Finding The Vertex Of Quadratic Function F(x) = (x-8)(x-2)
Understanding quadratic functions is fundamental in mathematics, and one of the key features of a quadratic function is its vertex. The vertex represents the minimum or maximum point of the parabola, which is the graphical representation of the quadratic function. In this article, we will delve into the process of finding the vertex of the quadratic function f(x) = (x-8)(x-2). We will explore different methods and provide a step-by-step guide to help you master this concept. Whether you are a student learning about quadratic functions for the first time or someone looking to refresh your knowledge, this comprehensive guide will equip you with the necessary skills to confidently determine the vertex of any quadratic function.
Understanding Quadratic Functions and Their Vertex
Before we dive into the specifics of finding the vertex of f(x) = (x-8)(x-2), let's establish a solid understanding of quadratic functions and the significance of the vertex. A quadratic function is a polynomial function of degree two, generally expressed in the standard form as:
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 upwards if a > 0 and downwards if a < 0. The vertex of the parabola is the point where the curve changes direction. It is the minimum point if the parabola opens upwards and the maximum point if the parabola opens downwards. Understanding the vertex is crucial for various applications, including optimization problems, physics, and engineering.
The vertex provides valuable information about the quadratic function, such as its range, symmetry, and extreme values. The x-coordinate of the vertex represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The y-coordinate of the vertex represents the minimum or maximum value of the function. In the context of real-world problems, the vertex can help determine the optimal solution, such as the maximum height of a projectile or the minimum cost of production.
Methods to Determine the Vertex of a Quadratic Function
There are several methods to find the vertex of a quadratic function. We will explore three common approaches:
- Using the Vertex Formula: This is a direct method that involves using a formula to calculate the coordinates of the vertex.
- Completing the Square: This method involves rewriting the quadratic function in vertex form, which directly reveals the coordinates of the vertex.
- Finding the Axis of Symmetry: This method involves finding the axis of symmetry first and then using it to determine the vertex.
Each method has its advantages and disadvantages, and the choice of method may depend on the specific form of the quadratic function and personal preference. We will demonstrate each method with the example function f(x) = (x-8)(x-2) to provide a clear and comprehensive understanding.
Method 1: Utilizing the Vertex Formula
The vertex formula is a powerful tool for finding the vertex of a quadratic function. It states that for a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex, denoted as h, is given by:
h = -b / 2a
Once we have the x-coordinate, we can find the y-coordinate of the vertex, denoted as k, by substituting h into the function:
k = f(h)
Therefore, the vertex of the parabola is the point (h, k). To apply this method to our example function, f(x) = (x-8)(x-2), we first need to expand the expression to get it into the standard form:
f(x) = (x-8)(x-2) = x² - 2x - 8x + 16 = x² - 10x + 16
Now we can identify the coefficients: a = 1, b = -10, and c = 16. Next, we use the vertex formula to find the x-coordinate of the vertex:
h = -b / 2a = -(-10) / (2 * 1) = 10 / 2 = 5
Now that we have the x-coordinate, we can find the y-coordinate by substituting h = 5 into the function:
k = f(5) = (5)² - 10(5) + 16 = 25 - 50 + 16 = -9
Therefore, the vertex of the quadratic function f(x) = (x-8)(x-2) is (5, -9). This means the parabola has a minimum point at (5, -9), and the axis of symmetry is the vertical line x = 5. This method provides a straightforward and efficient way to find the vertex, especially when the quadratic function is already in standard form or can be easily converted.
Method 2: Completing the Square
Completing the square is another powerful technique for finding the vertex of a quadratic function. This method involves rewriting the quadratic function in vertex form, which is given by:
f(x) = a(x - h)² + k
where (h, k) is the vertex of the parabola. The advantage of this form is that the vertex coordinates are directly visible. To complete the square for our example function, f(x) = (x-8)(x-2) = x² - 10x + 16, we follow these steps:
- Focus on the x² and x terms: x² - 10x
- Take half of the coefficient of the x term, square it, and add and subtract it: The coefficient of the x term is -10. Half of -10 is -5, and squaring it gives 25. So we add and subtract 25: x² - 10x + 25 - 25
- Rewrite the first three terms as a perfect square: (x - 5)² - 25
- Bring down the constant term from the original equation: (x - 5)² - 25 + 16
- Simplify: (x - 5)² - 9
Now the function is in vertex form: f(x) = (x - 5)² - 9. Comparing this to the vertex form f(x) = a(x - h)² + k, we can see that h = 5 and k = -9. Therefore, the vertex of the quadratic function f(x) = (x-8)(x-2) is (5, -9). This method provides a deeper understanding of the structure of the quadratic function and how it relates to the vertex. It is particularly useful when dealing with quadratic functions that are not in standard form.
Method 3: Finding the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola and divides it into two symmetrical halves. Finding the axis of symmetry can be a helpful intermediate step in determining the vertex. For a quadratic function in the form f(x) = ax² + bx + c, the equation of the axis of symmetry is given by:
x = -b / 2a
This is the same formula we used to find the x-coordinate of the vertex in the vertex formula method. However, this method emphasizes the geometric interpretation of the axis of symmetry. For our example function, f(x) = (x-8)(x-2) = x² - 10x + 16, we have a = 1 and b = -10. Therefore, the axis of symmetry is:
x = -(-10) / (2 * 1) = 10 / 2 = 5
So the axis of symmetry is the vertical line x = 5. Since the vertex lies on the axis of symmetry, the x-coordinate of the vertex is 5. To find the y-coordinate, we substitute x = 5 into the function:
f(5) = (5)² - 10(5) + 16 = 25 - 50 + 16 = -9
Therefore, the vertex of the quadratic function f(x) = (x-8)(x-2) is (5, -9). This method highlights the symmetry of the parabola and provides a visual understanding of the relationship between the axis of symmetry and the vertex. It is particularly useful when the focus is on the symmetry properties of the quadratic function.
Conclusion
In this article, we have explored the concept of the vertex of a quadratic function and demonstrated three different methods to find it. The vertex of the quadratic function f(x) = (x-8)(x-2) is (5, -9). We have shown how to find this vertex using the vertex formula, completing the square, and finding the axis of symmetry. Each method offers a unique perspective and can be applied depending on the specific form of the quadratic function and the desired level of understanding.
Mastering the concept of the vertex is crucial for understanding the behavior of quadratic functions and their applications. Whether you are solving mathematical problems or applying quadratic functions in real-world scenarios, the ability to confidently determine the vertex will be a valuable asset. By understanding the different methods and practicing with various examples, you can develop a strong foundation in quadratic functions and their properties. The vertex represents a critical point on the parabola, providing valuable information about the function's minimum or maximum value and its axis of symmetry. By mastering the techniques discussed in this article, you can confidently analyze and interpret quadratic functions in a variety of contexts.
