Complete The Table For F(x) = 2x² - 8x + 6 A Step-by-Step Guide

In the realm of mathematics, quadratic functions hold a significant position, serving as the building blocks for various applications across diverse fields. These functions, characterized by their parabolic curves, exhibit unique properties and patterns that are essential to understand. In this comprehensive guide, we embark on a journey to unravel the intricacies of the quadratic function f(x) = 2x² - 8x + 6, delving into its evaluation, graphical representation, and real-world applications.

Completing the Table: Unveiling the Function's Behavior

To gain a thorough understanding of the function f(x) = 2x² - 8x + 6, our initial step involves evaluating it for specific values of x. This process entails substituting the given x-values into the function's equation and meticulously calculating the corresponding f(x) values. By completing the table provided, we can effectively map the function's behavior and identify key characteristics.

Evaluating f(x) at x = 0: Unveiling the y-intercept

To determine the value of f(x) when x equals 0, we substitute x = 0 into the function's equation: f(0) = 2(0)² - 8(0) + 6 = 6. This calculation reveals that when x is 0, f(x) equals 6. This crucial point, (0, 6), represents the y-intercept of the parabola, where the curve intersects the vertical axis. The y-intercept serves as a fundamental landmark in visualizing the parabola's position and orientation on the coordinate plane.

Evaluating f(x) at x = 1: Navigating the Parabola's Trajectory

Next, we evaluate f(x) at x = 1 by substituting this value into the function's equation: f(1) = 2(1)² - 8(1) + 6 = 0. This calculation indicates that when x is 1, f(x) equals 0. The point (1, 0) signifies an x-intercept of the parabola, where the curve intersects the horizontal axis. X-intercepts play a pivotal role in determining the roots or solutions of the quadratic equation, representing the points where the function's value equals zero.

Evaluating f(x) at x = 2: Exploring the Parabola's Symmetry

To further explore the function's behavior, we evaluate f(x) at x = 2: f(2) = 2(2)² - 8(2) + 6 = -2. This calculation reveals that when x is 2, f(x) equals -2. The point (2, -2) lies on the parabola and contributes to our understanding of its overall shape and position. This point, along with others, helps us trace the curve's trajectory and identify its key features.

Evaluating f(x) at x = 3: Unveiling the Parabola's Vertex

Continuing our exploration, we evaluate f(x) at x = 3: f(3) = 2(3)² - 8(3) + 6 = 0. This calculation indicates that when x is 3, f(x) equals 0. Intriguingly, we encounter another x-intercept at the point (3, 0). This observation hints at the parabola's symmetry, suggesting that the vertex, the parabola's highest or lowest point, lies equidistant from these two x-intercepts. The vertex plays a crucial role in determining the parabola's maximum or minimum value and its axis of symmetry.

Evaluating f(x) at x = 4: Confirming the Parabola's Trajectory

Finally, we evaluate f(x) at x = 4: f(4) = 2(4)² - 8(4) + 6 = 6. This calculation reveals that when x is 4, f(x) equals 6. This point, (4, 6), mirrors the y-intercept (0, 6), further solidifying the parabola's symmetrical nature. By plotting this point, we gain additional insights into the parabola's curvature and its overall trajectory.

Summarizing the Results: Completing the Table

Now that we have meticulously evaluated f(x) for each given value of x, we can complete the table, summarizing our findings:

This completed table provides a comprehensive snapshot of the function's behavior, highlighting its key characteristics and paving the way for a deeper understanding of its graphical representation.

Unveiling the Secrets of Quadratic Functions: A Comprehensive Exploration

Having successfully completed the table and gained a preliminary understanding of the function f(x) = 2x² - 8x + 6, we now embark on a more in-depth exploration of quadratic functions, delving into their fundamental properties, graphical representations, and diverse applications.

Deciphering the Standard Form: Unlocking the Function's Structure

Quadratic functions are typically expressed in the standard form: f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. This standard form provides valuable insights into the function's characteristics. The coefficient 'a' determines the parabola's direction: if 'a' is positive, the parabola opens upwards, indicating a minimum value; if 'a' is negative, the parabola opens downwards, indicating a maximum value. The coefficient 'b' influences the parabola's horizontal position, while the constant 'c' represents the y-intercept.

Graphing Quadratic Functions: Visualizing the Parabola

The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola's vertex, the point where the curve changes direction, plays a crucial role in its graphical representation. The x-coordinate of the vertex can be calculated using the formula: x = -b / 2a. Once the x-coordinate is determined, the y-coordinate can be found by substituting this value back into the function's equation. The axis of symmetry, a vertical line passing through the vertex, divides the parabola into two symmetrical halves. The x-intercepts, also known as the roots or zeros of the function, are the points where the parabola intersects the x-axis. These intercepts can be found by setting f(x) = 0 and solving the resulting quadratic equation.

Applications of Quadratic Functions: From Projectiles to Optimization

Quadratic functions find widespread applications in various fields, including physics, engineering, economics, and computer science. In physics, they are used to model projectile motion, describing the trajectory of objects launched into the air. In engineering, they are employed in designing bridges, arches, and other structures. In economics, they can be used to model cost and revenue functions, optimizing production and pricing strategies. In computer science, they are utilized in various algorithms, including curve fitting and optimization problems.

Real-World Applications: Unveiling the Practical Significance of Quadratic Functions

To further illustrate the relevance of quadratic functions, let's delve into some real-world applications that highlight their practical significance:

Projectile Motion: Tracing the Path of a Thrown Object

Imagine throwing a ball into the air. The ball's trajectory, its path through the air, closely resembles a parabola. Quadratic functions can be used to model this projectile motion, allowing us to predict the ball's height and horizontal distance at any given time. This knowledge is invaluable in sports, where athletes can optimize their throws and kicks to achieve maximum distance and accuracy. Furthermore, in military applications, quadratic functions are used to calculate the trajectory of projectiles, ensuring precise targeting.

Bridge Design: Ensuring Structural Integrity and Stability

Engineers often employ quadratic functions in the design of bridges, particularly suspension bridges and arch bridges. The parabolic shape of these bridges distributes the load evenly, maximizing structural integrity and stability. The cables of a suspension bridge, for instance, form a parabolic curve, allowing them to effectively support the weight of the bridge deck and traffic. Similarly, the arch of an arch bridge distributes the load along the curve, ensuring the bridge's ability to withstand heavy loads and resist external forces.

Optimization Problems: Maximizing Profits and Minimizing Costs

In the realm of business and economics, quadratic functions play a crucial role in optimization problems. For example, a company might use a quadratic function to model its profit as a function of production quantity. By finding the vertex of the parabola, the company can determine the production quantity that maximizes its profit. Similarly, quadratic functions can be used to model costs, allowing businesses to identify the production level that minimizes expenses. These optimization techniques are essential for businesses to make informed decisions and achieve their financial goals.

Conclusion: Embracing the Power of Quadratic Functions

In this comprehensive guide, we have embarked on a journey to unravel the intricacies of the quadratic function f(x) = 2x² - 8x + 6. We began by completing the table, evaluating the function for specific values of x and gaining insights into its behavior. We then delved into the fundamental properties of quadratic functions, exploring their standard form, graphical representation, and diverse applications. Finally, we examined real-world examples, highlighting the practical significance of quadratic functions in various fields.

By understanding the properties and applications of quadratic functions, we equip ourselves with a powerful tool for solving real-world problems and making informed decisions. These functions serve as a testament to the beauty and practicality of mathematics, providing a framework for understanding and shaping the world around us.