Vertex Coordinates: Quadratic Function Example

Let's dive into how to find the vertex coordinates of a quadratic function. This is a fundamental concept in algebra, and understanding it can help you solve a variety of problems. In this article, we'll use the example function f(x) = -2x² + 8x - 9 to guide you through the process. So, grab your calculator, and let's get started!

Understanding Quadratic Functions and the Vertex

Before we jump into the calculations, let's quickly recap what a quadratic function is and what the vertex represents. Quadratic functions are polynomial functions of 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. Now, where does the vertex come into play? The vertex is the point where the parabola changes direction. If the parabola opens upwards (when a > 0), the vertex is the minimum point. If the parabola opens downwards (when a < 0), the vertex is the maximum point. Understanding this is key to understanding the behavior of quadratic equations. Think of the vertex as the peak or the valley of the parabola – the single most extreme point on the graph. It’s this extreme point that we're going to learn how to find.

Why is the vertex so important? Well, it gives us a lot of information about the quadratic function. For example, it tells us the maximum or minimum value of the function, which can be useful in optimization problems. Imagine you're trying to maximize profit in a business; a quadratic function could model your profit, and the vertex would tell you the point at which your profit is highest. The vertex also gives us the axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. This symmetry can simplify graphing and analyzing quadratic functions. So, finding the vertex is not just a mathematical exercise; it's a tool that unlocks deeper insights into the function's behavior and real-world applications. In summary, the vertex is a critical point on the parabola that reveals valuable information about the quadratic function, including its maximum or minimum value and its axis of symmetry.

Method 1: Using the Vertex Formula

The most direct way to find the vertex coordinates is by using the vertex formula. This formula is derived from completing the square and provides a straightforward method for determining the vertex. For a quadratic function in the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex, often denoted as h, is given by the formula: h = -b / 2a. Once you have the x-coordinate, you can find the y-coordinate, often denoted as k, by substituting h back into the original function: k = f(h). Let’s break down this formula a bit more. The a and b in the formula h = -b / 2a are the coefficients from your quadratic equation. The coefficient a is the number in front of the x² term, and b is the number in front of the x term. This formula is a shortcut that saves you from having to complete the square every time you want to find the vertex. The x-coordinate h is the axis of symmetry, the vertical line that divides the parabola into two identical halves. Then, substituting h back into the original function, k = f(h), gives you the y-coordinate of the vertex. This y-coordinate represents the maximum or minimum value of the function, depending on whether the parabola opens upwards or downwards. If a is positive, the parabola opens upwards, and k is the minimum value. If a is negative, the parabola opens downwards, and k is the maximum value. Therefore, using the vertex formula is a reliable and efficient method to pinpoint the most crucial point on the parabola.

Applying this formula to our example function, f(x) = -2x² + 8x - 9, we first identify the coefficients: a = -2 and b = 8. Plugging these values into the formula h = -b / 2a, we get: h = -8 / (2 * -2) = -8 / -4 = 2. So, the x-coordinate of the vertex is 2. Now, to find the y-coordinate, k, we substitute x = 2 back into the original function: f(2) = -2(2)² + 8(2) - 9 = -2(4) + 16 - 9 = -8 + 16 - 9 = -1. Therefore, the y-coordinate of the vertex is -1. Combining these results, we find that the vertex coordinates are (2, -1). This means that the highest point on the parabola represented by the function f(x) = -2x² + 8x - 9 is located at the point (2, -1). Because a is negative (-2), the parabola opens downwards, and this vertex represents the maximum value of the function. So, the vertex formula provides a clear and effective way to determine the critical point of a quadratic function, allowing us to analyze its behavior and properties. In summary, by identifying the coefficients a and b, applying the formula h = -b / 2a, and substituting h back into the function to find k, we can easily determine the vertex coordinates of any quadratic function.

Method 2: Completing the Square

Another method to find the vertex is by completing the square. This technique transforms the quadratic function into vertex form, which directly reveals the vertex coordinates. The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) are the coordinates of the vertex. Completing the square involves manipulating the original quadratic equation to fit this form. It's a bit more involved than using the vertex formula, but it's a powerful technique that can be used in other algebraic contexts as well. The process of completing the square might seem a bit abstract at first, but it’s a systematic way to rewrite the quadratic function. The goal is to create a perfect square trinomial, which can then be factored into a squared binomial. This transformation allows us to clearly see the vertex coordinates in the equation. Think of it as rearranging the terms and adding a specific constant to both sides of the equation to force it into the vertex form. The constant we add is determined by the coefficient of the x term and the coefficient of the x² term. Once we have the equation in vertex form, f(x) = a(x - h)² + k, identifying the vertex (h, k) is straightforward. The h value is the number being subtracted from x inside the parentheses, and the k value is the constant term added outside the parentheses. So, completing the square not only helps us find the vertex but also provides a deeper understanding of the structure of the quadratic function. This method highlights the relationship between the standard form and the vertex form of a quadratic equation, making it a valuable tool in your algebraic toolkit.

Let's apply this method to our function, f(x) = -2x² + 8x - 9. First, factor out the coefficient of the x² term, which is -2, from the first two terms: f(x) = -2(x² - 4x) - 9. Next, we need to complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (-4), which is -2, and square it: (-2)² = 4. Now, add and subtract this value inside the parentheses: f(x) = -2(x² - 4x + 4 - 4) - 9. Notice that we added and subtracted the same value, so we haven't changed the equation. We can now rewrite the expression inside the parentheses as a perfect square: f(x) = -2((x - 2)² - 4) - 9. Distribute the -2 to both terms inside the parentheses: f(x) = -2(x - 2)² + 8 - 9. Finally, simplify the equation: f(x) = -2(x - 2)² - 1. Now, the function is in vertex form, f(x) = a(x - h)² + k, where a = -2, h = 2, and k = -1. Therefore, the vertex coordinates are (2, -1), which matches the result we obtained using the vertex formula. Completing the square may seem like a longer process, but it provides a solid understanding of how quadratic functions can be transformed and analyzed. In summary, by factoring, completing the square, and rewriting the function in vertex form, we can directly identify the vertex coordinates, reinforcing the concept of quadratic function transformations and their graphical representations.

Conclusion

In this article, we explored two methods for finding the vertex coordinates of a quadratic function: using the vertex formula and completing the square. Both methods are valuable tools for analyzing quadratic functions and understanding their graphical representations. Whether you prefer the direct approach of the vertex formula or the step-by-step method of completing the square, mastering these techniques will enhance your understanding of quadratic functions and their applications. Remember, the vertex represents the maximum or minimum point of the parabola, providing crucial information about the function's behavior and properties. Keep practicing these methods, and you'll become proficient in finding the vertex of any quadratic function. For further learning on quadratic functions and their applications, you can check out resources like Khan Academy's Quadratic Functions Section.