Finding F(0) For F(x)=-3(x-2)^2+6 A Comprehensive Guide

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In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. Among the diverse family of functions, quadratic functions hold a special place due to their ubiquitous presence in various real-world phenomena, from the trajectory of a projectile to the shape of a suspension bridge. This article delves into the intricacies of a specific quadratic function, f(x) = -3(x-2)^2 + 6, with the primary goal of determining the value of f(0). However, our exploration extends beyond a simple calculation; we aim to provide a comprehensive understanding of the function's properties, its graphical representation, and its significance in broader mathematical contexts.

Understanding the Anatomy of the Quadratic Function

Before we embark on the journey of finding f(0), it is crucial to dissect the given quadratic function and understand its constituent parts. The function f(x) = -3(x-2)^2 + 6 is expressed in the vertex form of a quadratic equation, which is a particularly insightful representation. The general form of a vertex form quadratic equation is:

f(x) = a(x - h)^2 + k

where:

  • a determines the direction and steepness of the parabola.
  • (h, k) represents the coordinates of the vertex, the turning point of the parabola.

In our specific case, f(x) = -3(x-2)^2 + 6, we can readily identify the key parameters:

  • a = -3: The negative value of a indicates that the parabola opens downwards, and the magnitude of 3 suggests a steeper curve compared to the standard parabola.
  • h = 2: The x-coordinate of the vertex is 2.
  • k = 6: The y-coordinate of the vertex is 6.

Thus, the vertex of the parabola represented by f(x) = -3(x-2)^2 + 6 is located at the point (2, 6). This information is invaluable in visualizing the graph of the function and understanding its behavior.

The Significance of the Vertex

The vertex of a parabola holds paramount significance as it represents the maximum or minimum point of the quadratic function. In our case, since the parabola opens downwards (a < 0), the vertex (2, 6) corresponds to the maximum point of the function. This implies that the function attains its highest value at x = 2, and that maximum value is 6. The vertex form of the quadratic equation provides a direct way to identify this crucial feature of the function.

Furthermore, the vertex plays a pivotal role in determining the axis of symmetry of the parabola. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. For the function f(x) = -3(x-2)^2 + 6, the axis of symmetry is the vertical line x = 2. This symmetry property simplifies the process of sketching the graph of the parabola and understanding its overall shape.

Calculating f(0): A Step-by-Step Approach

Now that we have a solid grasp of the function's properties and its vertex form representation, we can proceed to the core task of calculating f(0). To find the value of the function at x = 0, we simply substitute 0 for x in the equation:

f(0) = -3(0 - 2)^2 + 6

Following the order of operations (PEMDAS/BODMAS), we first evaluate the expression within the parentheses:

f(0) = -3(-2)^2 + 6

Next, we square the term -2:

f(0) = -3(4) + 6

Now, we perform the multiplication:

f(0) = -12 + 6

Finally, we complete the addition:

f(0) = -6

Therefore, the value of the function f(x) = -3(x-2)^2 + 6 at x = 0 is -6. This means that the point (0, -6) lies on the graph of the parabola.

Visualizing the Solution: Graphing the Quadratic Function

To further solidify our understanding, let's visualize the solution by sketching the graph of the quadratic function f(x) = -3(x-2)^2 + 6. We already know that the vertex is at (2, 6) and the parabola opens downwards. We have also calculated that f(0) = -6, which gives us another point on the graph, (0, -6).

To get a better sense of the parabola's shape, we can find a few more points. For instance, we can calculate f(4):

f(4) = -3(4 - 2)^2 + 6 = -3(2)^2 + 6 = -3(4) + 6 = -12 + 6 = -6

This gives us the point (4, -6), which is symmetrical to the point (0, -6) with respect to the axis of symmetry x = 2. With these three points (0, -6), (2, 6), and (4, -6), we can sketch a reasonably accurate graph of the parabola. The graph will show a downward-opening curve with its peak at the vertex (2, 6) and passing through the points (0, -6) and (4, -6).

The graphical representation vividly illustrates that the function's value at x = 0 is indeed -6, confirming our algebraic calculation. The graph also provides a visual context for understanding the function's behavior and its key features, such as the vertex, axis of symmetry, and maximum value.

Connecting to Broader Mathematical Concepts

The exercise of finding f(0) for the quadratic function f(x) = -3(x-2)^2 + 6 extends beyond a mere numerical calculation. It serves as a gateway to understanding several fundamental mathematical concepts:

  • Function Evaluation: Substituting a specific value for the variable in a function is a core skill in mathematics, enabling us to determine the function's output for a given input.
  • Quadratic Functions: This exercise reinforces our understanding of quadratic functions, their vertex form representation, and the significance of the vertex and axis of symmetry.
  • Graphing Functions: Visualizing functions through their graphs is crucial for developing a deeper understanding of their behavior and properties. This example demonstrates how to sketch the graph of a quadratic function using key points like the vertex and intercepts.
  • Problem-Solving Strategies: The step-by-step approach we employed to calculate f(0) exemplifies a systematic problem-solving strategy that can be applied to a wide range of mathematical problems.

In conclusion, determining f(0) for the quadratic function f(x) = -3(x-2)^2 + 6 is not just about finding a numerical answer. It is an opportunity to delve into the intricacies of quadratic functions, understand their properties, and connect them to broader mathematical concepts. The process involves algebraic manipulation, graphical visualization, and a systematic problem-solving approach, all of which are essential skills in the realm of mathematics. The result, f(0) = -6, is a single data point that unveils a wealth of information about the behavior and characteristics of the given quadratic function.