Finding The Vertex Of F(x)=2x^2-8x+6 Using Completing The Square
In mathematics, quadratic functions play a vital role in modeling various real-world phenomena. Understanding the properties of these functions, such as their vertex, is crucial for analyzing their behavior and applications. This article delves into the method of completing the square to determine the vertex of a quadratic function, specifically focusing on the function $f(x) = 2x^2 - 8x + 6$. We will also discuss how to identify whether the vertex represents a minimum or maximum point and its coordinates.
Understanding Quadratic Functions and Their Vertex
A quadratic function is a polynomial function of degree two, generally expressed in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants and a ≠0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. The vertex of the parabola is the point where the curve changes direction. It represents either the minimum value of the function (if the parabola opens upwards) or the maximum value (if the parabola opens downwards).
For a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex can be found using the formula $x = -b / 2a$. However, to find the vertex by completing the square, we transform the quadratic function into vertex form, which directly reveals the vertex coordinates.
The Method of Completing the Square
Completing the square is a technique used to rewrite a quadratic expression in the form $a(x - h)^2 + k$, where (h, k) represents the vertex of the parabola. This method involves manipulating the quadratic expression to create a perfect square trinomial.
Let's apply this method to our given function, $f(x) = 2x^2 - 8x + 6$, step by step:
1. Factor out the coefficient of $x^2$ from the first two terms:
The first step in completing the square involves factoring out the coefficient of the $x^2$ term, which in our case is 2. This step is essential for creating a perfect square trinomial within the parentheses. Factoring out 2 from the first two terms of the function, we get:
This manipulation allows us to focus on completing the square for the expression inside the parentheses, making the process more manageable. The constant term, 6, remains outside the parentheses for now and will be incorporated later in the process.
2. Complete the square inside the parentheses:
To complete the square inside the parentheses, we need to add and subtract a specific value. This value is determined by taking half of the coefficient of the x term (which is -4 in this case), squaring it, and then adding and subtracting it within the parentheses. Half of -4 is -2, and squaring -2 gives us 4. Thus, we add and subtract 4 inside the parentheses:
By adding and subtracting the same value, we are not changing the overall value of the expression, but we are strategically creating a perfect square trinomial. The first three terms inside the parentheses, $x^2 - 4x + 4$, now form a perfect square.
3. Rewrite the perfect square trinomial as a squared binomial:
The perfect square trinomial, $x^2 - 4x + 4$, can be rewritten as a squared binomial. Recognizing that this trinomial is the result of squaring the binomial $(x - 2)$, we can rewrite the expression as:
This step is crucial because it transforms the quadratic expression into a form that directly reveals the vertex of the parabola. The squared binomial $(x - 2)^2$ represents the squared term in the vertex form of the quadratic function.
4. Distribute and simplify:
Now, we distribute the 2 back into the parentheses and simplify the expression:
Combining the constant terms, -8 and 6, we get:
This is the vertex form of the quadratic function, which is $f(x) = a(x - h)^2 + k$, where (h, k) is the vertex of the parabola.
Identifying the Vertex and Minimum/Maximum Point
From the vertex form, $f(x) = 2(x - 2)^2 - 2$, we can directly identify the vertex of the parabola. Comparing this to the general vertex form, $f(x) = a(x - h)^2 + k$, we can see that:
- h = 2
- k = -2
Therefore, the vertex of the parabola is at the point (2, -2).
To determine whether this vertex represents a minimum or maximum point, we look at the coefficient a in the vertex form. In our case, a = 2, which is a positive value. This indicates that the parabola opens upwards. When a parabola opens upwards, the vertex represents the minimum point of the function.
Therefore, the function $f(x) = 2x^2 - 8x + 6$ has a minimum at the point (2, -2).
Conclusion
In summary, by completing the square, we successfully transformed the quadratic function $f(x) = 2x^2 - 8x + 6$ into vertex form, which allowed us to easily identify the vertex as (2, -2). Furthermore, by analyzing the coefficient of the squared term, we determined that this vertex represents a minimum point. This method of completing the square provides a powerful tool for analyzing quadratic functions and understanding their graphical representation as parabolas. The vertex is a crucial feature of a parabola, indicating its extreme point, which can be either a minimum or a maximum depending on the parabola's orientation. Understanding how to find the vertex is essential for solving optimization problems and analyzing various real-world scenarios modeled by quadratic functions.
This comprehensive approach to finding the vertex of a quadratic function by completing the square not only provides a step-by-step solution but also reinforces the underlying concepts and significance of the vertex in the context of quadratic functions and their applications. Mastering this technique is fundamental for anyone studying mathematics, engineering, or any field that utilizes mathematical modeling.
