the visual representation of the function's behavior.

Conclusions

• Proficiency in Analyzing the Second Degree Function: Mastery of the concepts and tools associated with the second degree function, such as the Discriminant, the analysis of coefficients, and the location and analysis of the vertex, allows for precise inferences about the behavior and characteristics of the function.

• Effective Use of the Graph and Table: The ability to interpret the graph and table of the second degree function is essential in solving practical problems and understanding mathematical models.

• Connection with Other Mathematical Topics: The second degree function, with its parabolic graph, serves as a link between various areas of mathematics, contributing to an integrated understanding of the curriculum.

Exercises

1. Exercise 1: Determine the discriminant, vertex, and roots of the function f(x) = 2x² - 3x + 1.

2. Exercise 2: Given the function g(x) = -x² + 4x - 3, determine the concavity (facing upwards or downwards), the vertex, and the roots.

3. Exercise 3: Construct the table and graph of the function h(x) = x² - x - 2. From the graph, determine the value of h(3).