Convert G(x) = 40x + 4x^2 To Vertex Form A Step-by-Step Guide

In the realm of quadratic functions, understanding the vertex form is crucial for unlocking key insights into the function's behavior and graphical representation. This article delves into the process of converting the quadratic function g(x) = 40x + 4x^2 into its vertex form. We will break down each step, providing clear explanations and mathematical justifications along the way. Furthermore, we'll explore the significance of the vertex form and its applications in analyzing quadratic functions. So, if you're looking to master the art of vertex form conversion, you've come to the right place. This comprehensive guide will equip you with the knowledge and skills to tackle any quadratic function and express it in its vertex form, enabling you to easily identify the vertex, axis of symmetry, and other important features of the parabola. Whether you're a student learning about quadratic functions for the first time or a seasoned mathematician looking for a refresher, this article will serve as your go-to resource for understanding and applying the vertex form.

Before we dive into the conversion process, let's establish a solid foundation by understanding quadratic functions and the vertex form. A quadratic function is a polynomial function of degree two, generally expressed in the standard form as 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 (if a > 0) or downwards (if a < 0). The vertex form of a quadratic function provides a unique perspective by highlighting the vertex of the parabola, which is the point where the function reaches its minimum or maximum value. The vertex form is expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. The value of 'a' remains the same as in the standard form and determines the direction and steepness of the parabola. The vertex form is incredibly useful for quickly identifying the vertex, axis of symmetry (x = h), and the maximum or minimum value of the function (k). By understanding the relationship between the standard form and the vertex form, we can gain a deeper understanding of the function's properties and behavior. This transformation allows us to easily visualize the graph of the quadratic function and make informed predictions about its behavior. Furthermore, the vertex form simplifies many calculations and problem-solving tasks related to quadratic functions, making it an essential tool in various mathematical and real-world applications. The ability to seamlessly convert between standard form and vertex form empowers us to analyze and interpret quadratic functions with greater ease and efficiency.

The initial step in converting the given function, g(x) = 40x + 4x^2, into vertex form involves rearranging it into the standard form of a quadratic function. As mentioned earlier, the standard form is expressed as f(x) = ax^2 + bx + c. This rearrangement is crucial because it allows us to clearly identify the coefficients a, b, and c, which are essential for the subsequent steps in the conversion process. In our case, the function g(x) = 40x + 4x^2 is not yet in standard form because the terms are not arranged in descending order of their exponents. To rewrite it in standard form, we simply swap the positions of the terms, placing the term with x^2 first, followed by the term with x, and finally the constant term (which is 0 in this case). Therefore, rewriting g(x) = 40x + 4x^2 in standard form gives us g(x) = 4x^2 + 40x. Now, we can easily identify the coefficients: a = 4, b = 40, and c = 0. This step may seem straightforward, but it's a fundamental prerequisite for the following steps. A clear understanding of the standard form and the ability to quickly identify the coefficients are essential for mastering the vertex form conversion. By ensuring that the function is in standard form, we set the stage for a smooth and accurate transformation into vertex form. This methodical approach ensures that we don't miss any crucial details and that we're well-prepared for the subsequent steps in the process.

Having rewritten the function in standard form, g(x) = 4x^2 + 40x, the next crucial step is to factor out the leading coefficient, which is the coefficient of the x^2 term. In this case, the leading coefficient is 4. Factoring out the leading coefficient is a key step in completing the square, which is the core technique used to convert a quadratic function into vertex form. This process isolates the x^2 term within the parentheses, making it easier to manipulate and transform into a perfect square trinomial. To factor out 4 from the first two terms, we divide each of them by 4. This gives us: g(x) = 4(x^2 + 10x). Notice that we only factor out the coefficient from the x^2 and x terms; the constant term (if there were one) would remain outside the parentheses. This step is essential because it sets up the expression for the next stage, where we will complete the square. Factoring out the leading coefficient ensures that the coefficient of the x^2 term inside the parentheses is 1, which is a requirement for completing the square. By carefully factoring out the leading coefficient, we simplify the expression and pave the way for the subsequent steps in the conversion process. This meticulous approach is crucial for achieving an accurate and efficient transformation into vertex form, allowing us to easily identify the vertex and other key features of the parabola.

The heart of the vertex form conversion lies in the technique of completing the square. This algebraic manipulation transforms the quadratic expression inside the parentheses into a perfect square trinomial, which can then be easily expressed as a squared binomial. A perfect square trinomial is a trinomial that can be factored into the form (x + a)^2 or (x - a)^2. To complete the square for the expression x^2 + 10x inside the parentheses, we need to add and subtract a specific constant term. This constant is determined by taking half of the coefficient of the x term (which is 10), squaring it, and adding and subtracting the result inside the parentheses. The coefficient of the x term is 10, so half of it is 10/2 = 5, and squaring it gives us 5^2 = 25. Therefore, we add and subtract 25 inside the parentheses: g(x) = 4(x^2 + 10x + 25 - 25). Now, the first three terms inside the parentheses, x^2 + 10x + 25, form a perfect square trinomial. This trinomial can be factored as (x + 5)^2. So, we can rewrite the expression as: g(x) = 4((x + 5)^2 - 25). Completing the square is a powerful technique that allows us to rewrite a quadratic expression in a form that reveals the vertex of the parabola. By adding and subtracting the appropriate constant, we create a perfect square trinomial, which is the key to expressing the quadratic function in vertex form. This step requires careful attention to detail, but it is the cornerstone of the vertex form conversion process. Mastering this technique will enable you to confidently transform any quadratic function into its vertex form, unlocking its hidden properties and graphical representation.

With the square completed, the final step involves simplifying the expression and writing it in the vertex form, g(x) = a(x - h)^2 + k. We left off with the expression g(x) = 4((x + 5)^2 - 25). To simplify this, we need to distribute the 4 to both terms inside the outer parentheses. This gives us: g(x) = 4(x + 5)^2 - 4 * 25, which simplifies to g(x) = 4(x + 5)^2 - 100. Now, the function is in vertex form. By comparing this to the general vertex form, g(x) = a(x - h)^2 + k, we can identify the values of a, h, and k. In this case, a = 4, h = -5 (note the sign change because of the subtraction in the formula), and k = -100. Therefore, the vertex of the parabola is at the point (h, k) = (-5, -100). The vertex form provides a clear and concise representation of the quadratic function, highlighting the vertex and the vertical stretch or compression factor (a). The vertex is a crucial point on the parabola, representing either the minimum or maximum value of the function. By expressing the function in vertex form, we can easily determine the vertex and other key characteristics of the parabola, such as the axis of symmetry (x = h) and the direction of opening (upward if a > 0, downward if a < 0). This final step of simplification and writing in vertex form is the culmination of the entire conversion process, providing a powerful tool for analyzing and understanding quadratic functions.

In this comprehensive guide, we've walked through the step-by-step process of converting the quadratic function g(x) = 40x + 4x^2 into its vertex form. We began by understanding the importance of vertex form and its relationship to the standard form of a quadratic function. We then meticulously executed each step, including rewriting the function in standard form, factoring out the leading coefficient, completing the square, and finally, simplifying and expressing the function in vertex form. Through this process, we arrived at the vertex form: g(x) = 4(x + 5)^2 - 100, which revealed the vertex of the parabola to be at the point (-5, -100). The ability to convert quadratic functions into vertex form is a valuable skill in mathematics, providing a powerful tool for analyzing and understanding the behavior of parabolas. The vertex form allows us to quickly identify the vertex, axis of symmetry, and the maximum or minimum value of the function. By mastering this conversion process, you gain a deeper understanding of quadratic functions and their graphical representations. This knowledge is not only essential for academic success but also for various real-world applications, such as optimization problems and modeling physical phenomena. So, continue to practice and refine your skills in vertex form conversion, and you'll be well-equipped to tackle any quadratic function that comes your way. The journey of understanding quadratic functions is a rewarding one, and the vertex form is a key milestone in that journey.