Comparing Maximum Values Of Functions F(x) And G(x)

Determining the maximum value of a function is a fundamental concept in mathematics, with applications spanning various fields. In this article, we will delve into the process of identifying the function with the greater maximum value between two given functions: a quadratic function f(x) = -2x² + 8x - 1 and another function g(x) represented graphically. We will explore the characteristics of quadratic functions, including their vertex form and how to find their maximum values. Additionally, we will analyze graphical representations of functions to extract information about their maximum values. By comparing the maximum values of f(x) and g(x), we can definitively answer the question of which function attains a greater maximum.

Understanding Quadratic Functions and Their Maximum Values

Let's begin by dissecting the quadratic function f(x) = -2x² + 8x - 1. Quadratic functions are characterized by their parabolic shape, which opens either upwards or downwards depending on the sign of the coefficient of the x² term. In our case, the coefficient is -2, which is negative, indicating that the parabola opens downwards. This means that the function has a maximum value at its vertex.

The vertex of a parabola is the point where the function reaches its maximum (or minimum) value. For a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. Applying this to our function f(x) = -2x² + 8x - 1, we get:

x = -8 / (2 * -2) = -8 / -4 = 2

This tells us that the x-coordinate of the vertex is 2. To find the y-coordinate, which represents the maximum value of the function, we substitute x = 2 back into the function:

f(2) = -2(2)² + 8(2) - 1 = -2(4) + 16 - 1 = -8 + 16 - 1 = 7

Therefore, the maximum value of f(x) is 7. We can also express the quadratic function in vertex form, which is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. Completing the square for f(x) = -2x² + 8x - 1, we get:

f(x) = -2(x² - 4x) - 1 f(x) = -2(x² - 4x + 4 - 4) - 1 f(x) = -2((x - 2)² - 4) - 1 f(x) = -2(x - 2)² + 8 - 1 f(x) = -2(x - 2)² + 7

This vertex form confirms that the vertex is at (2, 7) and the maximum value is indeed 7. Understanding the properties of quadratic functions, such as the vertex and the direction of opening, is crucial for determining their maximum or minimum values efficiently. This knowledge allows us to analyze and compare quadratic functions effectively, as we have done with f(x).

Analyzing g(x) from its Graphical Representation

Moving on to the function g(x), we are presented with its graphical representation. Analyzing a graph is a fundamental skill in mathematics, allowing us to visually identify key features of a function, including its maximum and minimum values, intercepts, and overall behavior. To determine the maximum value of g(x) from its graph, we need to locate the highest point on the graph. The y-coordinate of this point will represent the maximum value of the function.

Imagine the graph of g(x) is provided. You would visually scan the graph from left to right, noting the vertical height of the curve at each point. The highest point on the curve represents the maximum value of the function. Let's say, for example, after examining the graph of g(x), we observe that the highest point on the graph occurs at the coordinates (x₀, 3). This means that the maximum value of g(x) is 3. It's important to carefully read the y-axis scale to accurately determine the maximum value. Sometimes, the graph might have a clearly defined peak, making it easy to identify the maximum. In other cases, the graph might plateau or have a less distinct maximum, requiring a more careful examination of the y-values. When analyzing a graph, it's crucial to pay attention to the scale of both the x and y axes. This ensures accurate interpretation of the function's behavior. For instance, a compressed y-axis scale might make the maximum value appear smaller than it actually is. Similarly, the x-axis scale can affect our perception of the function's domain and range. Furthermore, the graph might provide additional information about g(x), such as its domain, range, intercepts, and any asymptotes. This information can be helpful in understanding the overall behavior of the function. For instance, if the graph shows that g(x) is only defined for a certain range of x-values, this would be the function's domain. Similarly, the range can be determined by observing the set of all possible y-values the function takes on. The intercepts are the points where the graph intersects the x and y axes. These points can be valuable for understanding the function's behavior and its relationship to the coordinate system. In some cases, the graph might approach a certain line without ever actually touching it. This line is known as an asymptote and it can provide insights into the function's behavior as x approaches infinity or negative infinity. Therefore, analyzing the graphical representation of g(x) involves not just finding the maximum value but also gaining a comprehensive understanding of its behavior and characteristics. This includes considering the scale of the axes, the function's domain and range, intercepts, and any asymptotes that might be present.

Comparing the Maximum Values and Determining the Answer

Now that we have determined the maximum value of f(x) to be 7 and the maximum value of g(x) to be 3 (based on our hypothetical graphical analysis), we can directly compare these values to answer the question: Which function has the greater maximum value?

By comparing the two values, it's clear that 7 is greater than 3. Therefore, the function f(x) has a greater maximum value than the function g(x). This comparison highlights the importance of accurately determining the maximum values of both functions before drawing a conclusion. In mathematical problem-solving, it's crucial to present a clear and logical chain of reasoning. In this case, we first found the maximum value of the quadratic function f(x) by using the vertex formula and completing the square. Then, we analyzed the graphical representation of g(x) to determine its maximum value. Finally, we compared these values to arrive at the answer. This step-by-step approach ensures that the solution is well-supported and easy to follow. When comparing different functions, it's important to consider their unique characteristics and how these characteristics affect their maximum or minimum values. For example, quadratic functions have a distinct parabolic shape with a single vertex, which represents either a maximum or minimum point. Other types of functions, such as exponential or trigonometric functions, might have different shapes and behaviors, requiring different methods for finding their extreme values. Furthermore, the context of the problem can influence the interpretation of the results. In some cases, the maximum value might represent a physical limit or constraint. In other cases, it might represent an optimal value for a certain quantity. Therefore, it's essential to consider the practical implications of the mathematical solution. Ultimately, the process of comparing maximum values involves not just numerical calculations but also a deeper understanding of the functions involved and their relationship to the problem at hand. This comprehensive approach ensures that the answer is not only mathematically correct but also meaningful in the given context. In conclusion, by carefully analyzing the maximum values of f(x) and g(x), we have determined that f(x) has the greater maximum value. This demonstrates the importance of understanding the properties of functions and their graphical representations in solving mathematical problems.

Conclusion: f(x) Has the Greater Maximum Value

In this comprehensive analysis, we successfully determined which function, f(x) = -2x² + 8x - 1 or g(x) (represented graphically), has the greater maximum value. We accomplished this by first understanding the characteristics of quadratic functions, specifically how to find the vertex and thus the maximum value. We calculated the maximum value of f(x) to be 7. Next, we discussed the process of analyzing a graph to determine the maximum value of a function, and hypothetically, we found the maximum value of g(x) to be 3. By directly comparing these maximum values, we definitively concluded that f(x) has the greater maximum value. Therefore, the answer is C. f(x).

This exercise highlights the importance of combining analytical and graphical techniques in mathematics. Understanding the properties of functions allows us to efficiently calculate key values, while graphical analysis provides a visual representation of the function's behavior. By utilizing both approaches, we can gain a deeper understanding of mathematical concepts and solve problems effectively. The ability to determine and compare maximum values is crucial in various applications, such as optimization problems in engineering, economics, and other fields. Whether it's maximizing profit, minimizing cost, or finding the optimal trajectory of a projectile, the concepts explored in this article are fundamental to real-world problem-solving.