How Do I Find The Vertex Of The Function F(x) = 3x^2 - 6x + 2?

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In mathematics, particularly in algebra, quadratic functions play a vital role. These functions, characterized by the general form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0, create a parabolic curve when graphed. Understanding the properties of quadratic functions is crucial for solving a wide range of problems, from determining the trajectory of a projectile to optimizing business processes. One of the most important features of a parabola is its vertex, which represents either the minimum or maximum point of the function. The vertex provides critical information about the behavior of the quadratic function, making its determination a fundamental skill in algebra.

In this comprehensive guide, we will delve into the process of finding the vertex of a quadratic function. We will primarily focus on the function f(x) = 3x² - 6x + 2 as a practical example, illustrating the step-by-step methods and techniques involved. This article aims to equip you with the knowledge and skills necessary to identify the vertex of any quadratic function, enhancing your understanding of quadratic equations and their applications. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this guide will provide you with a clear and concise approach to mastering the concept of vertex determination. We will explore different methods, including using the vertex formula and completing the square, to ensure a thorough understanding of the topic. So, let's embark on this mathematical journey and unlock the secrets of finding the vertex of a quadratic function.

Quadratic functions, represented in the general form f(x) = ax² + bx + c, are fundamental in mathematics and have numerous applications in various fields. 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. When a is positive, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, when a is negative, the parabola opens downwards, and the vertex represents the maximum point. The vertex, therefore, is a critical feature of a quadratic function, providing essential information about its behavior and extreme values. It's the point where the function changes direction, making it a focal point in analysis and problem-solving.

The vertex of a parabola is the point where the curve changes direction. It is the extreme point of the parabola, meaning it is either the highest or the lowest point on the graph. The coordinates of the vertex are typically represented as (h, k), where h is the x-coordinate and k is the y-coordinate. These coordinates are significant because they provide crucial insights into the function's properties. The x-coordinate, h, represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The y-coordinate, k, represents the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards. Understanding the vertex and its coordinates is essential for sketching the graph of a quadratic function, determining its range, and solving optimization problems. The vertex, in essence, encapsulates the key characteristics of the parabola, making it a cornerstone in the study of quadratic functions.

In the context of our example function, f(x) = 3x² - 6x + 2, we can identify the coefficients as follows: a = 3, b = -6, and c = 2. Since a is positive (3 > 0), the parabola opens upwards, indicating that the vertex will be the minimum point of the function. This preliminary analysis sets the stage for the subsequent steps in finding the exact coordinates of the vertex. Recognizing the coefficients and their implications is a crucial first step in understanding the behavior of the quadratic function and predicting the nature of its graph. By determining that the parabola opens upwards, we know that the vertex will represent the lowest point on the curve, which is a valuable piece of information when interpreting the function's behavior and applications. This initial assessment lays the groundwork for a more detailed exploration of the function's properties and the precise location of its vertex.

The vertex formula is a direct and efficient method for finding the coordinates of the vertex of a quadratic function. Given the general form of a quadratic function, f(x) = ax² + bx + c, the x-coordinate (h) of the vertex can be calculated using the formula:

h = -b / 2a

This formula is derived from completing the square and represents the axis of symmetry of the parabola. Once we have the x-coordinate, we can find the y-coordinate (k) by substituting h back into the original function:

k = f(h)

The resulting coordinates (h, k) represent the vertex of the parabola. This formula provides a straightforward approach to locating the vertex without the need for graphing or other complex techniques. It is a powerful tool for analyzing quadratic functions and solving related problems. The vertex formula is particularly useful when dealing with quadratic functions where the coefficients are known, and a precise determination of the vertex is required.

Applying the vertex formula to our example function, f(x) = 3x² - 6x + 2, we first identify the coefficients: a = 3 and b = -6. We then substitute these values into the formula for the x-coordinate (h):

h = -(-6) / (2 * 3) = 6 / 6 = 1

This calculation gives us the x-coordinate of the vertex, which is 1. Next, we substitute this value back into the original function to find the y-coordinate (k):

k = f(1) = 3(1)² - 6(1) + 2 = 3 - 6 + 2 = -1

Therefore, the y-coordinate of the vertex is -1. Combining these results, we find that the vertex of the function f(x) = 3x² - 6x + 2 is (1, -1). This point represents the minimum value of the function, as the parabola opens upwards. The vertex formula provides a clear and concise method for determining the vertex, allowing for a precise understanding of the function's behavior and graph. This method is especially valuable when accuracy is paramount, and it eliminates the need for graphical approximations or other potentially less precise techniques. The calculated vertex (1, -1) gives us a key reference point for further analysis of the quadratic function.

Completing the square is another powerful algebraic technique for finding the vertex of a quadratic function. This method involves rewriting the quadratic function in vertex form, which directly reveals the coordinates of the vertex. The vertex form of a quadratic function is given by:

f(x) = a(x - h)² + k

where (h, k) represents the vertex of the parabola. By transforming the standard form of the quadratic function (f(x) = ax² + bx + c) into vertex form, we can easily identify the vertex coordinates. Completing the square involves a series of algebraic manipulations to create a perfect square trinomial within the equation. This method is particularly useful for understanding the structure of quadratic functions and provides a deeper insight into the relationship between the coefficients and the vertex.

To apply the completing the square method to our example function, f(x) = 3x² - 6x + 2, we follow these steps:

  1. Factor out the coefficient of the x² term (which is 3) from the first two terms:

    f(x) = 3(x² - 2x) + 2

  2. Complete the square inside the parentheses. To do this, take half of the coefficient of the x term (-2), square it ((-1)² = 1), and add and subtract it inside the parentheses:

    f(x) = 3(x² - 2x + 1 - 1) + 2

  3. Rewrite the perfect square trinomial as a squared binomial:

    f(x) = 3((x - 1)² - 1) + 2

  4. Distribute the 3 and simplify:

    f(x) = 3(x - 1)² - 3 + 2 f(x) = 3(x - 1)² - 1

Now, the function is in vertex form: f(x) = 3(x - 1)² - 1. Comparing this with the general vertex form f(x) = a(x - h)² + k, we can identify the vertex as (h, k) = (1, -1). This result confirms the vertex coordinates we found using the vertex formula. Completing the square not only provides the vertex but also offers a comprehensive understanding of how the quadratic function is transformed and positioned in the coordinate plane. The process highlights the importance of algebraic manipulation and its role in revealing the key features of quadratic functions.

Graphical representation provides a visual confirmation of our algebraic calculations and enhances our understanding of the quadratic function. By plotting the function f(x) = 3x² - 6x + 2, we can visually identify the vertex and verify our previous results. The graph of a quadratic function is a parabola, and the vertex is the point where the parabola changes direction. Using graphing tools or software, we can plot the function and observe the parabola's shape, direction, and the location of its vertex. This visual confirmation is an essential step in ensuring the accuracy of our calculations and solidifying our understanding of the function's behavior.

When we graph the function f(x) = 3x² - 6x + 2, we observe a parabola that opens upwards, which aligns with our earlier determination that a > 0. The lowest point on the parabola, which represents the vertex, is clearly visible on the graph. By examining the graph, we can see that the vertex is located at the point (1, -1), which matches the coordinates we calculated using both the vertex formula and the completing the square method. This graphical verification provides a strong confirmation of our algebraic results and reinforces our understanding of the function's properties. The visual representation allows us to see the parabola's symmetry and how the vertex represents the minimum value of the function. This comprehensive approach, combining algebraic calculations and graphical analysis, ensures a thorough understanding of the quadratic function and its key features.

In conclusion, finding the vertex of a quadratic function is a fundamental skill in algebra with numerous applications in mathematics and real-world scenarios. In this guide, we explored two primary methods for determining the vertex: the vertex formula and completing the square. Both methods provide accurate results and offer different perspectives on the structure and behavior of quadratic functions. We applied these methods to the example function f(x) = 3x² - 6x + 2, and both approaches consistently yielded the vertex coordinates (1, -1). Additionally, we emphasized the importance of graphical representation as a means of verifying our algebraic calculations and gaining a deeper understanding of the function's properties. The combination of algebraic techniques and visual analysis ensures a comprehensive grasp of quadratic functions and their key features.

The ability to find the vertex of a quadratic function is not only crucial for academic purposes but also for solving practical problems in various fields, such as physics, engineering, and economics. The vertex represents the maximum or minimum value of the function, which is essential in optimization problems, trajectory calculations, and modeling real-world phenomena. By mastering the techniques discussed in this guide, you will be well-equipped to analyze quadratic functions, interpret their graphs, and apply them to solve a wide range of problems. The vertex, as the turning point of the parabola, provides valuable information about the function's behavior and its extreme values, making its determination a vital skill in mathematical analysis and problem-solving. The knowledge and techniques presented here will empower you to confidently tackle quadratic functions and their applications in various contexts.