Convert G(x) = 4x^2 + 88x To Vertex Form A Step-by-Step Guide

In the realm of quadratic functions, the vertex form stands out as a particularly insightful representation. It provides a clear picture of the function's key features, such as its vertex (the point where the parabola reaches its maximum or minimum value) and its axis of symmetry. Transforming a quadratic function into vertex form allows for easy identification of these critical characteristics, which are essential for graphing, solving optimization problems, and understanding the function's behavior. In this article, we will delve into the process of converting the quadratic function g(x) = 4x² + 88x into its vertex form. This transformation involves a technique called completing the square, a powerful algebraic method that allows us to rewrite the quadratic expression in a way that reveals the vertex directly. By mastering this process, you will gain a deeper understanding of quadratic functions and their properties.

The vertex form of a quadratic function is expressed as g(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The coefficient 'a' determines the direction and steepness of the parabola. If 'a' is positive, the parabola opens upwards, indicating a minimum value at the vertex. Conversely, if 'a' is negative, the parabola opens downwards, indicating a maximum value at the vertex. The absolute value of 'a' dictates the parabola's width; a larger absolute value corresponds to a narrower parabola, while a smaller absolute value results in a wider parabola. Understanding these parameters is crucial for analyzing and interpreting quadratic functions in various contexts. The process of converting a quadratic function from its standard form to vertex form is a fundamental skill in algebra, with applications in numerous fields, including physics, engineering, and economics.

To effectively convert the given function into vertex form, it's crucial to first grasp the essence of this form. The vertex form of a quadratic equation is expressed as g(x) = a(x - h)² + k, where: * 'a' determines the direction and steepness of the parabola. * '(h, k)' represents the vertex of the parabola, which is the point where the parabola changes direction. The h value represents the x-coordinate of the vertex, and the k value represents the y-coordinate of the vertex. * The vertex is the minimum point if a > 0 and the maximum point if a < 0. Understanding these components is essential for both converting to vertex form and interpreting the resulting equation. The vertex form provides a direct way to identify the vertex of the parabola, which is a key feature for graphing and solving optimization problems. The value of 'a' also provides information about the shape of the parabola; a larger absolute value of 'a' indicates a narrower parabola, while a smaller absolute value indicates a wider parabola.

The significance of vertex form lies in its ability to readily reveal the vertex of the parabola. In the standard form of a quadratic equation, g(x) = ax² + bx + c, the vertex is not immediately apparent. However, in vertex form, the vertex coordinates (h, k) are directly visible, making it easier to sketch the graph of the parabola and understand its behavior. The h value indicates the horizontal shift of the parabola from the origin, while the k value indicates the vertical shift. This makes vertex form particularly useful in applications where the maximum or minimum value of a quadratic function is of interest, such as in projectile motion problems or optimization scenarios. Moreover, the vertex form allows for a more intuitive understanding of how the parameters 'a', 'h', and 'k' affect the shape and position of the parabola. This understanding is crucial for manipulating and analyzing quadratic functions in various mathematical and real-world contexts.

Converting the quadratic function g(x) = 4x² + 88x into vertex form involves a systematic process known as completing the square. This method allows us to rewrite the quadratic expression in a form that explicitly reveals the vertex of the parabola. Here's a step-by-step guide to converting the given function: 1. Factor out the coefficient of the x² term: In our case, the coefficient of x² is 4. Factoring this out from the first two terms gives us: g(x) = 4(x² + 22x). This step is crucial because it ensures that the expression inside the parentheses has a leading coefficient of 1, which is necessary for completing the square. By factoring out the leading coefficient, we simplify the process of finding the constant term that will complete the square. This step also allows us to maintain the correct scaling of the parabola, as the factored-out coefficient will become the 'a' value in the vertex form.

2. Complete the square: To complete the square for the expression inside the parentheses (x² + 22x), we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is 22, so half of it is 11, and the square of 11 is 121. Therefore, we add and subtract 121 inside the parentheses: g(x) = 4(x² + 22x + 121 - 121). Completing the square is the core of the conversion process, as it allows us to rewrite the quadratic expression as a perfect square trinomial plus a constant. This transformation is based on the algebraic identity (a + b)² = a² + 2ab + b². By adding and subtracting the appropriate constant, we can create a perfect square trinomial that can be factored into the form (x + h)². This step is essential for revealing the vertex of the parabola in the vertex form.

3. Rewrite as a squared term: The first three terms inside the parentheses (x² + 22x + 121) form a perfect square trinomial, which can be rewritten as (x + 11)². So, our equation becomes: g(x) = 4((x + 11)² - 121). This step is the culmination of the completing the square process, where we successfully transform the quadratic expression into a squared term. The squared term (x + 11)² represents a parabola that has been shifted horizontally. The value inside the parentheses, 11, indicates the horizontal shift, which is crucial for determining the x-coordinate of the vertex. By rewriting the expression in this form, we are one step closer to the vertex form, which will explicitly reveal the vertex coordinates.

4. Distribute and simplify: Distribute the 4 back into the parentheses: g(x) = 4(x + 11)² - 4 * 121. Simplify the expression: g(x) = 4(x + 11)² - 484. This final step involves distributing the factored-out coefficient back into the expression and simplifying the constant term. By distributing the 4, we ensure that the equation remains equivalent to the original function. The multiplication 4 * 121 results in 484, which represents the vertical shift of the parabola. This vertical shift is directly related to the y-coordinate of the vertex. After this simplification, the equation is in its vertex form, making it easy to identify the vertex and other key features of the parabola.

Following the steps outlined above, we have successfully converted the quadratic function g(x) = 4x² + 88x into its vertex form. The final result is: g(x) = 4(x + 11)² - 484. This form provides valuable insights into the function's behavior and characteristics. The vertex form clearly reveals the vertex of the parabola, which is the point where the parabola changes direction. In this case, the vertex is at the point (-11, -484). The x-coordinate of the vertex is the value that makes the squared term zero, which is -11. The y-coordinate of the vertex is the constant term, which is -484. The vertex represents the minimum point of the parabola since the coefficient 'a' (which is 4) is positive. This means that the parabola opens upwards, and the vertex is the lowest point on the graph. Understanding the vertex is crucial for sketching the graph of the parabola and for solving optimization problems related to the function.

Furthermore, the vertex form also provides information about the shape and orientation of the parabola. The coefficient 'a' (4 in this case) determines the steepness of the parabola. Since 'a' is positive, the parabola opens upwards. The magnitude of 'a' indicates how narrow or wide the parabola is; a larger value of 'a' results in a narrower parabola, while a smaller value results in a wider parabola. In this case, a = 4 indicates a relatively narrow parabola compared to a parabola with a smaller 'a' value. The vertex form also allows for easy identification of the axis of symmetry, which is the vertical line that passes through the vertex. The equation of the axis of symmetry is x = -11, which is the x-coordinate of the vertex. The axis of symmetry divides the parabola into two symmetrical halves, making it easier to visualize and analyze the graph of the function. By understanding the information provided by the vertex form, we can gain a comprehensive understanding of the quadratic function and its properties.

In conclusion, we have successfully transformed the quadratic function g(x) = 4x² + 88x into its vertex form, which is g(x) = 4(x + 11)² - 484. This transformation, achieved through the method of completing the square, is a fundamental technique in algebra. The vertex form provides a clear and concise representation of the function's key characteristics, most notably its vertex. The vertex, in this case, is located at the point (-11, -484), which represents the minimum value of the function. This information is invaluable for various applications, including graphing the parabola, solving optimization problems, and understanding the function's behavior in different contexts. The process of converting to vertex form not only reveals the vertex but also provides insights into the shape and orientation of the parabola. The coefficient 'a' (4 in this case) indicates that the parabola opens upwards and is relatively narrow. The axis of symmetry, which passes through the vertex, is given by the equation x = -11, dividing the parabola into two symmetrical halves.

The ability to convert quadratic functions into vertex form is a crucial skill in mathematics, with applications extending beyond the classroom. In various fields, such as physics, engineering, and economics, quadratic functions are used to model real-world phenomena. Understanding the vertex form allows professionals to easily identify critical points, such as maximum or minimum values, which are essential for making informed decisions and solving practical problems. For instance, 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 economics, quadratic functions can be used to model cost and revenue curves, and the vertex represents the point of maximum profit or minimum cost. Therefore, mastering the technique of converting to vertex form is not only beneficial for academic purposes but also for real-world applications, making it a valuable asset in various professional fields. By understanding the significance of the vertex form and the process of completing the square, you can effectively analyze and interpret quadratic functions in a wide range of situations.