Finding The Vertex Of F(x) = 2x^2 - 8x + 6 By Completing The Square

To find the vertex of the quadratic function $f(x) = 2x^2 - 8x + 6$, we will use the method of completing the square. This method allows us to rewrite the quadratic function in vertex form, which makes it easy to identify the vertex and determine whether it is a minimum or maximum point.

Understanding Quadratic Functions and Their Vertices

In this section, we delve deeper into the world of quadratic functions and the significance of their vertices. Quadratic functions, characterized by their parabolic shape, are expressed in the general form $f(x) = ax^2 + bx + c$, where a, b, and c are constants. The vertex of a parabola, a crucial point on the graph, represents either the minimum or maximum value of the function. Understanding how to locate the vertex is fundamental in various mathematical and real-world applications.

The vertex form of a quadratic function, $f(x) = a(x - h)^2 + k$, offers a clear advantage in identifying the vertex. In this form, the vertex is simply the point (h, k). The value of a determines whether the parabola opens upwards or downwards. If a > 0, the parabola opens upwards, indicating that the vertex is a minimum point. Conversely, if a < 0, the parabola opens downwards, and the vertex represents a maximum point.

The process of completing the square is a powerful technique used to transform a quadratic function from its general form to its vertex form. This method not only helps in finding the vertex but also provides insights into the symmetry and behavior of the quadratic function. By understanding the vertex, we can determine the axis of symmetry, which is the vertical line x = h that divides the parabola into two symmetrical halves. The vertex also helps in identifying the range of the function, as the k value represents the minimum or maximum value that the function can attain.

In practical applications, understanding the vertex of a quadratic function is crucial. For example, in physics, the trajectory of a projectile can be modeled by a quadratic function, and the vertex represents the maximum height reached by the projectile. In business and economics, quadratic functions can model cost, revenue, and profit, where the vertex helps in identifying the point of maximum profit or minimum cost. Therefore, mastering the technique of completing the square and understanding the vertex form is not only essential for mathematical problem-solving but also for real-world applications.

Step-by-Step Solution Using Completing the Square

In this section, we will walk through a detailed, step-by-step solution to find the vertex of the given quadratic function $f(x) = 2x^2 - 8x + 6$ using the method of completing the square. This method is a powerful algebraic technique that transforms a quadratic expression into a perfect square trinomial, making it easier to identify the vertex form of the quadratic function.

1. Factor out the coefficient of the $x^2$ term:

Our first step is to factor out the coefficient of the $x^2$ term, which in this case is 2, from the first two terms of the function. This gives us:

$f(x) = 2(x^2 - 4x) + 6$

By factoring out the 2, we simplify the expression inside the parentheses, making it easier to complete the square.

2. Complete the square:

Now, we focus on the expression inside the parentheses, $x^2 - 4x$. To complete the square, we need to add and subtract a value that will turn this expression into a perfect square trinomial. The value we need to add and subtract is (rac{b}{2})^2, where b is the coefficient of the x term. In this case, b = -4, so we calculate (rac{-4}{2})^2 = (-2)^2 = 4.

We add and subtract this value inside the parentheses:

$f(x) = 2(x^2 - 4x + 4 - 4) + 6$

Adding and subtracting the same value does not change the overall expression but allows us to rewrite it in a more convenient form.

3. Rewrite as a perfect square:

The expression $x^2 - 4x + 4$ is now a perfect square trinomial, which can be rewritten as $(x - 2)^2$. So, we have:

$f(x) = 2((x - 2)^2 - 4) + 6$

This step is crucial as it transforms the quadratic expression into a form that reveals the vertex of the parabola.

4. Distribute and simplify:

Next, we distribute the 2 back into the parentheses and simplify the expression:

$f(x) = 2(x - 2)^2 - 8 + 6$

$f(x) = 2(x - 2)^2 - 2$

This step brings us closer to the vertex form of the quadratic function.

5. Identify the vertex:

Now, the function is in vertex form, $f(x) = a(x - h)^2 + k$, where the vertex is the point (h, k). In our case, $h = 2$ and $k = -2$. Therefore, the vertex of the parabola is $(2, -2)$.

The vertex form provides a clear and direct way to identify the vertex of the parabola.

By following these steps, we have successfully transformed the given quadratic function into vertex form and identified the vertex as $(2, -2)$. This method is not only effective for this particular function but can be applied to any quadratic function to find its vertex.

Determining Minimum or Maximum Point

Once we have found the vertex of the quadratic function, the next crucial step is to determine whether this point represents a minimum or maximum value of the function. This determination is essential for understanding the behavior of the parabola and the nature of the function's extreme values. In the context of our example, we have found the vertex of the function $f(x) = 2x^2 - 8x + 6$ to be $(2, -2)$, but we still need to ascertain whether this point corresponds to a minimum or a maximum.

The key to determining whether the vertex is a minimum or maximum lies in the coefficient of the $x^2$ term, which we denote as a. In the general form of a quadratic function, $f(x) = ax^2 + bx + c$, the sign of a dictates the parabola's orientation. If a is positive, the parabola opens upwards, resembling a U-shape. In this case, the vertex is the lowest point on the graph, representing the minimum value of the function. Conversely, if a is negative, the parabola opens downwards, resembling an inverted U-shape. Here, the vertex is the highest point on the graph, representing the maximum value of the function.

In our specific function, $f(x) = 2x^2 - 8x + 6$, the coefficient a is 2, which is a positive number. This indicates that the parabola opens upwards, and the vertex $(2, -2)$ is indeed a minimum point. This means that the function $f(x)$ reaches its lowest value at $x = 2$, and this minimum value is $f(2) = -2$. Understanding this distinction between minimum and maximum points is critical in various applications, such as optimization problems where we seek to find the lowest cost, highest profit, or optimal value.

To further illustrate this concept, consider a scenario where the quadratic function represents the cost of production for a certain item. If the parabola opens upwards, the vertex represents the minimum cost required to produce the item. Conversely, if the function represents the profit from selling an item and the parabola opens downwards, the vertex represents the maximum profit that can be achieved. Therefore, by analyzing the sign of the coefficient a and identifying the vertex, we can gain valuable insights into the behavior of the quadratic function and its implications in real-world scenarios. In summary, since the coefficient of $x^2$ is positive in our example, the vertex $(2, -2)$ represents a minimum point of the function.

Conclusion

In this article, we have demonstrated how to find the vertex of the quadratic function $f(x) = 2x^2 - 8x + 6$ using the method of completing the square. By following the step-by-step process of factoring, completing the square, rewriting the function in vertex form, and identifying the vertex, we found that the vertex is located at the point $(2, -2)$. Additionally, we determined that this vertex represents a minimum point because the coefficient of the $x^2$ term is positive.

Understanding how to find the vertex of a quadratic function is a fundamental skill in algebra and calculus. The vertex provides valuable information about the function's behavior, including its minimum or maximum value and the axis of symmetry. The method of completing the square is a powerful technique that can be applied to any quadratic function to find its vertex, regardless of whether the function can be easily factored. This method not only helps in identifying the vertex but also provides a deeper understanding of the structure and properties of quadratic functions.

The ability to determine whether the vertex is a minimum or maximum point is equally important. This is achieved by examining the coefficient of the $x^2$ term. A positive coefficient indicates a minimum point, while a negative coefficient indicates a maximum point. This knowledge is crucial in various applications, such as optimization problems, where we seek to find the extreme values of a function. For instance, in business and economics, quadratic functions can model cost, revenue, and profit, and finding the vertex helps in identifying the point of maximum profit or minimum cost.

In summary, the process of finding the vertex and determining its nature (minimum or maximum) is essential for analyzing and understanding quadratic functions. The method of completing the square provides a systematic approach to achieve this, and the insights gained are applicable in a wide range of mathematical and real-world contexts. Mastering these concepts and techniques will undoubtedly enhance your problem-solving skills and deepen your understanding of quadratic functions.

Therefore, the correct answer is that the function has a minimum at $(2, -2)$.