Finding The Vertex Of F(x) = X² + 8x - 2 A Step-by-Step Guide

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The vertex of a graph plays a pivotal role in understanding the behavior and characteristics of functions, especially quadratic functions. In the realm of mathematics, quadratic functions, distinguished by their parabolic curves, hold significant importance. These functions, expressed in the standard form of f(x) = ax² + bx + c, reveal a wealth of information through their vertex, which represents either the minimum or maximum point on the parabola. This article delves deep into the process of finding the vertex of the quadratic function f(x) = x² + 8x - 2, offering a step-by-step guide and insightful explanations to empower readers with a comprehensive understanding.

Understanding the Significance of the Vertex

The vertex is not merely a point on the graph; it's a crucial indicator of the function's behavior. For a parabola that opens upwards (a > 0), the vertex represents the minimum value of the function, while for a parabola opening downwards (a < 0), it signifies the maximum value. This attribute makes the vertex indispensable in various applications, including optimization problems, physics, and engineering. Identifying the vertex allows us to determine the extreme values of the function, which is essential for making informed decisions and predictions. Moreover, the vertex provides valuable insights into the symmetry of the parabola, as it lies on the axis of symmetry, a vertical line that divides the parabola into two mirror-image halves.

Methods to Determine the Vertex

There are several methods to determine the vertex of a quadratic function, each offering its unique approach and advantages. We'll explore the two most commonly used methods:

  1. Using the Vertex Formula: The vertex formula provides a direct way to calculate the coordinates of the vertex. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex (h) is given by h = -b / 2a, and the y-coordinate (k) is found by substituting h back into the function, i.e., k = f(h). This formula is particularly useful when the function is given in standard form, as it avoids the need for completing the square.

  2. Completing the Square: Completing the square is a technique that transforms the quadratic function into vertex form, f(x) = a(x - h)² + k, where (h, k) represents the vertex. This method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored. Completing the square not only reveals the vertex but also provides valuable information about the parabola's transformations, such as horizontal and vertical shifts.

Step-by-Step Guide: Finding the Vertex of f(x) = x² + 8x - 2

Let's now apply these methods to find the vertex of the given quadratic function, f(x) = x² + 8x - 2. This step-by-step guide will walk you through both the vertex formula and completing the square methods, providing a clear understanding of each approach.

Method 1: Using the Vertex Formula

  1. Identify a, b, and c: In the function f(x) = x² + 8x - 2, we have a = 1, b = 8, and c = -2. Identifying these coefficients is the first step in applying the vertex formula.

  2. Calculate the x-coordinate (h): Using the formula h = -b / 2a, we substitute the values of a and b: h = -8 / (2 * 1) = -4. The x-coordinate of the vertex is -4.

  3. Calculate the y-coordinate (k): Substitute h = -4 back into the function to find k: k = f(-4) = (-4)² + 8(-4) - 2 = 16 - 32 - 2 = -18. The y-coordinate of the vertex is -18.

Therefore, the vertex of the graph of the function f(x) = x² + 8x - 2 is (-4, -18). This point represents the minimum value of the function since the coefficient of x² is positive, indicating that the parabola opens upwards.

Method 2: Completing the Square

  1. Rewrite the function: Start by rewriting the function as f(x) = (x² + 8x) - 2. This step isolates the terms containing x, preparing them for completing the square.

  2. Complete the square: To complete the square, take half of the coefficient of the x term (which is 8), square it (which is 16), and add and subtract it inside the parentheses: f(x) = (x² + 8x + 16 - 16) - 2. Adding and subtracting the same value ensures that the expression remains equivalent to the original.

  3. Factor the perfect square trinomial: Factor the perfect square trinomial: f(x) = (x + 4)² - 16 - 2. This step transforms the quadratic expression into a form that reveals the vertex directly.

  4. Simplify: Combine the constant terms: f(x) = (x + 4)² - 18. Now the function is in vertex form, f(x) = a(x - h)² + k, where a = 1, h = -4, and k = -18.

From the vertex form, we can directly identify the vertex as (-4, -18). This method not only provides the vertex but also showcases the transformations applied to the basic parabola y = x², such as a horizontal shift of 4 units to the left and a vertical shift of 18 units downwards.

Visualizing the Parabola and Its Vertex

To further solidify your understanding, let's visualize the parabola represented by the function f(x) = x² + 8x - 2. The parabola opens upwards, as the coefficient of x² is positive (a = 1). The vertex, (-4, -18), is the lowest point on the graph, representing the minimum value of the function. The axis of symmetry is a vertical line passing through the vertex, x = -4, dividing the parabola into two symmetrical halves. By plotting a few additional points, such as the y-intercept (0, -2) and its symmetrical point (-8, -2), we can sketch the parabola and visually confirm the location of the vertex and the overall shape of the curve.

Practical Applications of Vertex Determination

The ability to find the vertex of a quadratic function extends beyond theoretical mathematics and finds practical applications in various fields. Here are a few examples:

  • Optimization Problems: In business and economics, the vertex can be used to determine the maximum profit or minimum cost. For instance, a company might use a quadratic function to model the relationship between the price of a product and the quantity sold. The vertex would then represent the price that maximizes revenue.
  • Projectile Motion: In physics, the trajectory of a projectile, such as a ball thrown into the air, can be modeled by a quadratic function. The vertex represents the highest point reached by the projectile, allowing us to calculate the maximum height and the time it takes to reach that height.
  • Engineering Design: Engineers often use quadratic functions to design structures, such as bridges and arches. The vertex helps determine the optimal shape and dimensions to ensure stability and minimize stress.

Conclusion: Mastering the Vertex

Finding the vertex of a quadratic function is a fundamental skill in mathematics with far-reaching applications. Whether you choose to use the vertex formula or complete the square, mastering these techniques will empower you to analyze and interpret quadratic functions effectively. The vertex provides valuable insights into the function's behavior, extreme values, and symmetry, making it an indispensable tool in various fields. By understanding the significance of the vertex and practicing the methods to determine it, you'll gain a deeper appreciation for the power and versatility of quadratic functions.

In the case of f(x) = x² + 8x - 2, we've definitively established that the vertex is (-4, -18). This point serves as a cornerstone for understanding the parabola's shape, position, and behavior, highlighting the importance of vertex determination in mathematical analysis and problem-solving.