Exploring Quadratic Functions: From Theory to Practice

Objectives

1. Recognize and describe the characteristics of a quadratic function.

2. Identify the inputs (x values) and outputs (y values) in a quadratic function.

Contextualization

Quadratic functions, also known as second-degree functions, are used in various fields of knowledge. From physics, in the description of projectile motion, to economics, in profit and cost analysis, these functions are fundamental for modeling and solving real-world problems. Understanding how to identify the inputs (x values) and outputs (y values) of a quadratic function is essential for applying this knowledge in practical situations.

Relevance of the Theme

Did you know that quadratic functions are widely used in engineering to calculate the trajectory of rockets and satellites? Additionally, in the financial market, these functions are employed to forecast stock behavior and optimize investment portfolios. The skills associated with understanding and manipulating these functions are highly valued in careers such as data science, engineering, and economics.

Definition of Quadratic Function

A quadratic function is a polynomial function of the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The graph of this function is a parabola that can open upwards or downwards depending on the sign of 'a'.

• General Form: f(x) = ax² + bx + c

• Constants: 'a', 'b', and 'c' are real numbers, with 'a' ≠ 0

• Graph: The graph is a parabola

• Direction of the Parabola: The parabola opens upwards if 'a' > 0 and downwards if 'a' < 0

Identification of Inputs (x) and Outputs (y)

The inputs of a quadratic function are the 'x' values that we insert into the function, and the outputs are the corresponding 'y' values obtained. The relationship between x and y is expressed by the function's formula.

• Input: 'x' value entered into the function

• Output: 'y' value obtained after calculating the function

• Relationship: The output 'y' is obtained by substituting 'x' into the formula f(x)

• Example: For f(x) = x² - 4x + 3, if x = 2, then y = f(2) = 2² - 4*2 + 3 = -1

Calculation of the Vertex of the Parabola

The vertex of a parabola is the point where it reaches its maximum or minimum value. For a quadratic function f(x) = ax² + bx + c, the vertex can be found using the formulas x_v = -b/(2a) and y_v = f(x_v).

• Vertex Formula: x_v = -b/(2a)

• Calculation of y_v: y_v = f(x_v)

• Critical Point: The vertex is the maximum or minimum point of the parabola

• Example: For f(x) = x² - 4x + 3, x_v = 2 and y_v = f(2) = -1

Practical Applications

• Aerospace Engineering: Calculating the trajectory of rockets and satellites.
• Economics: Analyzing profit and cost to optimize business operations.
• Financial Market: Predicting stock behavior and optimizing investment portfolios.

Key Terms

• Quadratic Function: A polynomial function of the form f(x) = ax² + bx + c.

• Parabola: The graph of a quadratic function.

• Vertex: The maximum or minimum point of a parabola.

• Roots: 'x' values where f(x) = 0.

Questions

• How can understanding quadratic functions help in solving real problems, such as cost optimization in a business?

• What are the applications of quadratic functions in your daily life?

• In what ways are quadratic functions relevant to the careers you wish to pursue?

Conclusion

To Reflect

Throughout this lesson, we explored quadratic functions, also known as second-degree functions, and their wide-ranging applications in various fields of knowledge. We understood how to identify the inputs (x values) and outputs (y values) of these functions, calculated the vertex of the parabola and the roots of the function. Through practical activities and mini-challenges, we saw how these theoretical concepts apply in real situations, such as calculating rocket trajectories in aerospace engineering or predicting stock behavior in the financial market. Reflecting on these practical applications helps us recognize the relevance and importance of quadratic functions in our daily lives and our future careers.

Mini Challenge - Modeling the Trajectory of a Rocket

In this mini-challenge, you will apply the concepts of quadratic functions to model the trajectory of a rocket. Working in groups, you will build a simplified model of a rocket and calculate the parabolic trajectory using a quadratic function.

• Divide into groups of 4-5 people.
• Use cardstock, scissors, tape, and a ruler to build a simplified model of a rocket.
• Define a quadratic function that represents the rocket's trajectory.
• Calculate the input points (x values) and output points (y values) of the function, as well as the vertex and the roots.
• Draw the trajectory on a poster board or whiteboard.
• Each group must present their model and explain the calculations performed.