Finding The Vertex Of Parabola Y = X² - 6x + 5

Hey guys! Ever wondered how to pinpoint the vertex of a parabola? It's a crucial point that reveals a lot about the parabola's behavior. Today, we're diving deep into the function y = x² - 6x + 5 to uncover its vertex and understand the methods to find it. Let's solve this together!

Understanding the Parabola and Its Vertex

Before we jump into the math, let's get a grip on what a parabola actually is. Think of it as a U-shaped curve, a graphical representation of a quadratic function. This curve opens upwards if the coefficient of x² is positive and downwards if it's negative. The vertex? That's the turning point of the parabola – the minimum point if the parabola opens upwards, or the maximum point if it opens downwards. This single point holds the key to the parabola's symmetry and its range of values. When we talk about finding the vertex, we're essentially looking for the coordinates (x, y) that pinpoint this crucial spot on the graph. Knowing the vertex not only helps us visualize the parabola but also provides valuable information about the function's behavior, such as its minimum or maximum value. Understanding parabolas and their vertices is fundamental in various fields, including physics, engineering, and economics, where quadratic functions are used to model a wide range of phenomena. So, grasping this concept is not just about solving math problems; it's about understanding the world around us in a more profound way.

Methods to Determine the Vertex

Now, let's talk strategies. How do we actually find this elusive vertex? There are a couple of main methods, and we'll explore both. The first involves using a formula, a direct and efficient way to calculate the vertex coordinates. The second method involves completing the square, a technique that transforms the quadratic equation into a vertex form, making the vertex coordinates immediately apparent. Let's dive into the first method, using the formula. This approach leverages the standard form of a quadratic equation, which is ax² + bx + c. From this form, we can extract the coefficients a, b, and c, which are the building blocks for our vertex formula. The x-coordinate of the vertex is given by -b/2a, a simple yet powerful formula. Once we have the x-coordinate, we can plug it back into the original quadratic equation to find the corresponding y-coordinate. This method is straightforward and reliable, especially when you need a quick solution. However, it's essential to remember the formula and apply it correctly, paying close attention to the signs of the coefficients. Now, let's consider the second method: completing the square. This technique might seem a bit more involved, but it offers a deeper understanding of the parabola's structure. Completing the square involves rewriting the quadratic equation in the form a(x - h)² + k, where (h, k) represents the vertex coordinates. This form directly reveals the vertex, making it easy to identify. The process involves manipulating the equation by adding and subtracting specific terms to create a perfect square trinomial. While it requires a bit more algebraic manipulation, completing the square provides valuable insights into the parabola's symmetry and its transformation from the basic quadratic function. So, whether you prefer the directness of the formula or the structural clarity of completing the square, understanding both methods will equip you with a robust toolkit for finding the vertex of any parabola.

Applying the Vertex Formula to y = x² - 6x + 5

Alright, let's get our hands dirty with the function y = x² - 6x + 5. Our mission is to find the vertex, and we're going to use the vertex formula method. Remember, the x-coordinate of the vertex is given by -b/2a. In our function, a is the coefficient of x², which is 1, and b is the coefficient of x, which is -6. Plugging these values into our formula, we get x = -(-6) / (2 * 1), which simplifies to x = 6 / 2, and finally, x = 3. So, the x-coordinate of our vertex is 3. Now, we need to find the y-coordinate. To do this, we substitute x = 3 back into the original equation: y = (3)² - 6(3) + 5. This simplifies to y = 9 - 18 + 5, which gives us y = -4. Therefore, the y-coordinate of our vertex is -4. Putting it all together, the vertex of the parabola represented by the function y = x² - 6x + 5 is (3, -4). This means the parabola's turning point is located at the point (3, -4) on the coordinate plane. If the parabola opens upwards, this point represents the minimum value of the function, and if it opens downwards, it represents the maximum value. In our case, since the coefficient of x² is positive (1), the parabola opens upwards, and (3, -4) is the minimum point. This also tells us that the axis of symmetry of the parabola is the vertical line x = 3, which passes through the vertex. The vertex and the axis of symmetry are fundamental features that help us understand and visualize the behavior of the parabola. Knowing the vertex allows us to sketch the graph of the parabola accurately and predict its values for different inputs.

Step-by-Step Calculation

Let's break down the calculation into simple steps, just to make sure we've got this nailed. First, we identify the coefficients in our quadratic equation: y = x² - 6x + 5. Here, a = 1, b = -6, and c = 5. Next, we apply the formula for the x-coordinate of the vertex: x = -b / 2a. Substituting our values, we get x = -(-6) / (2 * 1) = 6 / 2 = 3. So far, so good! Now, we need to find the y-coordinate. We do this by plugging x = 3 back into the original equation: y = (3)² - 6(3) + 5. Let's simplify this: y = 9 - 18 + 5. Doing the math, we get y = -4. And there you have it! The vertex of the parabola is (3, -4). This step-by-step approach helps to avoid errors and ensures that we understand each part of the process. It's like building a house: you lay the foundation first, then the walls, and finally the roof. Each step is crucial, and skipping one can lead to problems down the line. In this case, correctly identifying the coefficients and applying the formula in the right order are the foundation, and substituting the x-coordinate back into the equation is the roof that completes the process. By breaking it down, we can tackle even the most complex problems with confidence and precision. So, remember, when faced with a quadratic equation, take it one step at a time, and you'll find the vertex in no time!

Why (3, -4) is the Correct Vertex

So, we've landed on (3, -4) as the vertex. But why is this the correct answer? Let's think about what the vertex represents. It's the point where the parabola changes direction. For our parabola, y = x² - 6x + 5, this is the minimum point because the coefficient of x² is positive, meaning the parabola opens upwards. If we were to graph this function, we'd see that the lowest point on the curve is indeed at x = 3. Any value of x less than or greater than 3 will result in a higher y-value. This is a crucial characteristic of parabolas: they are symmetrical around a vertical line that passes through the vertex. This line is called the axis of symmetry, and its equation is x = h, where h is the x-coordinate of the vertex. In our case, the axis of symmetry is x = 3. This means that if we pick two x-values that are equidistant from x = 3, the corresponding y-values will be the same. For example, if we choose x = 2 (one unit to the left of 3) and x = 4 (one unit to the right of 3), we'll find that the y-values are the same. This symmetry reinforces the idea that (3, -4) is the turning point of the parabola. Furthermore, we can verify our result by completing the square. If we rewrite the equation y = x² - 6x + 5 in vertex form, which is y = a(x - h)² + k, where (h, k) is the vertex, we should arrive at the same vertex coordinates. Completing the square involves manipulating the equation to create a perfect square trinomial. This process confirms that (3, -4) is indeed the vertex, reinforcing our confidence in our solution.

Eliminating Other Options

Now, let's talk about the other options presented: B) (3, -1), C) (3, 4), and D) (3, 1). We know that the x-coordinate of the vertex is 3, so all these options have that part right. However, the y-coordinates are different, and only one can be correct. Let's consider option B) (3, -1). We already calculated that when x = 3, y = -4. So, (3, -1) is incorrect. Similarly, for option C) (3, 4), substituting x = 3 into the equation gives us y = -4, not 4. So, this option is also incorrect. Option D) (3, 1) suffers from the same fate. When x = 3, y is not 1; it's -4. So, option D) is incorrect as well. This process of elimination highlights the importance of accurate calculation and understanding the properties of a parabola. By carefully substituting values and comparing them with our calculated vertex, we can confidently rule out incorrect options. This is a valuable strategy in problem-solving, not just in mathematics but in many other areas of life. When faced with multiple choices, systematically evaluating each option and comparing it with known facts or calculated results can lead us to the correct answer. This approach not only helps us find the solution but also deepens our understanding of the underlying concepts. So, remember, elimination is a powerful tool in your problem-solving arsenal.

Conclusion: The Vertex is (3, -4)

In conclusion, guys, the vertex of the parabola represented by the function y = x² - 6x + 5 is indeed (3, -4). We found this by using the vertex formula, a reliable and efficient method. We also discussed why this is the correct answer, emphasizing the properties of a parabola and its vertex. And we even went through the process of eliminating the other options, highlighting the importance of careful calculation and logical reasoning. Understanding how to find the vertex of a parabola is a fundamental skill in algebra and calculus. It allows us to analyze and graph quadratic functions, which have numerous applications in real-world scenarios. From physics to engineering to economics, parabolas are used to model various phenomena, such as the trajectory of a projectile, the shape of a satellite dish, and the cost curves of a business. So, mastering this concept is not just about acing your math exams; it's about developing a valuable tool for understanding and solving problems in many different fields. Remember, practice makes perfect. The more you work with quadratic functions and parabolas, the more comfortable you'll become with finding the vertex and interpreting its significance. So, keep practicing, keep exploring, and keep asking questions. The world of mathematics is full of fascinating concepts waiting to be discovered, and the vertex of a parabola is just one small piece of the puzzle.

A) (3, -4) is the correct answer.