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

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In mathematics, quadratic functions are expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve. A crucial feature of a parabola is its vertex, which represents either the minimum or maximum point of the function. Determining the vertex is essential for understanding the behavior and characteristics of the quadratic function. This article provides a comprehensive guide on how to find the vertex of a quadratic function using the completing-the-square method, a powerful algebraic technique. We will delve into the step-by-step process, explain the underlying concepts, and illustrate the method with a concrete example. We will also discuss how to determine whether the vertex represents a minimum or maximum point, further enhancing our understanding of the quadratic function.

Understanding Quadratic Functions and Their Vertex

Before diving into the completing-the-square method, let's first establish a clear understanding of quadratic functions and their vertex. As mentioned earlier, a quadratic function is defined as f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The coefficient a plays a significant role in determining the shape and direction of the parabola. If a > 0, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if a < 0, the parabola opens downwards, and the vertex represents the maximum point of the function.

The vertex of a parabola is the point where the function changes direction. It is the minimum value if the parabola opens upward, and the maximum value if the parabola opens downward. The vertex is represented by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate (or the function's value) at the vertex. The x-coordinate, h, can be found using the formula h = -b / 2a. However, the completing-the-square method provides an alternative and often more insightful way to find the vertex, as it transforms the quadratic function into vertex form, which directly reveals the vertex coordinates.

The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. Transforming a quadratic function into vertex form allows us to easily identify the vertex and determine whether it is a minimum or maximum point. The completing-the-square method is the key to achieving this transformation.

The Completing-the-Square Method: A Step-by-Step Guide

The completing-the-square method is a powerful algebraic technique used to rewrite a quadratic expression in a form that allows us to easily identify the vertex of the corresponding parabola. This method is based on the algebraic identity (x + p)^2 = x^2 + 2px + p^2. The core idea is to manipulate the given quadratic expression to create a perfect square trinomial, which can then be factored into the form (x + p)^2. Let's break down the method into a series of steps:

Step 1: Factor out the coefficient of the x² term (if not 1).

This step is crucial when the coefficient of the x² term, a, is not equal to 1. Factoring out a ensures that we have a leading coefficient of 1 within the parentheses, which is necessary for completing the square. For example, if we have the expression 2x² + 8x + 6, we would factor out the 2, resulting in 2(x² + 4x + 3). This step sets the stage for creating a perfect square trinomial within the parentheses.

Step 2: Take half of the coefficient of the x term, square it, and add and subtract it inside the parentheses.

This is the heart of the completing-the-square method. We focus on the expression inside the parentheses, which now has a leading coefficient of 1. We take half of the coefficient of the x term, which we'll call b', square it ((b'/2)²), and then both add and subtract this value inside the parentheses. Adding and subtracting the same value doesn't change the overall expression but allows us to manipulate it into a perfect square trinomial. For instance, in the example 2(x² + 4x + 3), the coefficient of the x term is 4. Half of 4 is 2, and squaring it gives us 4. So, we add and subtract 4 inside the parentheses: 2(x² + 4x + 4 - 4 + 3).

Step 3: Rewrite the perfect square trinomial as a squared binomial.

The first three terms inside the parentheses should now form a perfect square trinomial, which can be factored into the form (x + p)². The value of p is simply half of the coefficient of the x term from the previous step (b'/2). In our example, x² + 4x + 4 is a perfect square trinomial that can be factored as (x + 2)². So, we rewrite the expression as 2((x + 2)² - 4 + 3).

Step 4: Simplify the expression.

Now, we simplify the expression by combining the constant terms inside the parentheses and distributing the coefficient a (if we factored it out in Step 1). In our example, we have 2((x + 2)² - 1). Distributing the 2, we get 2(x + 2)² - 2.

Step 5: Identify the vertex.

At this point, the quadratic function is in vertex form: f(x) = a(x - h)² + k. The vertex is (h, k). Note that in the expression (x - h)², h is the value that makes the expression inside the parentheses equal to zero. In our example, 2(x + 2)² - 2 can be rewritten as 2(x - (-2))² + (-2). Therefore, the vertex is (-2, -2).

By following these five steps, you can effectively use the completing-the-square method to transform any quadratic function into vertex form and easily identify its vertex.

Example: Finding the Vertex of f(x) = 2x² - 8x + 6

Let's apply the completing-the-square method to find the vertex of the quadratic function f(x) = 2x² - 8x + 6. This example will solidify your understanding of the steps involved and demonstrate how to apply them in practice.

Step 1: Factor out the coefficient of the x² term (if not 1).

In this case, the coefficient of the x² term is 2, so we factor it out: f(x) = 2(x² - 4x + 3).

Step 2: Take half of the coefficient of the x term, square it, and add and subtract it inside the parentheses.

The coefficient of the x term inside the parentheses is -4. Half of -4 is -2, and squaring it gives us 4. We add and subtract 4 inside the parentheses: f(x) = 2(x² - 4x + 4 - 4 + 3).

Step 3: Rewrite the perfect square trinomial as a squared binomial.

The terms x² - 4x + 4 form a perfect square trinomial, which can be factored as (x - 2)². So, we rewrite the expression as: f(x) = 2((x - 2)² - 4 + 3).

Step 4: Simplify the expression.

We simplify the expression by combining the constant terms inside the parentheses and distributing the 2: f(x) = 2((x - 2)² - 1) = 2(x - 2)² - 2.

Step 5: Identify the vertex.

Now the function is in vertex form: f(x) = 2(x - 2)² - 2. The vertex is (2, -2).

Therefore, by using the completing-the-square method, we have successfully found that the vertex of the quadratic function f(x) = 2x² - 8x + 6 is (2, -2).

Determining Minimum or Maximum

Once we have found the vertex, the next important step is to determine whether it represents a minimum or maximum point of the quadratic function. This is crucial for understanding the overall behavior of the parabola. The key to this determination lies in the coefficient a of the x² term in the original quadratic function (or the vertex form). As mentioned earlier, the sign of a dictates the direction in which the parabola opens:

  • If a > 0, the parabola opens upwards, resembling a U shape. In this case, the vertex represents the minimum point of the function. The y-coordinate of the vertex is the smallest value the function can attain.
  • If a < 0, the parabola opens downwards, resembling an inverted U shape. In this case, the vertex represents the maximum point of the function. The y-coordinate of the vertex is the largest value the function can attain.

In our example, f(x) = 2x² - 8x + 6, the coefficient a is 2, which is positive (a > 0). Therefore, the parabola opens upwards, and the vertex (2, -2) represents the minimum point of the function. This means that the smallest value the function f(x) can achieve is -2, and it occurs when x = 2.

Understanding whether the vertex is a minimum or maximum is vital for various applications, such as optimization problems where we seek to find the maximum or minimum value of a function under certain constraints. By analyzing the sign of the coefficient a, we can quickly determine the nature of the vertex and gain valuable insights into the behavior of the quadratic function.

Conclusion

In conclusion, the completing-the-square method is a powerful technique for finding the vertex of a quadratic function. By transforming the function into vertex form, we can easily identify the coordinates of the vertex and determine whether it represents a minimum or maximum point. This method provides a deeper understanding of quadratic functions and their parabolic graphs. The step-by-step approach outlined in this article, along with the illustrative example, equips you with the knowledge and skills to confidently apply the completing-the-square method to any quadratic function. Remember to factor out the leading coefficient, complete the square within the parentheses, rewrite the perfect square trinomial as a squared binomial, simplify the expression, and finally, identify the vertex from the vertex form. By mastering this technique, you gain a valuable tool for analyzing and solving problems involving quadratic functions in various mathematical and real-world contexts.