Analyzing Quadratic Function F(x) = 9x^2 - 9x Minimum Maximum Domain And Range

In the realm of mathematics, quadratic functions hold a prominent position due to their versatile nature and wide-ranging applications. These functions, characterized by their parabolic curves, often represent real-world phenomena such as projectile motion, optimization problems, and various engineering designs. In this article, we delve into the analysis of a specific quadratic function, f(x) = 9x² - 9x, with the aim of uncovering its key characteristics, including whether it possesses a minimum or maximum value, the location of this extreme value, and its domain and range. Our exploration will be entirely analytical, relying on mathematical principles rather than graphical representations.

The analysis of quadratic functions is a cornerstone of mathematical education, providing students with essential tools for problem-solving and critical thinking. Understanding the behavior of these functions, including their minimum or maximum values, allows us to model and optimize various real-world scenarios. From determining the trajectory of a projectile to maximizing profit in business, quadratic functions offer a powerful framework for analysis and decision-making.

This article will serve as a comprehensive guide to understanding the properties of the given quadratic function. We will embark on a step-by-step journey, employing established mathematical techniques to dissect the function and reveal its underlying behavior. Our focus will be on clarity and conciseness, ensuring that readers of all backgrounds can grasp the concepts and appreciate the elegance of mathematical analysis.

Our initial task is to ascertain whether the function f(x) = 9x² - 9x has a minimum or maximum value, without resorting to graphical methods. To accomplish this, we will leverage the properties of quadratic functions and their coefficients. The key lies in the coefficient of the x² term. In our case, this coefficient is 9, which is a positive number.

Recall that a quadratic function in the general form f(x) = ax² + bx + c exhibits different behaviors depending on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, resembling a 'U' shape. This indicates that the function has a minimum value at its vertex. Conversely, if 'a' is negative, the parabola opens downwards, resembling an inverted 'U' shape, implying that the function has a maximum value at its vertex.

Since the coefficient of our x² term, 9, is positive, we can confidently conclude that the parabola represented by the function f(x) = 9x² - 9x opens upwards. This implies that the function has a minimum value. This minimum value corresponds to the y-coordinate of the vertex of the parabola.

To further solidify our understanding, let's consider the intuitive explanation behind this behavior. When the coefficient of x² is positive, as x moves away from the vertex in either direction (towards positive or negative infinity), the x² term dominates the function's value. Since x² is always non-negative, the function's value increases as x moves away from the vertex, creating the characteristic upward-opening parabolic shape. Conversely, when the coefficient of x² is negative, the x² term makes the function's value decrease as x moves away from the vertex, resulting in a downward-opening parabola.

Therefore, by examining the coefficient of the x² term, we have successfully determined that the function f(x) = 9x² - 9x has a minimum value, without the need for graphical analysis. This simple yet powerful technique allows us to quickly ascertain the fundamental behavior of a quadratic function.

Having established that the function f(x) = 9x² - 9x has a minimum value, our next objective is to pinpoint this minimum value and determine the x-coordinate at which it occurs. This involves finding the vertex of the parabola represented by the function. The vertex is a crucial point on the parabola, as it represents either the minimum or maximum value of the function, depending on the parabola's orientation.

To find the vertex, we can employ a couple of methods. One common approach involves completing the square. However, for this particular function, a more direct method is to use the vertex formula. The vertex formula provides a straightforward way to calculate the x-coordinate of the vertex, which we'll denote as _x_v. The formula is given by:

_x_v = -b / 2a

where 'a' and 'b' are the coefficients of the x² and x terms, respectively, in the quadratic function f(x) = ax² + bx + c. In our case, a = 9 and b = -9. Plugging these values into the formula, we get:

x_v = -(-9) / (2 * 9) = 9 / 18 = 1/2

Thus, the x-coordinate of the vertex is 1/2. This tells us that the minimum value of the function occurs at x = 1/2.

Now, to find the minimum value itself, we need to evaluate the function at _x_v = 1/2. This means substituting 1/2 for x in the function f(x) = 9x² - 9x:

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 is -9/4. This is the y-coordinate of the vertex, representing the lowest point on the parabola.

In summary, we have determined that the minimum value of the function f(x) = 9x² - 9x is -9/4, and it occurs at x = 1/2. This information provides a complete picture of the function's behavior around its extreme point. The vertex (1/2, -9/4) serves as a critical reference point for understanding the function's graph and its overall characteristics.

To fully characterize the function f(x) = 9x² - 9x, we need to identify its domain and range. The domain of a function represents the set of all possible input values (x-values) for which the function is defined. The range, on the other hand, represents the set of all possible output values (y-values) that the function can produce.

For polynomial functions, such as quadratic functions, the domain is typically all real numbers. This means that we can input any real number into the function, and it will produce a valid output. There are no restrictions on the values of x that we can use. Therefore, the domain of f(x) = 9x² - 9x is all real numbers, which can be expressed in interval notation as (-∞, ∞).

Determining the range requires a bit more consideration. Recall that we have already established that the function has a minimum value of -9/4. Since the parabola opens upwards (due to the positive coefficient of the x² term), the function's values will extend upwards from this minimum value. In other words, the function can take on any value greater than or equal to -9/4.

Therefore, the range of f(x) = 9x² - 9x is all real numbers greater than or equal to -9/4. In interval notation, this can be expressed as [-9/4, ∞). The square bracket indicates that -9/4 is included in the range, as it is the minimum value of the function.

In summary, the domain of the function f(x) = 9x² - 9x is all real numbers (-∞, ∞), and its range is [-9/4, ∞). These characteristics provide a complete description of the set of possible input and output values for the function. Understanding the domain and range is crucial for interpreting the function's behavior and its applications in various contexts.

In this comprehensive analysis, we have successfully explored the key characteristics of the quadratic function f(x) = 9x² - 9x. We began by determining, without graphing, that the function has a minimum value due to the positive coefficient of the x² term. This fundamental property dictates the upward-opening nature of the parabola represented by the function.

Next, we calculated the minimum value and its location using the vertex formula. We found that the minimum value is -9/4, and it occurs at x = 1/2. This precise determination of the vertex provides a crucial reference point for understanding the function's behavior and its graphical representation.

Finally, we identified the function's domain and range. The domain, encompassing all possible input values, is all real numbers (-∞, ∞). The range, representing all possible output values, is [-9/4, ∞), reflecting the function's minimum value and its upward-extending nature.

This analysis demonstrates the power of mathematical techniques in dissecting and understanding the behavior of quadratic functions. By employing established principles and formulas, we have gained a complete picture of the function f(x) = 9x² - 9x, including its minimum value, its location, and its domain and range. This knowledge is invaluable for various applications, from optimizing real-world scenarios to modeling physical phenomena.

The study of quadratic functions is a cornerstone of mathematical education, providing students with essential tools for problem-solving and critical thinking. By mastering the techniques presented in this article, readers can confidently analyze and interpret quadratic functions, unlocking their potential for diverse applications.

This exploration serves as a testament to the elegance and power of mathematics in revealing the hidden properties of functions and their role in shaping our understanding of the world around us.