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. It represents either the minimum or maximum value of the function, making it essential in various applications, from optimization problems to understanding the trajectory of projectiles. In this comprehensive guide, 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, including expanding the function, using the vertex formula, and completing the square. By the end of this guide, you will have a solid understanding of how to determine the vertex of any quadratic function.

Understanding Quadratic Functions and Their Properties

Before we dive into finding the vertex, let's first establish a firm understanding of quadratic functions and their key properties. 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 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.

The vertex of a parabola is the point where the curve changes direction. It is the turning point of the quadratic function. The x-coordinate of the vertex is the axis of symmetry, which is 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, depending on the parabola's orientation. Understanding these properties is crucial for accurately determining and interpreting the vertex of a quadratic function. The vertex form of a quadratic equation, f(x) = a(x - h)² + k, directly reveals the vertex coordinates as (h, k), making it a particularly useful form for analysis. Recognizing the relationship between the coefficients of the quadratic equation and the parabola's shape and position is fundamental for solving various mathematical problems and real-world applications. For example, in physics, understanding quadratic functions helps in analyzing projectile motion, where the vertex represents the maximum height reached by the projectile. Similarly, in economics, quadratic functions can model cost and revenue curves, where the vertex indicates the point of maximum profit or minimum cost. Therefore, mastering the concepts of quadratic functions and their properties, including the vertex, is essential for success in mathematics and its applications.

Method 1: Expanding the Function and Using the Vertex Formula

The first method we'll explore involves expanding the given quadratic function and then applying the vertex formula. The given function is f(x) = (x - 8)(x - 2). To begin, we need to expand this expression into the standard quadratic form f(x) = ax² + bx + c. This can be achieved by using the distributive property (also known as the FOIL method):

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

Now that we have the quadratic 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 x-coordinate of the vertex, denoted as h, using the formula h = -b / 2a. Plugging in the values of a and b, we get:

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

To find the y-coordinate of the vertex, denoted as k, we substitute the value of h back into the function f(x):

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 method is particularly useful when the quadratic function is given in factored form because it systematically transforms the function into standard form, allowing for the easy application of the vertex formula. The vertex formula is a cornerstone in quadratic function analysis, providing a straightforward approach to finding the vertex coordinates. By understanding and applying this method, you can efficiently determine the vertex of any quadratic function given in standard or factored form. This process not only helps in identifying the maximum or minimum value of the function but also provides critical insights into the parabola's symmetry and overall behavior. Mastering this technique is essential for various applications, including optimization problems, graph analysis, and real-world modeling scenarios.

Method 2: Completing the Square

Another powerful method for finding the vertex of a quadratic function is completing the square. This technique transforms the quadratic expression into vertex form, f(x) = a(x - h)² + k, where (h, k) directly represents the vertex coordinates. Let's apply this method to our function, f(x) = x² - 10x + 16, which we obtained by expanding the original expression. The first step in completing the square is to focus on the x² and x terms. We need to create a perfect square trinomial. To do this, we take half of the coefficient of the x term, which is -10, and square it:

(-10 / 2)² = (-5)² = 25

Now, we add and subtract this value (25) inside the function to maintain the equation's balance:

f(x) = x² - 10x + 25 - 25 + 16

Notice that the first three terms, x² - 10x + 25, form a perfect square trinomial. We can rewrite this as (x - 5)². The remaining terms, -25 and +16, can be combined:

f(x) = (x - 5)² - 9

Now the function is in vertex form, f(x) = a(x - h)² + k, where a = 1, h = 5, and k = -9. The vertex is therefore (5, -9). Completing the square is a versatile method that not only helps in finding the vertex but also in solving quadratic equations and rewriting quadratic expressions in a more manageable form. The process involves creating a perfect square trinomial, which allows us to express the quadratic function in vertex form. This form directly reveals the vertex coordinates, making it easy to identify the maximum or minimum point of the parabola. Additionally, completing the square provides a deeper understanding of the structure of quadratic functions and their graphical representations. It is a fundamental technique in algebra and calculus, with applications in various fields, including physics, engineering, and economics. Mastering this method enhances your ability to analyze and manipulate quadratic functions effectively.

Graphical Representation and Verification

To further solidify our understanding and verify our results, let's consider the graphical representation of the quadratic function f(x) = (x - 8)(x - 2). We have determined that the vertex is at the point (5, -9). Additionally, the roots of the function, which are the x-intercepts, can be found by setting f(x) = 0:

(x - 8)(x - 2) = 0

This gives us x = 8 and x = 2. These are the points where the parabola intersects the x-axis. The y-intercept can be found by setting x = 0 in the original function:

f(0) = (0 - 8)(0 - 2) = (-8)(-2) = 16

So, the y-intercept is 16. Now, we have enough information to sketch the graph of the parabola. The vertex (5, -9) is the lowest point on the graph, as the coefficient of x² is positive (a = 1), indicating that the parabola opens upwards. The parabola passes through the points (2, 0), (8, 0), and (0, 16). Plotting these points and sketching the curve confirms that the vertex is indeed at (5, -9). Graphical representation provides a visual confirmation of our algebraic calculations. By plotting the key points of the quadratic function, such as the vertex, x-intercepts, and y-intercept, we can visualize the parabola and verify the accuracy of our results. This approach enhances our understanding of the relationship between the algebraic expression and its geometric representation. Moreover, graphing the function helps in identifying the axis of symmetry, which is the vertical line passing through the vertex. The symmetry of the parabola around this line further reinforces the properties of quadratic functions. Therefore, graphical analysis is an invaluable tool for verifying and interpreting the solutions obtained through algebraic methods.

Conclusion

In conclusion, we have successfully determined the vertex of the quadratic function f(x) = (x - 8)(x - 2) using multiple methods. We expanded the function and applied the vertex formula, and we also completed the square to rewrite the function in vertex form. Both methods yielded the same result: the vertex is located at (5, -9). Additionally, we verified our result through graphical representation, plotting the key points of the parabola and confirming that the vertex is indeed at (5, -9). Understanding how to find the vertex of a quadratic function is a fundamental skill in algebra and calculus. It allows us to identify the maximum or minimum value of the function, which is crucial in various applications, such as optimization problems and real-world modeling. Mastering different methods, such as using the vertex formula and completing the square, provides flexibility and a deeper understanding of quadratic functions. Furthermore, graphical verification reinforces the connection between algebraic expressions and their geometric representations. By combining these techniques, you can confidently analyze and solve problems involving quadratic functions. The vertex is a key feature of a parabola, and its determination is essential for understanding the function's behavior and properties. This comprehensive guide has equipped you with the knowledge and skills to find the vertex of any quadratic function, enhancing your mathematical proficiency and problem-solving abilities.