Finding The Vertex Of The Quadratic Function F(x)=(x-8)(x-2)

The vertex of a quadratic function is a crucial point that reveals significant information about the parabola's shape and behavior. In this comprehensive guide, we will delve deep into the quadratic function f(x) = (x - 8)(x - 2), meticulously unraveling the steps to pinpoint its vertex. Understanding the vertex is paramount in various mathematical and real-world applications, from optimizing projectile trajectories to designing parabolic reflectors. We will explore multiple methods to determine the vertex, ensuring a thorough grasp of the underlying concepts. This exploration will empower you to confidently analyze and manipulate quadratic functions, unlocking their hidden potential and practical relevance. So, let's embark on this mathematical journey, demystifying the vertex and its significance in the realm of quadratic functions.

Understanding Quadratic Functions and the Significance of the Vertex

Before diving into the specifics of finding the vertex, let's establish a solid foundation by understanding the basics of quadratic functions. A quadratic function is a polynomial function 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 (if a > 0) or downwards (if a < 0). The vertex is the turning point of the parabola, representing either the minimum value of the function (for upward-opening parabolas) or the maximum value (for downward-opening parabolas).

The vertex holds immense significance in analyzing the behavior of the quadratic function. Its coordinates, represented as (h, k), provide valuable insights:

• h: The x-coordinate of the vertex represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. This line is defined by the equation x = h.
• k: The y-coordinate of the vertex represents the minimum or maximum value of the function. If the parabola opens upwards, k is the minimum value; if it opens downwards, k is the maximum value.

Knowing the vertex allows us to quickly determine the range of the function, identify intervals of increasing and decreasing behavior, and solve optimization problems. For instance, in physics, the vertex can represent the maximum height reached by a projectile, while in economics, it can represent the point of maximum profit. Therefore, mastering the techniques to find the vertex is crucial for a comprehensive understanding of quadratic functions and their applications.

Method 1: Expanding and Using the Vertex Formula

The first method we'll explore involves expanding the given quadratic function f(x) = (x - 8)(x - 2) into its standard form and then applying the vertex formula. The standard form of a quadratic function is f(x) = ax² + bx + c. Expanding the given function, we get:

f(x) = (x - 8)(x - 2) f(x) = x² - 2x - 8x + 16 f(x) = x² - 10x + 16

Now that we have the function in standard form, we can identify the coefficients: a = 1, b = -10, and c = 16. The vertex formula provides a direct way to calculate the coordinates of the vertex (h, k):

• h = -b / 2a
• k = f(h)

Let's apply the formula to our function. First, we calculate h:

h = -(-10) / (2 * 1) h = 10 / 2 h = 5

Next, we calculate k by substituting h = 5 into the function f(x):

k = f(5) k = (5)² - 10(5) + 16 k = 25 - 50 + 16 k = -9

Therefore, the vertex of the quadratic function f(x) = (x - 8)(x - 2) is (5, -9). This method is straightforward and widely applicable, making it a valuable tool in your mathematical arsenal.

Method 2: Averaging the Roots

Another elegant method to find the vertex of a quadratic function in factored form, like f(x) = (x - 8)(x - 2), involves averaging the roots. The roots of a quadratic function are the x-values where the function equals zero, i.e., the points where the parabola intersects the x-axis. In the factored form f(x) = (x - r₁)(x - r₂), the roots are simply r₁ and r₂.

In our case, f(x) = (x - 8)(x - 2), so the roots are x = 8 and x = 2. The x-coordinate of the vertex, h, lies exactly midway between the roots. Therefore, we can find h by averaging the roots:

h = (r₁ + r₂) / 2 h = (8 + 2) / 2 h = 10 / 2 h = 5

This gives us the same h-value as we found using the vertex formula. Now, to find the y-coordinate of the vertex, k, we substitute h = 5 back into the original function:

k = f(5) k = (5 - 8)(5 - 2) k = (-3)(3) k = -9

Again, we arrive at the vertex (5, -9). This method offers a quicker alternative when the quadratic function is presented in factored form. It leverages the symmetry of the parabola, making the calculation of the vertex's x-coordinate particularly efficient.

Method 3: Completing the Square

Completing the square is a powerful technique that not only helps in finding the vertex but also transforms the quadratic function into vertex form. The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex. This form directly reveals the vertex coordinates, making it a highly advantageous representation.

Let's apply this method to f(x) = x² - 10x + 16. The first step is to focus on the x² and x terms and create a perfect square trinomial. To do this, we take half of the coefficient of the x term (-10), square it ((-5)² = 25), and add and subtract it within the expression:

f(x) = x² - 10x + 25 - 25 + 16

Now, we can group the first three terms, which form a perfect square trinomial:

f(x) = (x² - 10x + 25) - 25 + 16 f(x) = (x - 5)² - 9

Comparing this to the vertex form f(x) = a(x - h)² + k, we can directly identify the vertex as (5, -9). Here, a = 1, h = 5, and k = -9. Completing the square provides a structured approach to transforming the quadratic function into a form that readily reveals the vertex. Moreover, it reinforces the connection between algebraic manipulation and geometric representation, deepening your understanding of quadratic functions.

Visualizing the Vertex and the Parabola

Understanding the algebraic methods to find the vertex is crucial, but visualizing the parabola and its vertex adds another layer of comprehension. When we plot the graph of f(x) = (x - 8)(x - 2) = x² - 10x + 16, we observe a parabola opening upwards (since a = 1 > 0). The vertex (5, -9) is the lowest point on the parabola, representing the minimum value of the function.

The axis of symmetry, x = 5, is a vertical line passing through the vertex, dividing the parabola into two mirror images. The roots, x = 2 and x = 8, are the points where the parabola intersects the x-axis. The y-intercept, where the parabola intersects the y-axis, is found by setting x = 0 in the function, which gives us f(0) = 16.

Visualizing these features enhances our understanding of the relationship between the equation and the graph of a quadratic function. It reinforces the concept of the vertex as a critical point, highlighting its role in determining the parabola's shape and position. Graphing tools, whether online calculators or graphing software, can be invaluable in solidifying this visual understanding.

Applications of the Vertex in Real-World Scenarios

The vertex of a quadratic function isn't just a theoretical concept; it has practical applications in various real-world scenarios. Understanding the vertex allows us to solve optimization problems, predict trajectories, and model various phenomena. Let's explore some examples:

• Projectile Motion: In physics, the path of a projectile, such as a ball thrown into the air, can be modeled by a quadratic function. The vertex represents the maximum height the projectile reaches and the time it takes to reach that height. Engineers and physicists use this knowledge to calculate the trajectory of projectiles, optimizing launch angles and velocities.
• Optimization Problems: Many real-world problems involve finding the maximum or minimum value of a quantity. For example, a business might want to maximize profit or minimize cost. If the profit or cost function can be modeled as a quadratic function, the vertex will represent the optimal point. This principle is applied in various fields, including economics, engineering, and operations research.
• Parabolic Reflectors: Parabolic reflectors, used in satellite dishes, telescopes, and headlights, rely on the property that parallel rays of light or radio waves converge at the focus of the parabola. The vertex plays a crucial role in the design of these reflectors, ensuring efficient signal collection or beam projection.
• Bridge Design: The cables of suspension bridges often form a parabolic shape. Engineers use quadratic functions to model the shape of the cables, ensuring structural integrity and load distribution. The vertex of the parabola is a key point in these calculations.

These examples illustrate the versatility of quadratic functions and the practical significance of the vertex. By mastering the techniques to find the vertex, you equip yourself with a valuable tool for problem-solving in diverse fields.

Conclusion

In this comprehensive guide, we've explored the concept of the vertex of a quadratic function in detail, focusing on the function f(x) = (x - 8)(x - 2). We've examined three distinct methods for finding the vertex: expanding and using the vertex formula, averaging the roots, and completing the square. Each method offers a unique perspective and reinforces the understanding of quadratic functions. We've also emphasized the importance of visualizing the parabola and its vertex, as well as the numerous real-world applications where this concept proves invaluable.

By understanding the significance of the vertex and mastering the techniques to find it, you gain a powerful tool for analyzing and manipulating quadratic functions. Whether you're solving mathematical problems, modeling real-world phenomena, or simply seeking a deeper understanding of the world around you, the vertex of a quadratic function is a fundamental concept that will serve you well. So, embrace this knowledge, practice the methods, and unlock the potential of quadratic functions in your academic and professional endeavors.