Analyzing The Quadratic Function F(x) = -3x^2 + 30x - 1 A Comprehensive Guide

This article delves into the characteristics of the quadratic function f(x) = -3x² + 30x - 1. We will explore its key features, including whether it has a minimum or maximum value, the location and value of this extremum, and the function's domain and range. By analyzing the function's equation, we can gain a comprehensive understanding of its behavior without relying on a graph. This approach highlights the power of algebraic techniques in revealing the properties of quadratic functions.

a. Determining the Existence of a Minimum or Maximum Value

To determine whether the quadratic function f(x) = -3x² + 30x - 1 has a minimum or maximum value, we need to examine the coefficient of the x² term. In this case, the coefficient is -3. The sign of this coefficient plays a crucial role in determining the function's concavity. A negative coefficient indicates that the parabola opens downwards, resembling an inverted U-shape. This means that the function will have a maximum value at its vertex, as the vertex represents the highest point on the graph. Conversely, if the coefficient of the x² term were positive, the parabola would open upwards, and the function would have a minimum value at its vertex.

Understanding the relationship between the coefficient of the x² term and the concavity of the parabola is fundamental to analyzing quadratic functions. The negative coefficient in our function, f(x) = -3x² + 30x - 1, immediately tells us that the parabola opens downwards, guaranteeing the existence of a maximum value. This initial observation allows us to proceed with finding the vertex, which will provide us with the location and value of this maximum. The process of identifying the concavity based on the leading coefficient is a cornerstone of quadratic function analysis, enabling us to quickly grasp the function's overall behavior and potential extrema. Furthermore, this concept extends to higher-degree polynomials, where the leading coefficient influences the end behavior of the graph. Thus, mastering this principle provides a solid foundation for understanding a broader range of functions.

b. Finding the Minimum or Maximum Value and Its Location

Since we've established that the function f(x) = -3x² + 30x - 1 has a maximum value, our next step is to find this value and determine where it occurs. The maximum value of a downward-opening parabola occurs at its vertex. There are two primary methods to find the vertex: completing the square and using the vertex formula. Let's explore both methods to solidify our understanding. Firstly, the vertex form of a quadratic equation is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. Transforming the given equation into this form involves completing the square. We start by factoring out the coefficient of the x² term (-3) from the first two terms: f(x) = -3(x² - 10x) - 1. Now, we complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (-10), square it ((-5)² = 25), and add and subtract it inside the parentheses: f(x) = -3(x² - 10x + 25 - 25) - 1. This allows us to rewrite the expression as f(x) = -3((x - 5)² - 25) - 1. Distributing the -3 and simplifying, we get f(x) = -3(x - 5)² + 75 - 1, which simplifies to f(x) = -3(x - 5)² + 74. From this vertex form, we can directly identify the vertex as (5, 74).

Alternatively, we can use the vertex formula, which provides a direct way to calculate the x-coordinate of the vertex. For a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by h = -b / 2a. In our case, a = -3 and b = 30, so h = -30 / (2 * -3) = 5. This confirms that the x-coordinate of the vertex is 5. To find the y-coordinate (the maximum value), we substitute x = 5 back into the original function: f(5) = -3(5)² + 30(5) - 1 = -75 + 150 - 1 = 74. Therefore, the vertex is indeed at (5, 74), and the maximum value of the function is 74. This maximum value occurs when x = 5. Both methods, completing the square and using the vertex formula, lead us to the same conclusion: the function f(x) = -3x² + 30x - 1 attains its maximum value of 74 at x = 5. Understanding these methods provides a robust toolkit for analyzing quadratic functions and identifying their key features.

c. Identifying the Function's Domain and Range

The domain and range are fundamental properties of any function, defining the set of possible input values (domain) and the set of possible output values (range). For a quadratic function, the domain is typically all real numbers, while the range is restricted based on whether the parabola opens upwards or downwards and the location of its vertex. Let's analyze the domain and range of f(x) = -3x² + 30x - 1. Firstly, the domain of a quadratic function is the set of all real numbers. This is because we can input any real number into the quadratic expression and obtain a valid output. There are no restrictions on the values of x that we can use. Therefore, the domain of f(x) = -3x² + 30x - 1 is (-∞, ∞). Now, let's consider the range. As we determined earlier, this function has a maximum value of 74, which occurs at the vertex (5, 74). Since the parabola opens downwards (due to the negative coefficient of the x² term), the function's values will never exceed this maximum. This means that the range consists of all real numbers less than or equal to 74. We can express this range in interval notation as (-∞, 74].

The range is directly influenced by the vertex and the direction in which the parabola opens. For a downward-opening parabola, the range extends from negative infinity up to the y-coordinate of the vertex, including the y-coordinate itself. Conversely, for an upward-opening parabola, the range extends from the y-coordinate of the vertex (the minimum value) to positive infinity. Understanding the relationship between the parabola's orientation, the vertex, and the range is crucial for completely characterizing quadratic functions. In summary, for the function f(x) = -3x² + 30x - 1, the domain is all real numbers, and the range is (-∞, 74]. This comprehensive analysis of the function's domain and range, combined with our earlier findings about its maximum value and location, provides a complete picture of its behavior.

In conclusion, by analyzing the quadratic function f(x) = -3x² + 30x - 1, we have successfully determined that it has a maximum value, found the maximum value and its location, and identified the function's domain and range. This exploration demonstrates the power of algebraic techniques in understanding the properties of quadratic functions without relying on graphical representations.