Analyzing The Quadratic Function G(x) = X² - 14x - 1 Vertex Axis Of Symmetry And Graphing

In mathematics, analyzing functions is a fundamental skill. Understanding the properties of a function allows us to predict its behavior, graph it accurately, and solve related problems. In this article, we will delve into the function g(x) = x² - 14x - 1, exploring its vertex, axis of symmetry, and how its graph relates to the basic quadratic function f(x) = x². This comprehensive analysis will provide a clear understanding of the function's characteristics and graphical representation.

Determining the Vertex of g(x) = x² - 14x - 1

The vertex of a parabola, which is the graph of a quadratic function, is a crucial point. It represents either the minimum or maximum value of the function. For a quadratic function in the standard form g(x) = ax² + bx + c, the vertex's x-coordinate can be found using the formula x = -b / 2a. In our case, g(x) = x² - 14x - 1, so a = 1, b = -14, and c = -1. Applying the formula, we get:

x = -(-14) / (2 * 1) = 14 / 2 = 7

This tells us that the x-coordinate of the vertex is 7. To find the y-coordinate, we substitute this value back into the original function:

g(7) = (7)² - 14(7) - 1 = 49 - 98 - 1 = -50

Therefore, the vertex of the function g(x) = x² - 14x - 1 is (7, -50). This point represents the minimum value of the function since the coefficient of the x² term (a) is positive, indicating an upward-opening parabola. Understanding the vertex is essential for visualizing the graph and determining the function's range. The vertex form of a quadratic equation, g(x) = a(x - h)² + k, where (h, k) is the vertex, provides another way to represent the function and easily identify the vertex. Converting the given function to vertex form can further solidify our understanding of its properties. For g(x) = x² - 14x - 1, completing the square gives us g(x) = (x - 7)² - 50, which directly shows the vertex at (7, -50). The vertex not only gives us the minimum or maximum point of the parabola but also serves as a reference point for understanding the symmetry of the graph. The axis of symmetry, which we will discuss next, passes through the vertex, further emphasizing its importance in analyzing quadratic functions. Furthermore, the vertex helps in determining the intervals where the function is increasing or decreasing. To the left of the vertex, the function decreases, and to the right, it increases. This information is invaluable in various applications, such as optimization problems where we aim to find the maximum or minimum value of a quantity. In summary, the vertex is a cornerstone in the analysis of quadratic functions, providing insights into the function's behavior, graph, and applications.

Identifying the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. Its equation is given by x = h, where h is the x-coordinate of the vertex. Since we found the vertex to be (7, -50), the axis of symmetry is the vertical line x = 7. This line acts as a mirror, reflecting the parabola onto itself. Knowing the axis of symmetry is crucial for sketching the graph of the function, as it provides a central reference point. Any point on the parabola has a corresponding point on the other side of the axis of symmetry, equidistant from it. This symmetry simplifies the process of plotting points and visualizing the overall shape of the parabola. The axis of symmetry is not just a geometrical feature; it also has implications for the function's properties. For instance, the function's values are the same for x-values that are equidistant from the axis of symmetry. This property can be useful in solving equations and understanding the function's behavior. The axis of symmetry, along with the vertex, provides a framework for understanding the parabola's orientation and position in the coordinate plane. The distance between the vertex and any other point on the parabola is mirrored on the opposite side of the axis of symmetry, reinforcing the symmetrical nature of the graph. Moreover, the axis of symmetry helps in identifying the intervals where the function is increasing or decreasing. As we move away from the axis of symmetry in either direction, the function's values change consistently, either increasing or decreasing, depending on the parabola's orientation. In practical applications, the axis of symmetry can be used to solve problems involving symmetry, such as finding the optimal placement of objects or designing structures with balanced proportions. Its role in defining the symmetry of the parabola makes it an essential tool in the analysis and application of quadratic functions. Understanding the axis of symmetry not only enhances our ability to graph the function but also deepens our understanding of its fundamental properties and behavior. The interplay between the vertex and the axis of symmetry highlights the elegance and predictability of quadratic functions, making them a cornerstone of mathematical modeling and problem-solving.

Graph Transformations: Shifting f(x) = x² to Obtain g(x)

Understanding how to transform basic functions is essential for graphing more complex functions. The function g(x) = x² - 14x - 1 can be seen as a transformation of the basic quadratic function f(x) = x². To determine the transformations, we can rewrite g(x) in vertex form, which we found to be g(x) = (x - 7)² - 50. This form reveals two key transformations:

1. Horizontal Shift: The term (x - 7)² indicates a horizontal shift of 7 units to the right. This is because replacing x with (x - 7) in the function f(x) = x² shifts the graph horizontally. A negative value inside the parenthesis shifts the graph to the right, while a positive value shifts it to the left.
2. Vertical Shift: The term -50 indicates a vertical shift of 50 units downward. This is because subtracting 50 from the function (x - 7)² moves the entire graph vertically down. A negative value outside the squared term shifts the graph downward, while a positive value shifts it upward.

Therefore, to graph g(x) = x² - 14x - 1, we start with the graph of f(x) = x², shift it 7 units to the right, and then shift it 50 units downward. These transformations preserve the shape of the parabola but change its position in the coordinate plane. Understanding transformations is a powerful tool in graphing functions, as it allows us to visualize complex functions by relating them to simpler ones. The horizontal and vertical shifts, along with stretches, compressions, and reflections, form the basic vocabulary of graph transformations. By mastering these transformations, we can quickly sketch the graphs of a wide variety of functions. In the case of quadratic functions, the vertex form directly reveals the horizontal and vertical shifts, making it an invaluable tool for graphing. The transformations not only help us visualize the graph but also provide insights into the function's behavior. The horizontal shift affects the position of the axis of symmetry, while the vertical shift affects the minimum or maximum value of the function. Combining these transformations, we can accurately predict the graph of any quadratic function in vertex form. Furthermore, understanding transformations is crucial in various applications, such as modeling physical phenomena where shifts and scaling are common. For example, in physics, the trajectory of a projectile can be modeled using a quadratic function, and transformations can help us understand how changes in initial conditions affect the projectile's path. In conclusion, graph transformations provide a powerful framework for understanding and visualizing functions. By recognizing the basic transformations, we can easily graph complex functions and gain insights into their behavior and applications. The transformation from f(x) = x² to g(x) = x² - 14x - 1 illustrates the power of these techniques in simplifying the process of graphing and analyzing functions.

Conclusion

In this article, we have thoroughly analyzed the function g(x) = x² - 14x - 1. We determined that the vertex is at (7, -50), the axis of symmetry is the line x = 7, and the graph of g(x) can be obtained by shifting the graph of f(x) = x² 7 units to the right and 50 units downward. This comprehensive analysis provides a deep understanding of the function's properties and its graphical representation. By understanding these concepts, you can confidently analyze and graph quadratic functions, a fundamental skill in mathematics and various applications.