Find The Vertex Of A Parabola: 5x^2 - 30x + 49

Understanding Parabolas and Their Vertices

Parabolas are fascinating curves that appear in many areas of mathematics and physics, from the trajectory of a thrown ball to the shape of satellite dishes. At the heart of every parabola lies its vertex, a crucial point that represents either the minimum or maximum value of the function. For a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the vertex is the point where the parabola changes direction. If the coefficient 'a' is positive, the parabola opens upwards, and the vertex is the lowest point. If 'a' is negative, the parabola opens downwards, and the vertex is the highest point. Understanding how to find the vertex is fundamental to analyzing and graphing quadratic functions. This article will guide you through the process of finding the vertex for the specific equation $f(x) = 5x^2 - 30x + 49$, and we'll explore the different methods you can use. We'll break down each step clearly, ensuring that you can confidently determine the vertex of any parabola. Whether you're a student tackling algebra homework or simply curious about the properties of quadratic equations, this explanation is designed to be accessible and informative. We'll cover the formulaic approach and touch upon the intuition behind it, making the concept of the vertex more concrete. So, let's dive in and unravel the mystery of the parabolic vertex!

Method 1: Using the Vertex Formula

One of the most direct ways to find the vertex of a parabola given by the equation $f(x) = ax^2 + bx + c$ is by using the vertex formula. This formula is derived from the process of completing the square, but it offers a shortcut for pinpointing the coordinates of the vertex. The x-coordinate of the vertex is given by the formula $x = -b / (2a)$. Once you have the x-coordinate, you can find the corresponding y-coordinate by substituting this value back into the original function, i.e., $y = f(-b / (2a))$. Let's apply this to our specific equation: $f(x) = 5x^2 - 30x + 49$. In this equation, we can identify our coefficients: $a = 5$, $b = -30$, and $c = 49$. Now, let's plug these values into the x-coordinate formula: $x = -(-30) / (2 * 5)$. Simplifying this, we get $x = 30 / 10$, which equals $x = 3$. So, the x-coordinate of our vertex is 3. To find the y-coordinate, we substitute $x = 3$ back into our original function: $f(3) = 5(3)^2 - 30(3) + 49$. Let's calculate this: $f(3) = 5(9) - 90 + 49$. This simplifies to $f(3) = 45 - 90 + 49$. Performing the subtraction and addition, we get $f(3) = -45 + 49$, which results in $f(3) = 4$. Therefore, the vertex of the parabola $f(x) = 5x^2 - 30x + 49$ is at the coordinates (3, 4). This method is efficient and widely applicable for any quadratic function in standard form. Remember to pay close attention to the signs of your coefficients, as a simple sign error can lead to an incorrect vertex.

Method 2: Completing the Square

Another powerful method for finding the vertex of a parabola is by completing the square. This technique transforms the standard form of the quadratic equation, $f(x) = ax^2 + bx + c$, into the vertex form, $f(x) = a(x - h)^2 + k$. In the vertex form, the coordinates of the vertex are directly given by $(h, k)$. While this method might seem more involved than the formulaic approach, it provides a deeper understanding of the parabola's structure and is essential for other mathematical concepts. Let's start with our equation: $f(x) = 5x^2 - 30x + 49$. The first step is to factor out the coefficient 'a' (which is 5 in this case) from the terms involving x: $f(x) = 5(x^2 - 6x) + 49$. Now, we focus on the expression inside the parentheses, $x^2 - 6x$. To complete the square for this expression, we take half of the coefficient of the x term (-6), square it, and add and subtract it inside the parentheses. Half of -6 is -3, and squaring it gives us 9. So, we add and subtract 9 inside the parentheses: $f(x) = 5(x^2 - 6x + 9 - 9) + 49$. Now, we can group the first three terms inside the parentheses, which form a perfect square trinomial: $f(x) = 5((x - 3)^2 - 9) + 49$. Next, we distribute the 5 back into the parentheses: $f(x) = 5(x - 3)^2 - 5(9) + 49$. This simplifies to $f(x) = 5(x - 3)^2 - 45 + 49$. Finally, combine the constant terms: $f(x) = 5(x - 3)^2 + 4$. We have successfully transformed the equation into vertex form, $f(x) = a(x - h)^2 + k$. By comparing $f(x) = 5(x - 3)^2 + 4$ with the general vertex form, we can identify $a = 5$, $h = 3$, and $k = 4$. Therefore, the vertex of the parabola is at the coordinates (3, 4). This method not only reveals the vertex but also shows how the original function is a transformation of the basic $y = x^2$ graph.

Analyzing the Options and Confirming the Vertex

Now that we have meticulously calculated the vertex of the parabola $f(x) = 5x^2 - 30x + 49$ using two different methods, it's time to examine the provided options and confirm our findings. The options given were: a. $(3,0)$, b. $(3,4)$, c. $(4,0)$, d. $(4,3)$. Our calculations consistently showed that the vertex of the parabola is at the coordinates (3, 4). Let's review why this is the correct answer. In Method 1, using the vertex formula $x = -b / (2a)$, we found the x-coordinate to be 3. Substituting this value back into the function $f(x)$ yielded a y-coordinate of 4. Thus, the vertex is (3, 4). In Method 2, by completing the square, we transformed the equation into its vertex form $f(x) = 5(x - 3)^2 + 4$. In this form, the vertex is directly observable as $(h, k)$, where $h = 3$ and $k = 4$. Both approaches converge on the same result, providing strong evidence that (3, 4) is indeed the correct vertex. Comparing this with the given options, we can see that option b. (3,4) perfectly matches our calculated vertex. The other options are incorrect because their x or y coordinates do not align with the values derived from the quadratic equation's coefficients. For instance, option a. $(3,0)$ has the correct x-coordinate but an incorrect y-coordinate. Option c. $(4,0)$ and d. $(4,3)$ have incorrect x-coordinates altogether. It's always a good practice to double-check your work, especially when multiple methods yield the same result, as it significantly increases the confidence in your answer. This confirmation step is crucial in problem-solving to ensure accuracy.

Conclusion: Mastering Parabola Vertex Calculation

In conclusion, finding the vertex of a parabola is a fundamental skill in understanding quadratic functions. We've explored two reliable methods for determining the vertex of $f(x) = 5x^2 - 30x + 49$: the vertex formula and completing the square. Both methods led us to the same correct answer: the vertex is located at (3, 4). The vertex formula, $x = -b / (2a)$, provides a quick way to find the x-coordinate, which is then substituted back into the function to find the y-coordinate. Completing the square transforms the quadratic equation into its vertex form, $f(x) = a(x - h)^2 + k$, from which the vertex $(h, k)$ can be directly read. Understanding these methods empowers you to analyze the shape and position of any parabola. The vertex is a key feature that helps in sketching the graph and identifying the minimum or maximum value of the function. We confirmed that option b. (3,4) is the correct answer among the given choices. Practice with various quadratic equations will further solidify your understanding and speed up your ability to find the vertex. For more in-depth exploration of quadratic functions and their properties, you can visit Khan Academy's section on quadratic functions or delve into resources on Paul's Online Math Notes for Algebra.