Finding The Vertex Of The Quadratic Function F(x) = (x-8)(x-2)
Introduction to Quadratic Functions and Their Vertices
In the realm of mathematics, quadratic functions play a pivotal role, particularly in algebra and calculus. Understanding the properties and characteristics of these functions is crucial for solving a myriad of problems. One of the most significant features of a quadratic function is its vertex, which represents either the minimum or maximum point on the parabola that the function describes. This article delves into the process of finding the vertex of a specific quadratic function, f(x) = (x-8)(x-2), providing a step-by-step guide and illuminating the underlying concepts.
Before we dive into the specifics, let's establish a solid foundation. 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 is not equal to zero. 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 is positive, the parabola opens upwards, indicating a minimum point; if a is negative, the parabola opens downwards, indicating a maximum point. The vertex is this extreme point, either the lowest point on the curve (minimum) or the highest point on the curve (maximum).
The vertex of a parabola is not just a point on a graph; it holds significant information about the function's behavior. It represents the point where the function changes direction, from decreasing to increasing (if the parabola opens upwards) or from increasing to decreasing (if the parabola opens downwards). This makes the vertex crucial in optimization problems, where we seek to find the maximum or minimum value of a function. Furthermore, the vertex is the axis of symmetry for the parabola, meaning the graph is symmetrical around the vertical line passing through the vertex. This symmetry can be exploited to simplify problem-solving and gain a deeper understanding of the function's characteristics. The location of the vertex is defined by its coordinates (h, k), where 'h' is the x-coordinate and 'k' is the y-coordinate. Determining these coordinates is our primary goal when analyzing a quadratic function.
Methods to Determine the Vertex of a Quadratic Function
There are several methods to determine the vertex of a quadratic function. Each method offers a unique approach and may be more suitable depending on the form in which the quadratic function is presented. We will explore three primary methods: using the vertex form, completing the square, and using the formula derived from the standard form. Understanding these methods equips us with a versatile toolkit for tackling a variety of quadratic function problems.
1. Using the Vertex Form
The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. This form is particularly advantageous because the vertex is immediately apparent. By simply inspecting the equation, we can identify the values of 'h' and 'k', thereby determining the vertex. To utilize this method, we need to transform the given quadratic function into vertex form. This often involves algebraic manipulation, such as completing the square, which we will discuss later. However, once the function is in vertex form, the process of identifying the vertex becomes straightforward and efficient. For instance, if we have a function in the form f(x) = 2(x - 3)² + 5, we can directly see that the vertex is at the point (3, 5).
2. Completing the Square
Completing the square is a powerful algebraic technique used to transform a quadratic function from its standard form (f(x) = ax² + bx + c) into vertex form. This method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a squared binomial. The process typically involves taking half of the coefficient of the x term, squaring it, and adding and subtracting it within the expression. This allows us to rewrite the quadratic function in the form f(x) = a(x - h)² + k, from which the vertex (h, k) can be easily identified. Completing the square is a versatile method that can be applied to any quadratic function, regardless of its initial form. It provides a systematic approach to finding the vertex, making it a valuable tool in the study of quadratic functions.
3. Using the Formula Derived from the Standard Form
For a quadratic function in the standard form f(x) = ax² + bx + c, there's a direct formula to calculate the x-coordinate ('h') of the vertex: h = -b / 2a. Once we have the value of 'h', we can substitute it back into the original function to find the y-coordinate ('k') of the vertex, i.e., k = f(h). This formula provides a quick and efficient way to determine the vertex without the need for algebraic manipulation like completing the square. It's particularly useful when dealing with quadratic functions in standard form, as it offers a straightforward path to finding the vertex. Understanding the derivation of this formula from the process of completing the square can provide a deeper appreciation for its validity and utility.
Step-by-Step Solution for f(x) = (x-8)(x-2)
Now, let's apply these methods to find the vertex of the specific quadratic function f(x) = (x-8)(x-2). This will involve expanding the function, identifying the coefficients, and then employing one or more of the methods discussed above.
1. Expanding the Function
First, we need to expand the given function from its factored form to the standard form f(x) = ax² + bx + c. This involves multiplying the two binomials (x-8) and (x-2):
f(x) = (x - 8)(x - 2) f(x) = x² - 2x - 8x + 16 f(x) = x² - 10x + 16
Now, we have the function in standard form, where a = 1, b = -10, and c = 16. This form allows us to readily apply the formula for finding the vertex.
2. Using the Formula h = -b / 2a
Next, we'll use the formula h = -b / 2a to find the x-coordinate of the vertex. We have a = 1 and b = -10, so:
h = -(-10) / (2 * 1) h = 10 / 2 h = 5
Thus, the x-coordinate of the vertex is 5.
3. Finding the y-coordinate, k = f(h)
To find the y-coordinate ('k') of the vertex, we substitute h = 5 back into the original function f(x) = x² - 10x + 16:
k = f(5) = (5)² - 10(5) + 16 k = 25 - 50 + 16 k = -9
Therefore, the y-coordinate of the vertex is -9.
4. The Vertex Coordinates
Combining the x and y coordinates, we find that the vertex of the quadratic function f(x) = (x-8)(x-2) is (5, -9). This point represents the minimum value of the function since the coefficient a is positive (a = 1), indicating that the parabola opens upwards.
Verification and Graphical Interpretation
To ensure the accuracy of our calculations, it's beneficial to verify the result using alternative methods or graphical interpretations. We can use completing the square method to verify the vertex, and we can also visualize the parabola to confirm that the vertex aligns with our calculated point.
1. Verification by Completing the Square
Let's verify our result by completing the square for the function f(x) = x² - 10x + 16:
f(x) = x² - 10x + 16 f(x) = (x² - 10x) + 16
To complete the square, we take half of the coefficient of the x term (-10), which is -5, and square it, which gives us 25. We then add and subtract 25 inside the parenthesis:
f(x) = (x² - 10x + 25 - 25) + 16 f(x) = (x² - 10x + 25) - 25 + 16 f(x) = (x - 5)² - 9
This is the vertex form of the quadratic function, f(x) = a(x - h)² + k, where h = 5 and k = -9. This confirms that the vertex is indeed (5, -9), matching our previous result.
2. Graphical Interpretation
Visualizing the graph of the quadratic function f(x) = x² - 10x + 16 can provide further confirmation of our result. The graph of this function is a parabola that opens upwards. The vertex, (5, -9), is the lowest point on the parabola. The parabola is symmetrical around the vertical line x = 5, which passes through the vertex. The x-intercepts of the graph are the solutions to the equation f(x) = 0, which are x = 2 and x = 8, as given in the original factored form f(x) = (x - 8)(x - 2). These observations align with our calculated vertex and the properties of quadratic functions, reinforcing the accuracy of our solution.
Conclusion: Significance of the Vertex
In conclusion, we have successfully determined the vertex of the quadratic function f(x) = (x-8)(x-2) using both the formula h = -b / 2a and the method of completing the square. The vertex, (5, -9), represents the minimum point of the parabola described by this function. This exercise demonstrates the importance of understanding quadratic functions and their properties, particularly the significance of the vertex.
The vertex is a critical feature of a quadratic function, offering insights into its behavior and characteristics. It represents the point where the function attains its minimum or maximum value, making it invaluable in optimization problems. Moreover, the vertex defines the axis of symmetry of the parabola, providing a sense of balance and predictability to the graph. The ability to find the vertex efficiently, whether through formulaic methods or algebraic manipulation, is a fundamental skill in mathematics.
Understanding the vertex also allows us to sketch the graph of a quadratic function more accurately. Knowing the vertex, the direction the parabola opens (upwards or downwards), and the x-intercepts (if any) provides a comprehensive picture of the function's behavior. This graphical understanding can be particularly useful in visualizing solutions to quadratic equations and inequalities.
The methods and concepts discussed in this article extend beyond the specific function f(x) = (x-8)(x-2). They are applicable to any quadratic function, regardless of its form or coefficients. Mastering these techniques empowers us to analyze and solve a wide range of mathematical problems involving quadratic relationships. The vertex, therefore, serves as a cornerstone in the study of quadratic functions, providing both practical utility and theoretical insights into the nature of these essential mathematical constructs.
