How To Find The Vertex Of The Function F(x) = 6(x + 5)^2 - 10?

In mathematics, understanding the properties of functions is crucial, and quadratic functions hold a special place due to their widespread applications. One of the most important features of a quadratic function is its vertex, which represents the point where the function reaches its minimum or maximum value. In this article, we will delve into the process of finding the vertex of a quadratic function, specifically focusing on the function f(x) = 6(x + 5)² - 10. This comprehensive guide will not only provide the steps to find the vertex but also explain the underlying concepts and the significance of the vertex in various contexts.

Understanding Quadratic Functions and the Vertex

Quadratic functions are polynomial functions of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. If a > 0, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if a < 0, the parabola opens downwards, and the vertex represents the maximum point of the function. The vertex is a critical point as it helps determine the range, axis of symmetry, and overall behavior of the quadratic function.

The vertex of a parabola is the point where the parabola changes direction. It is the extreme point of the quadratic function. The vertex form of a quadratic function, which is given by f(x) = a(x - h)² + k, provides a straightforward way to identify the vertex. In this form, the vertex is the point (h, k). The value of h represents the x-coordinate of the vertex, and k represents the y-coordinate, which is the minimum or maximum value of the function. Understanding the vertex form is essential for quickly determining the vertex and sketching the graph of the parabola. It also simplifies the process of solving optimization problems where finding the maximum or minimum value of a quadratic function is necessary. The vertex not only provides the extreme value but also helps in understanding the symmetry of the parabola. The vertical line passing through the vertex is the axis of symmetry, dividing the parabola into two symmetrical halves. This symmetry is a key characteristic of quadratic functions and is used in various applications, such as physics and engineering, to model parabolic trajectories and shapes. Moreover, the vertex plays a crucial role in understanding the transformations of quadratic functions. By knowing the vertex, one can easily translate, reflect, or stretch the parabola, making it easier to analyze and compare different quadratic functions. In practical applications, the vertex can represent various critical points, such as the maximum height of a projectile, the minimum cost in a business model, or the optimal design parameter in engineering. Therefore, mastering the concept of the vertex is fundamental to understanding and applying quadratic functions in real-world scenarios.

Identifying the Vertex Form

The given function, f(x) = 6(x + 5)² - 10, is already in the vertex form, which is f(x) = a(x - h)² + k. This form is incredibly useful because it directly reveals the coordinates of the vertex. By comparing the given function with the vertex form, we can easily identify the values of a, h, and k. The coefficient a determines the direction and steepness of the parabola, while h and k define the position of the vertex in the coordinate plane. The process of converting a quadratic function from the standard form ax² + bx + c to the vertex form involves completing the square, a technique that can be slightly more complex. However, when the function is already in vertex form, the task of finding the vertex becomes straightforward. In our example, the function f(x) = 6(x + 5)² - 10 clearly shows the structure of the vertex form, making the identification of the vertex a simple task. The advantage of having a function in vertex form extends beyond just finding the vertex. It also provides insights into the transformations applied to the basic parabola y = x². The value of a indicates whether the parabola is stretched or compressed vertically and whether it is reflected across the x-axis. The value of h represents a horizontal shift, and the value of k represents a vertical shift. Together, these transformations allow us to visualize and analyze the graph of the quadratic function more effectively. For instance, in the given function, the coefficient 6 indicates that the parabola is vertically stretched by a factor of 6, making it narrower than the basic parabola. The term (x + 5) indicates a horizontal shift of 5 units to the left, and the constant term -10 indicates a vertical shift of 10 units downward. These transformations are crucial in understanding how the graph of the function is positioned and shaped in the coordinate plane. Understanding these shifts and stretches is invaluable in various applications, such as optimizing the design of parabolic mirrors or analyzing the trajectory of projectiles. The vertex form not only simplifies the identification of the vertex but also enhances our ability to interpret and manipulate quadratic functions in a broader context.

Determining the Vertex Coordinates

To determine the vertex coordinates from the vertex form f(x) = 6(x + 5)² - 10, we need to identify the values of h and k. Recall that the vertex form is generally written as f(x) = a(x - h)² + k, where the vertex is the point (h, k). By comparing the given function with the general vertex form, we can extract the values of h and k. In the function f(x) = 6(x + 5)² - 10, we can see that the term (x + 5)² can be rewritten as (x - (-5))². This allows us to identify h as -5. The constant term outside the squared expression is k, which in this case is -10. Therefore, the vertex of the parabola is the point (-5, -10). This straightforward method of identifying the vertex coordinates is one of the primary reasons why the vertex form is so valuable. It eliminates the need for more complex calculations or algebraic manipulations, providing a direct and efficient way to find the vertex. Understanding the significance of the negative sign in the vertex form is crucial. The h value in the vertex form a(x - h)² represents a horizontal shift of h units from the origin. If h is negative, as in our case where h = -5, it indicates a shift to the left. If h were positive, it would indicate a shift to the right. Similarly, the k value represents a vertical shift of k units from the x-axis. A negative k value, like -10 in our example, indicates a downward shift, while a positive k value would indicate an upward shift. These horizontal and vertical shifts are fundamental transformations that help us understand how the parabola is positioned in the coordinate plane relative to the basic parabola y = x². By correctly identifying the values of h and k, we can accurately plot the vertex and use it as a reference point to sketch the graph of the quadratic function. The vertex, being the extreme point of the parabola, provides valuable information about the function's behavior, such as its minimum or maximum value and its axis of symmetry. The ease with which we can determine the vertex from the vertex form highlights the importance of recognizing and utilizing this form in analyzing quadratic functions.

The Vertex: Minimum or Maximum Point?

Once we have determined the vertex of the function f(x) = 6(x + 5)² - 10 as (-5, -10), the next step is to understand whether this point represents a minimum or maximum value of the function. This is determined by the coefficient a in the vertex form f(x) = a(x - h)² + k. In our function, a = 6. Since a is positive (6 > 0), the parabola opens upwards. This means that the vertex represents the lowest point on the graph, and the function has a minimum value. Conversely, if a were negative, the parabola would open downwards, and the vertex would represent the highest point on the graph, indicating a maximum value. Understanding this relationship between the sign of a and the nature of the vertex is crucial for interpreting the behavior of quadratic functions. It allows us to quickly determine whether a quadratic function has a minimum or maximum value and to identify that value as the y-coordinate of the vertex. In the case of f(x) = 6(x + 5)² - 10, the positive value of a indicates that the parabola opens upwards, and the vertex (-5, -10) is the minimum point. The minimum value of the function is -10, which occurs at x = -5. This information is valuable in various applications, such as optimization problems where we seek to find the minimum or maximum value of a quantity described by a quadratic function. For example, in business applications, a quadratic function might represent the cost of production as a function of the number of units produced. Finding the vertex would help determine the minimum cost. Similarly, in physics, a quadratic function might describe the height of a projectile as a function of time. The vertex would then represent the maximum height reached by the projectile. The ability to quickly determine whether the vertex is a minimum or maximum point and to identify the corresponding value is a powerful tool in solving a wide range of problems. It allows us to analyze the behavior of quadratic functions and to extract meaningful information that can be applied in various real-world contexts. The sign of the leading coefficient a is therefore a key indicator of the overall shape and behavior of the parabola, and its relationship to the vertex provides valuable insights into the function's properties.

Graphing the Function

To visualize the function f(x) = 6(x + 5)² - 10, we can use the information we have gathered so far. We know that the vertex is (-5, -10) and that the parabola opens upwards because the coefficient a is positive (a = 6). This gives us a good starting point for sketching the graph. The vertex is the key point around which the parabola is symmetric. To get a more accurate graph, we can find additional points by plugging in values of x into the function and calculating the corresponding y values. For instance, we can find the y-intercept by setting x = 0: f(0) = 6(0 + 5)² - 10 = 6(25) - 10 = 150 - 10 = 140. So, the y-intercept is (0, 140). Due to the symmetry of the parabola, we can find another point that is equidistant from the vertex on the other side. Since the y-intercept is 5 units to the right of the vertex (x = -5), there will be a corresponding point 5 units to the left of the vertex, which is at x = -10. The y-value at x = -10 will be the same as the y-intercept, which is 140. Thus, we have another point (-10, 140). We can also find the x-intercepts, if they exist, by setting f(x) = 0 and solving for x: 0 = 6(x + 5)² - 10 10 = 6(x + 5)² 10/6 = (x + 5)² 5/3 = (x + 5)² Taking the square root of both sides, we get: ±√(5/3) = x + 5 x = -5 ± √(5/3) So, the x-intercepts are approximately x ≈ -5 + 1.29 = -3.71 and x ≈ -5 - 1.29 = -6.29. These x-intercepts give us two more points on the graph. With the vertex, the y-intercept, and the x-intercepts, we have enough points to sketch a reasonably accurate graph of the function. The graph will be a U-shaped parabola with its vertex at (-5, -10), opening upwards, and passing through the calculated points. Visualizing the graph helps to confirm our understanding of the function's behavior and its key features. The vertex is the lowest point, and the parabola extends upwards from there, demonstrating the minimum value of the function. The symmetry of the parabola is also evident in the graph, with the axis of symmetry passing through the vertex. Graphing the function is a valuable step in the process of analyzing quadratic functions as it provides a visual representation of the function's properties and behavior.

Conclusion

In summary, finding the vertex of the quadratic function f(x) = 6(x + 5)² - 10 involves recognizing the vertex form of the quadratic equation and identifying the h and k values. The vertex is (-5, -10), and since the coefficient a is positive, this point represents the minimum value of the function. Understanding the vertex is crucial for analyzing and graphing quadratic functions, as it provides key information about the function's behavior and extreme values. This comprehensive guide has walked through the process step-by-step, from understanding the basic concepts of quadratic functions and the vertex form to identifying the vertex coordinates and determining whether it represents a minimum or maximum point. Additionally, we have discussed how to use the vertex and other key points to graph the function, providing a visual representation of its behavior. The ability to find the vertex and interpret its significance is a valuable skill in mathematics and has applications in various fields, including physics, engineering, and economics. Whether you are solving optimization problems, analyzing trajectories, or modeling real-world phenomena, understanding the properties of quadratic functions and their vertices is essential. By mastering these concepts, you can gain a deeper understanding of mathematical functions and their applications, making you a more proficient problem solver and analytical thinker. The process of finding the vertex is not just a mathematical exercise; it is a gateway to understanding the broader applications of quadratic functions in the world around us. From the trajectory of a ball to the shape of a satellite dish, quadratic functions and their vertices play a crucial role in describing and analyzing various phenomena. Therefore, the time and effort spent in mastering these concepts are well worth the investment, as they provide a foundation for further exploration and application in diverse fields. The vertex, in particular, serves as a cornerstone in understanding the behavior and properties of quadratic functions, making it a key concept for anyone studying mathematics or related disciplines. Understanding this article will give you a solid grasp of quadratic functions, their properties, and their practical implications.