Analyzing The Quadratic Function G(x) = -2x^2 + 20x - 53

In this in-depth discussion, we will delve into the intricacies of the quadratic function g(x) = -2x^2 + 20x - 53. Quadratic functions, characterized by their parabolic shape, play a pivotal role in various mathematical and real-world applications. Understanding their properties, such as vertex, axis of symmetry, intercepts, and concavity, is crucial for problem-solving and modeling. This article aims to provide a comprehensive analysis of the given function, covering these key aspects and illustrating their significance. We will explore how to determine the vertex form of the quadratic, which provides valuable insights into the function's behavior and characteristics. Furthermore, we will discuss the process of finding the roots (if any) and the intercepts of the parabola, offering a complete picture of the function's graphical representation. Through detailed explanations and examples, we will unravel the properties of g(x), empowering you to tackle similar quadratic functions with confidence.

Before diving into the specifics of g(x) = -2x^2 + 20x - 53, let's establish a firm foundation in the general characteristics of quadratic functions. A quadratic function is defined as a polynomial function of degree two, generally expressed in the standard 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 symmetrical U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. The vertex, the turning point of the parabola, is a crucial feature, representing the minimum value of the function when a > 0 and the maximum value when a < 0. The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into two symmetrical halves. The x-intercepts, also known as roots or zeros, are the points where the parabola intersects the x-axis, representing the solutions to the equation f(x) = 0. The y-intercept is the point where the parabola intersects the y-axis, obtained by setting x = 0 in the function. Understanding these key components allows us to effectively analyze and interpret the behavior of any quadratic function. By recognizing the relationship between the coefficients and the parabola's characteristics, we can predict its shape, position, and overall behavior. This knowledge is essential for solving quadratic equations, optimization problems, and modeling real-world phenomena that exhibit parabolic patterns.

Now, let's focus specifically on the given quadratic function, g(x) = -2x^2 + 20x - 53. Our goal is to dissect this function and uncover its key properties. First, we observe that the coefficient of the x^2 term, a, is -2. This immediately tells us that the parabola opens downwards, indicating a maximum value. The negative sign signifies a concave-down shape, which is crucial for understanding the function's overall behavior. Next, we identify the other coefficients: b = 20 and c = -53. These coefficients, along with a, play a significant role in determining the vertex, axis of symmetry, and intercepts of the parabola. To find the vertex, we can use the formula x = -b / 2a to find the x-coordinate of the vertex. Plugging in the values, we get x = -20 / (2 * -2) = 5. This means the axis of symmetry is the vertical line x = 5. To find the y-coordinate of the vertex, we substitute x = 5 back into the function: g(5) = -2(5)^2 + 20(5) - 53 = -50 + 100 - 53 = -3. Therefore, the vertex of the parabola is at the point (5, -3). This vertex represents the maximum point of the function, as the parabola opens downwards. By understanding the vertex and the direction of opening, we gain a significant understanding of the function's behavior. Further analysis will involve finding the intercepts and exploring the vertex form of the equation, providing a complete picture of the function's graphical representation.

The vertex of a parabola is a pivotal point, representing either the maximum or minimum value of the quadratic function. For the function g(x) = -2x^2 + 20x - 53, determining the vertex is a crucial step in understanding its behavior. As mentioned earlier, the x-coordinate of the vertex can be found using the formula x = -b / 2a. In this case, a = -2 and b = 20, so x = -20 / (2 * -2) = 5. This indicates that the axis of symmetry is the vertical line x = 5. To find the y-coordinate of the vertex, we substitute this x-value back into the original function: g(5) = -2(5)^2 + 20(5) - 53. Evaluating this expression, we get g(5) = -2(25) + 100 - 53 = -50 + 100 - 53 = -3. Thus, the vertex of the parabola is located at the point (5, -3). Since the coefficient of the x^2 term is negative, the parabola opens downwards, and the vertex represents the maximum value of the function. The vertex form of a quadratic function, f(x) = a(x - h)^2 + k, where (h, k) is the vertex, provides a convenient way to visualize and analyze the parabola. By completing the square, we can rewrite g(x) in vertex form, further solidifying our understanding of its characteristics. This process not only helps in identifying the vertex but also provides insights into the transformations applied to the basic parabola y = x^2, such as stretches, reflections, and translations. The vertex form is a powerful tool for graphing and analyzing quadratic functions, making it an essential concept in algebra.

Converting a quadratic function from standard form to vertex form is a valuable technique for understanding its graphical representation and properties. The vertex form, f(x) = a(x - h)^2 + k, directly reveals the vertex (h, k) and the direction of opening, which is determined by the coefficient a. To convert g(x) = -2x^2 + 20x - 53 to vertex form, we use the method of completing the square. First, factor out the coefficient of the x^2 term, which is -2, from the first two terms: g(x) = -2(x^2 - 10x) - 53. Next, complete the square inside the parentheses. Take half of the coefficient of the x term (-10), which is -5, and square it (-5)^2 = 25. Add and subtract this value inside the parentheses: g(x) = -2(x^2 - 10x + 25 - 25) - 53. Now, rewrite the expression inside the parentheses as a squared term: g(x) = -2((x - 5)^2 - 25) - 53. Distribute the -2: g(x) = -2(x - 5)^2 + 50 - 53. Finally, simplify the constant terms: g(x) = -2(x - 5)^2 - 3. Now we have the function in vertex form. From this form, we can clearly see that the vertex is (5, -3), which confirms our earlier calculation. The coefficient a = -2 indicates that the parabola opens downwards and is vertically stretched by a factor of 2. Converting to vertex form provides a clear picture of the transformations applied to the basic parabola y = x^2, allowing for easy graphing and analysis. This technique is essential for solving optimization problems and understanding the maximum or minimum values of quadratic functions.

Intercepts are the points where the parabola intersects the coordinate axes, providing valuable information about the function's behavior and position. To find the y-intercept of g(x) = -2x^2 + 20x - 53, we set x = 0 and evaluate the function: g(0) = -2(0)^2 + 20(0) - 53 = -53. Therefore, the y-intercept is the point (0, -53). This point indicates where the parabola crosses the y-axis. To find the x-intercepts, also known as roots or zeros, we set g(x) = 0 and solve for x: -2x^2 + 20x - 53 = 0. This is a quadratic equation that can be solved using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = -2, b = 20, and c = -53. Plugging these values into the quadratic formula, we get: x = (-20 ± √(20^2 - 4(-2)(-53))) / (2(-2)). Simplifying the expression under the square root: 20^2 - 4(-2)(-53) = 400 - 424 = -24. Since the discriminant (the value under the square root) is negative, there are no real roots. This means that the parabola does not intersect the x-axis. The absence of real roots indicates that the entire parabola lies below the x-axis, which is consistent with the fact that it opens downwards and has a vertex at (5, -3). Understanding the intercepts, or lack thereof, provides crucial information about the function's graph and its relationship to the coordinate axes. This analysis, combined with the vertex and direction of opening, gives a complete picture of the quadratic function's behavior.

The discriminant, denoted as Δ, is a critical component of the quadratic formula that provides valuable information about the nature of the roots of a quadratic equation. For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant is given by Δ = b^2 - 4ac. The discriminant determines whether the quadratic equation has two distinct real roots, one real root (a repeated root), or no real roots (complex roots). If Δ > 0, the equation has two distinct real roots, meaning the parabola intersects the x-axis at two different points. If Δ = 0, the equation has one real root (a repeated root), meaning the vertex of the parabola lies on the x-axis. If Δ < 0, the equation has no real roots, meaning the parabola does not intersect the x-axis. For the function g(x) = -2x^2 + 20x - 53, we found that the discriminant is 20^2 - 4(-2)(-53) = 400 - 424 = -24. Since Δ < 0, we concluded that the equation -2x^2 + 20x - 53 = 0 has no real roots. This confirms that the parabola does not intersect the x-axis. The discriminant is a powerful tool for quickly determining the number and nature of the roots of a quadratic equation without actually solving the equation. It provides a shortcut for understanding the relationship between the parabola and the x-axis, which is essential for graphing and analyzing quadratic functions. Understanding the discriminant is crucial for solving quadratic equations and interpreting their solutions in various mathematical and real-world contexts.

Graphing a quadratic function provides a visual representation of its behavior and characteristics. To graph g(x) = -2x^2 + 20x - 53, we can utilize the information we have gathered about its vertex, axis of symmetry, and intercepts. We know that the vertex is at the point (5, -3), which represents the maximum point of the parabola since it opens downwards. The axis of symmetry is the vertical line x = 5, which divides the parabola into two symmetrical halves. We also found that the y-intercept is (0, -53) and that there are no x-intercepts. To sketch the graph, start by plotting the vertex (5, -3). Draw the axis of symmetry x = 5 as a dashed line. Plot the y-intercept (0, -53). Since the parabola is symmetrical about the axis of symmetry, we can find another point on the parabola by reflecting the y-intercept across the axis of symmetry. The point symmetrical to (0, -53) about x = 5 is (10, -53). Now, we have three points: the vertex (5, -3), the y-intercept (0, -53), and the symmetrical point (10, -53). Sketch a smooth, downward-opening curve that passes through these points. The parabola should be symmetrical about the axis of symmetry and should not intersect the x-axis, as we determined earlier. Graphing provides a visual confirmation of our analysis and helps in understanding the function's behavior over its entire domain. It allows us to see the maximum value, the symmetry, and the overall shape of the parabola, making it an invaluable tool for analyzing quadratic functions. A well-drawn graph enhances our understanding and facilitates problem-solving in various contexts.

Quadratic functions are not just abstract mathematical concepts; they have numerous real-world applications in various fields. Their parabolic shape makes them ideal for modeling phenomena such as projectile motion, the trajectory of a ball thrown in the air, or the path of a satellite orbiting the Earth. The maximum or minimum value of a quadratic function can be used to solve optimization problems, such as determining the maximum height reached by a projectile or finding the dimensions of a rectangular garden that maximize the enclosed area for a given perimeter. In physics, quadratic functions are used to describe the relationship between distance, time, and acceleration in uniformly accelerated motion. The equation of a parabola can represent the shape of a satellite dish or a suspension bridge cable. In economics, quadratic functions can model cost, revenue, and profit functions, allowing businesses to analyze their financial performance and make informed decisions. For example, a quadratic cost function can help determine the level of production that minimizes costs, while a quadratic revenue function can help find the price that maximizes revenue. In engineering, quadratic functions are used in the design of lenses, reflectors, and other optical devices. The reflective property of parabolas is used in spotlights and telescopes to focus light or radio waves. Understanding quadratic functions and their properties is essential for solving a wide range of practical problems in science, engineering, economics, and other disciplines. Their versatility and wide-ranging applications make them a fundamental concept in mathematics and a valuable tool for problem-solving.

In this comprehensive analysis, we have explored the quadratic function g(x) = -2x^2 + 20x - 53 in detail. We began by understanding the general properties of quadratic functions, including their standard form, parabolic shape, and key characteristics such as vertex, axis of symmetry, and intercepts. We then delved into the specifics of g(x), identifying the coefficients and their implications for the parabola's direction of opening. We calculated the vertex using the formula x = -b / 2a and substituted this value back into the function to find the y-coordinate. We converted the function to vertex form by completing the square, which directly revealed the vertex and the transformations applied to the basic parabola. We found the y-intercept by setting x = 0 and attempted to find the x-intercepts by setting g(x) = 0 and using the quadratic formula. The negative discriminant indicated that there are no real roots, meaning the parabola does not intersect the x-axis. We discussed the significance of the discriminant in determining the nature of the roots. We then sketched the graph of the function, utilizing the vertex, axis of symmetry, and y-intercept. Finally, we highlighted the numerous real-world applications of quadratic functions in various fields. Through this detailed analysis, we have gained a thorough understanding of g(x) = -2x^2 + 20x - 53 and its graphical representation. This knowledge empowers us to tackle similar quadratic functions with confidence and apply them to solve a wide range of problems. The ability to analyze and interpret quadratic functions is a valuable skill in mathematics and various other disciplines, making this a crucial topic to master.