Introduction

The Relevance of the Topic

The second degree function (or quadratic function) is a central concept in mathematics. The ability to model and solve real-world problems with second degree functions is widely applied in natural sciences, social sciences, and computer sciences. This is a topic of great importance for the development of logical and mathematical reasoning, which are fundamental skills for any citizen.

Contextualization

The second degree function is inserted in the field of polynomial functions. After the introduction of linear functions in Elementary School, the study of quadratic functions marks a deepening in students' understanding of the variation between quantities, providing the ability to model complex situations. From here, it is possible to explore more advanced topics, such as graph theory, space geometry, and mathematical physics. Therefore, this is a crucial starting point for the future study of mathematics.

Theoretical Development

Components

• General Characteristics: A quadratic function is characterized by the squared variable, which can be represented in the form f(x) = ax² + bx + c, where a, b, and c are real constants and a is not zero.

  • 'a' is the coefficient of the highest degree term and defines the concavity of the graph.
  • 'b' is the linear coefficient and indicates the shift of the graph to the left or right.
  • 'c' is the constant, which determines the vertical shift of the graph.

• Canonical Form: A crucial aspect to be discussed is the transformation of the quadratic equation from the general form to the canonical form f(x) = a(x - h)² + k. In this form, (h, k) is the vertex of the graph and 'a' is the dilation/compression coefficient.

  • 'h' and 'k' determine the shift of the graph on the Cartesian plane.
  • 'a' indicates the opening direction of the graph.

• Sign of the Coefficient 'a': This topic delves into the influence of the coefficient 'a' on the graph of the quadratic function.

  • If 'a' is positive, the parabola opens upwards; if 'a' is negative, the parabola opens downwards.
  • The greater the absolute value of 'a', the narrower the parabola.

Key Terms

• Quadratic Function: A mathematical function whose most general form is f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0.
• Vertex: Minimum or maximum point of a parabola. Given by the coordinate (h, k) in the canonical form of the quadratic function.
• Coefficient 'a': This term in a quadratic function determines the opening direction of the parabola.

Examples and Cases

• Graph of y = x² - 3x + 2: With this example, it is possible to visualize how the coefficients 'a', 'b', and 'c' affect the graph of the quadratic function.

  • From the canonical form f(x) = a(x - h)² + k, it can be identified that the parabola opens upwards, as a = 1.
  • The vertex of the parabola is located at the point (1.5, -1.25).
  • The intersection with the y-axis occurs at (0, 2).

• Modeling a Real-World Problem: Modeling real-world situations in quadratic functions is a practical application of this concept. For example, predicting the height of a ball as a function of time when vertically thrown upwards can be modeled by a quadratic function.

  • The equation of height as a function of time can be expressed in the form f(t) = -4.9t² + vt + h, where -4.9 is half the acceleration due to gravity, v is the initial velocity, and h is the initial height. Here, the second degree function arises as a direct result of constant acceleration.

Detailed Summary

Relevant Points

• Understanding the General Characteristics: Mastering the knowledge of the characteristics of a second degree function in the general form (ax² + bx + c) is the first step to understand how these functions behave graphically. The coefficients a, b, and c are key components in describing these functions and how they relate to the Cartesian plane.

• The Importance of the Canonical Form: The concept of the canonical form of the second degree function (f(x) = a(x - h)² + k) is crucial to interpret the role of a, h, and k in the graph of the function. Especially, understanding how the variation of these parameters translates into movements of the graph on the Cartesian plane is essential.

• The Sign of the Coefficient 'a': The opening direction of the parabola is determined by the coefficient 'a'. It is important to highlight that 'a' does not necessarily need to be a positive number, it can be negative or zero. This characteristic has direct consequences on the interpretation and resolution of problems involving second degree functions.

Conclusions

• The second degree function is a powerful and versatile concept in mathematics and has applications in a variety of disciplines and real-world situations. From the perspective of mathematical modeling, the quadratic function is the basis for understanding the movement of bodies subject to gravitational force (falls and vertical launches, for example).

• The analysis of the coefficients a, b, and c, as well as the interpretation of the resulting graph of a second degree function, allow valuable conclusions to be drawn about the behavior of variables in a given context.

Suggested Exercises

1. Vertex Calculation: Given the function y = 2x² - 8x + 9, find the coordinates of the vertex of the parabola.
2. Interpretation of Coefficients: Compare the graph of the function y = -x² with the graph of the function y = 2x². What can be deduced about the relationship between the coefficient 'a' and the graph's behavior?
3. Modeling Real-World Problems: Model the height, as a function of time, of a ball thrown vertically upwards, considering that the ball was thrown from a height of 5 meters with an initial velocity of 10 m/s.