Inequality In Vertex Form For Region Less Than Quadratic Function

In this article, we will delve into the process of identifying the inequality in vertex form that accurately represents a region less than a given quadratic function. This is a fundamental concept in algebra and is crucial for understanding the behavior of quadratic functions and their graphical representations. We will tackle the problem step by step, focusing on the key elements such as the vertex form of a quadratic equation, the significance of the vertex, and how to determine the inequality that defines the region below the curve. By the end of this exploration, you'll have a clear understanding of how to solve this type of problem and be able to apply the same principles to similar scenarios. Understanding quadratic inequalities is essential not only for academic success but also for practical applications in fields like physics, engineering, and economics, where quadratic models are frequently used to describe real-world phenomena. So, let's embark on this mathematical journey and unravel the intricacies of quadratic inequalities.

The vertex form of a quadratic function is expressed as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is a critical point, as it signifies either the minimum or maximum value of the function. The coefficient a determines the direction the parabola opens: if a > 0, the parabola opens upwards, indicating a minimum value at the vertex; if a < 0, the parabola opens downwards, indicating a maximum value at the vertex. The vertex form provides a direct way to identify the vertex, which is essential for sketching the graph of the parabola and understanding its behavior. To fully grasp the concept, let's consider a scenario where we have a vertex at point (2, 3) and a coefficient a = 1. The equation in vertex form would be y = 1(x - 2)² + 3. This equation immediately tells us that the parabola has its vertex at (2, 3) and opens upwards. By manipulating the values of a, h, and k, we can change the shape and position of the parabola, allowing us to model a variety of real-world scenarios. For instance, in projectile motion, the vertex represents the maximum height reached by the projectile, and the equation can help us determine the trajectory and range of the object. Therefore, mastering the vertex form is crucial for both theoretical understanding and practical applications of quadratic functions.

To determine the inequality representing the region less than the quadratic function, we must consider the direction the parabola opens and the location of the region relative to the curve. If the parabola opens upwards, the region less than the function lies below the curve. Conversely, if the parabola opens downwards, the region less than the function lies above the curve. The inequality symbol plays a crucial role in defining this region. The symbol “<” indicates that the region includes all points strictly below the curve, while the symbol “≤” includes the points on the curve itself. Similarly, “>” indicates points strictly above the curve, and “≥” includes the points on the curve. To illustrate, consider a parabola opening upwards with the inequality y < a(x - h)² + k. This inequality represents the region below the parabola, excluding the parabola itself. If the inequality were y ≤ a(x - h)² + k, the region would include the parabola. Therefore, to accurately represent the region less than a quadratic function, we must carefully select the appropriate inequality symbol based on whether the boundary (the parabola) is included or excluded. In practical applications, this distinction is vital. For example, in optimization problems, we might need to find the maximum or minimum value of a function within a certain region. The correct inequality will define the feasible region, and the solution must satisfy this condition. Thus, understanding the relationship between the inequality symbol and the region it represents is fundamental to solving real-world problems involving quadratic functions.

Now, let's apply the given information: the vertex (8, 24) and the point (3, 99), to find the specific inequality. The vertex form of the quadratic function is y = a(x - h)² + k. We know the vertex (h, k) is (8, 24), so we can substitute these values into the equation: y = a(x - 8)² + 24. Next, we use the point (3, 99) to solve for the coefficient a. Substituting x = 3 and y = 99, we get: 99 = a(3 - 8)² + 24. Simplifying this equation, we have 99 = a(-5)² + 24, which becomes 99 = 25a + 24. Subtracting 24 from both sides, we get 75 = 25a. Dividing by 25, we find a = 3. So, the quadratic function is y = 3(x - 8)² + 24. Since we are looking for the region less than this function, the inequality will be of the form y < 3(x - 8)² + 24. This inequality represents all the points below the parabola defined by the function. In real-world contexts, this could represent a variety of scenarios. For instance, if the quadratic function models the height of a projectile, the inequality could represent the region where the projectile is below a certain height at a given time. Similarly, in business applications, the quadratic function might model cost or revenue, and the inequality could represent the region where cost is below a certain level or revenue is above a certain threshold. Therefore, finding the correct inequality is crucial for making informed decisions and solving practical problems.

With the quadratic function y = 3(x - 8)² + 24 and the condition that we are looking for the region less than the function, we can now analyze the provided options to select the correct answer. Option A, y < 1/3(x + 8)² + 24, has a coefficient of 1/3 and a vertex at (-8, 24), which does not match our calculated function. Option B, y < 3(x + 8)² + 24, has the correct coefficient of 3, but the vertex is at (-8, 24), which is also incorrect. Option C, y < 1/3(x - 8)² + 24, has the correct vertex at (8, 24) but an incorrect coefficient of 1/3. Option D, y < 3(x - 8)² + 24, has both the correct coefficient of 3 and the correct vertex at (8, 24). Therefore, Option D is the correct answer. This meticulous process of elimination ensures that we have not only found an answer that seems plausible but have also verified that it aligns perfectly with the given conditions. In mathematical problem-solving, this level of rigor is essential for achieving accuracy and confidence in our solutions. Moreover, the ability to analyze options and eliminate incorrect ones is a valuable skill that extends beyond mathematics. It is a critical thinking skill that is applicable in various fields and everyday situations, where we are often faced with multiple choices and need to make informed decisions based on available information. Therefore, practicing this skill in the context of mathematical problems helps develop a broader analytical mindset.

In conclusion, the inequality in vertex form that represents the region less than the quadratic function with vertex (8, 24) and the point (3, 99) on the boundary is y < 3(x - 8)² + 24. This solution was derived by first understanding the vertex form of a quadratic equation, then substituting the given vertex and point to find the specific coefficients, and finally, selecting the appropriate inequality symbol to represent the region below the curve. Throughout this process, we emphasized the importance of each step, from understanding the significance of the vertex form to meticulously analyzing the options and eliminating incorrect ones. This approach ensures not only that we arrive at the correct answer but also that we develop a deeper understanding of the underlying mathematical concepts. The ability to solve this type of problem is crucial for various applications in mathematics, science, and engineering, where quadratic functions are used to model a wide range of phenomena. Furthermore, the problem-solving skills honed in this exercise, such as logical reasoning, attention to detail, and the ability to analyze and synthesize information, are valuable assets in any field. Therefore, mastering the concepts and techniques presented in this article is an investment in both mathematical proficiency and broader problem-solving capabilities.