Analyzing Quadratic Function F(x) = 9x² - 9x For Minimum Value, Domain And Range

Introduction

In this article, we delve into the analysis of the quadratic function f(x) = 9x² - 9x. Our primary goal is to understand the behavior of this function, specifically focusing on identifying whether it has a minimum or maximum value and determining the location and magnitude of this extremum. Additionally, we will explore the function's domain and range, providing a comprehensive overview of its properties. Quadratic functions, characterized by their parabolic shapes, are fundamental in mathematics and have wide-ranging applications in physics, engineering, and economics. Understanding their properties, such as finding minima and maxima, is crucial for solving optimization problems and modeling real-world phenomena.

a. Determining Minimum or Maximum Value Without Graphing

To determine whether the quadratic function f(x) = 9x² - 9x has a minimum or maximum value without resorting to graphing, we need to analyze the coefficient of the x² term. In this case, the coefficient is 9, which is a positive number. The sign of this coefficient is a key indicator of the parabola's concavity. When the coefficient of the x² term is positive, the parabola opens upwards, resembling a U-shape. This upward-opening shape implies that the function has a minimum value at its vertex. Conversely, if the coefficient were negative, the parabola would open downwards (an inverted U-shape), indicating a maximum value at the vertex. This property stems from the fundamental nature of quadratic functions; the squared term dominates the function's behavior for large values of x, either positive or negative, causing the function to increase without bound when the coefficient is positive and decrease without bound when the coefficient is negative. Understanding this connection between the coefficient of the x² term and the concavity of the parabola allows us to quickly ascertain the presence of a minimum or maximum value without the need for graphical representation. Furthermore, this analytical approach provides a more precise understanding of the function's behavior compared to visual inspection, which can be limited by the accuracy of the graph. This method is particularly valuable in applications where precise determination of extrema is essential, such as in optimization problems where even slight inaccuracies can lead to suboptimal solutions. In summary, the positive coefficient of the x² term in f(x) = 9x² - 9x definitively tells us that the function has a minimum value.

b. Finding the Minimum Value and Where It Occurs

Having established that the function f(x) = 9x² - 9x has a minimum value, we now proceed to find this minimum value and the x-coordinate at which it occurs. The minimum value of a parabola that opens upwards occurs at its vertex. There are several methods to find the vertex of a parabola. One common approach is to complete the square, which transforms the quadratic function into vertex form, f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Alternatively, we can use the formula for the x-coordinate of the vertex, which is given by x = -b / 2a, where a and b are the coefficients of the x² and x terms, respectively, in the standard form of the quadratic equation, f(x) = ax² + bx + c. In our case, a = 9 and b = -9, so applying the formula, we get x = -(-9) / (2 * 9) = 9 / 18 = 1/2. This x-coordinate represents the location of the vertex and thus where the minimum value occurs. To find the minimum value itself, we substitute this x-coordinate back into the original function: f(1/2) = 9(1/2)² - 9(1/2) = 9(1/4) - 9/2 = 9/4 - 18/4 = -9/4. Therefore, the minimum value of the function f(x) = 9x² - 9x is -9/4, and it occurs at x = 1/2. This minimum value represents the lowest point on the parabola, and its accurate determination is crucial in various applications, such as finding the optimal operating point in engineering systems or the minimum cost in economic models. The vertex form of the quadratic function provides further insights into the parabola's symmetry and its relationship to the x and y axes, allowing for a more complete understanding of its behavior. In summary, by applying the vertex formula and substituting the result back into the function, we have precisely determined the minimum value and its location for the given quadratic function.

c. Identifying the Function's Domain and Range

The domain and range are fundamental aspects of understanding any function, including the quadratic function f(x) = 9x² - 9x. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For quadratic functions, there are generally no restrictions on the input values, as we can square and multiply any real number. Therefore, the domain of f(x) = 9x² - 9x is all real numbers, which can be expressed in interval notation as (-∞, ∞). This means that we can input any real number into the function and obtain a valid output. The range, on the other hand, is the set of all possible output values (y-values) that the function can produce. Since we've already established that this parabola opens upwards and has a minimum value, the range will be bounded below by this minimum value. We calculated the minimum value to be -9/4. Therefore, the range of f(x) = 9x² - 9x includes all real numbers greater than or equal to -9/4. In interval notation, this is expressed as [-9/4, ∞). The lower bound of the range is inclusive because the function actually attains the value of -9/4 at the vertex. Understanding the domain and range provides a complete picture of the function's behavior. The domain tells us the allowable inputs, while the range tells us the possible outputs. In the context of real-world applications, the domain might represent physical limitations on the input variable, while the range might represent the possible outcomes or results of the modeled phenomenon. For example, if the quadratic function represents the height of a projectile, the domain might be restricted to non-negative time values, and the range would represent the possible heights the projectile can reach. In summary, the domain of f(x) = 9x² - 9x is all real numbers, and the range is all real numbers greater than or equal to -9/4, providing a comprehensive understanding of the function's input and output behavior.

Conclusion

In conclusion, we have thoroughly analyzed the quadratic function f(x) = 9x² - 9x. We determined that the function has a minimum value because the coefficient of the x² term is positive. We then calculated the minimum value to be -9/4, which occurs at x = 1/2. Furthermore, we identified the domain of the function as all real numbers and the range as all real numbers greater than or equal to -9/4. This comprehensive analysis provides a complete understanding of the function's behavior and properties. The techniques and concepts explored in this article are fundamental in the study of quadratic functions and have broad applications in various fields. Understanding how to find minima and maxima, as well as domain and range, is crucial for solving optimization problems, modeling real-world phenomena, and making informed decisions based on mathematical analysis. The ability to analyze quadratic functions is a valuable skill in mathematics and its applications, enabling us to solve problems ranging from simple parabolic trajectories to complex optimization scenarios.