Comparing Minimum Values Of Quadratic Functions F And G

In the realm of mathematics, quadratic functions hold a significant position due to their widespread applications in various fields, ranging from physics to engineering. Understanding the properties and behavior of quadratic functions is crucial for solving real-world problems and making informed decisions. This article delves into the comparison of two quadratic functions, f and g, to determine the truthfulness of a given statement. We will analyze their vertices, direction of opening, and minimum values to arrive at a conclusive answer. The ability to analyze and compare quadratic functions is a fundamental skill in algebra and calculus, and this article aims to provide a comprehensive understanding of these concepts.

A quadratic function is a polynomial function of degree two, generally expressed in the form f(x) = ax² + 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 can open upwards or downwards depending on the sign of the coefficient a. If a > 0, the parabola opens upwards, indicating a minimum value. Conversely, if a < 0, the parabola opens downwards, indicating a maximum value. The vertex of the parabola is the point where the function attains its minimum or maximum value, and it plays a crucial role in understanding the function's behavior.

Understanding the vertex form of a quadratic equation is essential for quickly identifying the vertex and the direction in which the parabola opens. The vertex form is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The value of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). The vertex form provides a clear and concise way to analyze the key features of a quadratic function, making it a valuable tool in mathematical problem-solving. By understanding the relationship between the coefficients and the graph, we can easily determine the minimum or maximum value of the function and its location.

The quadratic function f is characterized by its vertex at (3, 4) and its upward opening. This information provides us with valuable insights into the function's behavior. Since the parabola opens upwards, the vertex represents the minimum point of the function. Therefore, the minimum value of f is 4, which occurs at x = 3. The upward opening of the parabola also implies that the coefficient of the x² term is positive. While we don't have the explicit equation for f, we can infer its general form as f(x) = a(x - 3)² + 4, where a > 0. The value of a determines the steepness of the parabola, but the minimum value remains at 4.

The characteristics of function f are crucial for comparison with function g. Knowing that the vertex is at (3, 4) and the parabola opens upwards allows us to establish a baseline for the function's behavior. The minimum value of 4 is a key piece of information that will be used to compare with the minimum value of function g. The upward opening also confirms that the function will increase as x moves away from the vertex in either direction. This understanding of function f lays the foundation for a comprehensive comparison with function g. By analyzing the given information, we can deduce important properties of the function without having its explicit equation.

The quadratic function g is given by the equation g(x) = 2(x - 4)² + 3. This equation is in vertex form, which makes it easy to identify the vertex and the direction of opening. The vertex of g is (4, 3), and since the coefficient of the (x - 4)² term is 2, which is positive, the parabola opens upwards. This indicates that the vertex represents the minimum point of the function. Therefore, the minimum value of g is 3, which occurs at x = 4. The vertex form of the equation provides a clear and concise representation of the function's key features.

Function g's equation g(x) = 2(x - 4)² + 3 provides a wealth of information about its behavior. The coefficient 2 indicates that the parabola is steeper than the standard parabola y = x². The vertex (4, 3) represents the lowest point on the graph, and the upward opening confirms that the function will increase as x moves away from 4 in either direction. The minimum value of 3 is a crucial characteristic that will be compared with the minimum value of function f. By analyzing the equation, we can gain a complete understanding of the function's properties and behavior. This analysis is essential for comparing and contrasting the two quadratic functions.

To determine which statement is true, we need to compare the minimum values of the quadratic functions f and g. We established that the minimum value of f is 4, and the minimum value of g is 3. Therefore, the minimum value of f is greater than the minimum value of g. This comparison is crucial for evaluating the truthfulness of the given statement. The difference in minimum values arises from the different vertices and coefficients of the two quadratic functions.

The comparison of minimum values is the key to answering the question. The minimum value of f being 4 and the minimum value of g being 3 clearly demonstrates that f has a higher minimum value. This seemingly simple comparison is a fundamental concept in understanding the behavior of quadratic functions. By identifying and comparing the minimum values, we can draw conclusions about the relative positions and characteristics of the two parabolas. This comparison is a direct application of the understanding of vertex form and the concept of minimum values in quadratic functions.

Based on our analysis, the minimum value of f is 4, and the minimum value of g is 3. Therefore, the statement