Find The Vertex Of F(x) = 2x² - 12x + 7 A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of quadratic functions and tackling a common problem: finding the vertex. Specifically, we're going to break down how to find the vertex of the quadratic function f(x) = 2x² - 12x + 7. Don't worry, it's not as scary as it looks! We'll go through it step-by-step, making sure you understand the process inside and out. Whether you're a student prepping for a test, a math enthusiast, or just curious, this guide is for you. Let's get started!

Understanding Quadratic Functions and the Vertex

Before we jump into the calculations, let's make sure we're all on the same page about quadratic functions and the vertex. At its core, a quadratic function is a polynomial function of degree two. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic function is expressed as f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (otherwise, it wouldn't be quadratic!). Our specific function, f(x) = 2x² - 12x + 7, perfectly fits this form, with a = 2, b = -12, and c = 7. Identifying these coefficients is the first crucial step in our journey to find the vertex.

Now, what exactly is the vertex? Think of a quadratic function's graph as a parabola – a U-shaped curve. This parabola can either open upwards (if 'a' is positive) or downwards (if 'a' is negative). The vertex is the turning point of this parabola; it's the point where the curve changes direction. If the parabola opens upwards, the vertex is the lowest point, representing the minimum value of the function. Conversely, if the parabola opens downwards, the vertex is the highest point, representing the maximum value. Understanding this visual representation is key to grasping the significance of the vertex.

The vertex is not just a point on a graph; it holds valuable information about the quadratic function. Its x-coordinate tells us the axis of symmetry – the vertical line that divides the parabola into two symmetrical halves. The y-coordinate of the vertex tells us the minimum or maximum value of the function, depending on whether the parabola opens upwards or downwards. In many real-world applications, finding the vertex is essential for optimization problems, such as maximizing profit, minimizing costs, or determining the trajectory of a projectile. So, learning how to find the vertex is a powerful tool in your mathematical arsenal.

Understanding the roles of 'a', 'b', and 'c' in the quadratic equation is paramount. The coefficient 'a' dictates the parabola's direction and how wide or narrow it is. A positive 'a' means the parabola opens upwards, and a larger absolute value of 'a' makes the parabola narrower. A negative 'a' flips the parabola downwards. The coefficient 'b' influences the position of the parabola's axis of symmetry, and 'c' represents the y-intercept – the point where the parabola crosses the y-axis. All these elements work together to define the shape and position of the quadratic function's graph, and mastering their individual contributions is crucial for working with quadratics effectively. Now that we have a solid understanding of quadratic functions and the vertex, let's move on to the methods for finding it.

Method 1: Using the Vertex Formula

The most direct method for finding the vertex of a quadratic function is by using the vertex formula. This formula provides a straightforward way to calculate the coordinates of the vertex directly from the coefficients of the quadratic function. Remember our general form: f(x) = ax² + bx + c? The vertex formula leverages these coefficients to pinpoint the vertex's location. The vertex formula is given by: Vertex = (h, k), where h = -b / 2a and k = f(h). Let's break down what this means and how to apply it to our specific function, f(x) = 2x² - 12x + 7.

First, let's calculate the x-coordinate of the vertex (h). Looking at the formula h = -b / 2a, we need to identify 'a' and 'b' from our function. As we established earlier, a = 2 and b = -12. Plugging these values into the formula, we get: h = -(-12) / (2 * 2) = 12 / 4 = 3. So, the x-coordinate of our vertex is 3. This tells us that the axis of symmetry for our parabola is the vertical line x = 3. Remember, the axis of symmetry divides the parabola perfectly in half, so the vertex lies directly on this line.

Next, we need to find the y-coordinate of the vertex (k). The formula tells us that k = f(h), which means we need to substitute the value we just found for 'h' (which is 3) back into our original function, f(x) = 2x² - 12x + 7. So, k = f(3) = 2(3)² - 12(3) + 7. Let's simplify this: k = 2(9) - 36 + 7 = 18 - 36 + 7 = -11. Therefore, the y-coordinate of our vertex is -11. This means that the vertex point is located 3 units to the right on the x-axis and 11 units down on the y-axis. Because 'a' is positive (a = 2), this vertex represents the minimum point of the parabola.

Now that we have both the x-coordinate (h = 3) and the y-coordinate (k = -11), we can confidently state that the vertex of the quadratic function f(x) = 2x² - 12x + 7 is (3, -11). This method, using the vertex formula, is often the quickest and most efficient way to find the vertex, especially when you're comfortable with algebraic manipulations. It's a valuable tool to have in your mathematical toolkit. The vertex formula offers a direct path to the solution, minimizing the steps required and reducing the chances of errors. By understanding the formula and practicing its application, you can confidently tackle vertex-finding problems. Now, let's explore another method for finding the vertex: completing the square.

Method 2: Completing the Square

Another powerful technique for finding the vertex of a quadratic function is completing the square. This method involves rewriting the quadratic function in vertex form, which directly reveals the vertex coordinates. While it might seem a bit more involved than the vertex formula at first, completing the square is a fundamental algebraic technique with applications beyond just finding vertices. It helps deepen your understanding of quadratic functions and their properties. Let's walk through the process for f(x) = 2x² - 12x + 7.

The first step in completing the square is to factor out the coefficient of the x² term (our 'a' value) from the first two terms of the quadratic expression. In our case, a = 2, so we factor out 2 from 2x² - 12x, giving us: f(x) = 2(x² - 6x) + 7. Notice that we've left the constant term (+7) outside the parentheses for now. This step is crucial because it sets us up to create a perfect square trinomial inside the parentheses.

Now comes the core of the completing the square method: turning the expression inside the parentheses (x² - 6x) into a perfect square trinomial. To do this, we take half of the coefficient of our x term (-6), square it, and add it inside the parentheses. Half of -6 is -3, and (-3)² is 9. So, we need to add 9 inside the parentheses. However, because we're inside the parentheses that are being multiplied by 2, we're actually adding 2 * 9 = 18 to the overall expression. To keep the equation balanced, we must also subtract 18 outside the parentheses. This gives us: f(x) = 2(x² - 6x + 9) + 7 - 18.

Next, we rewrite the expression inside the parentheses as a squared binomial. The expression x² - 6x + 9 is a perfect square trinomial because it can be factored as (x - 3)². So, our function now looks like this: f(x) = 2(x - 3)² - 11. This is the vertex form of the quadratic function! The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) is the vertex. By simply comparing our transformed function to the vertex form, we can directly read off the vertex coordinates.

In our case, f(x) = 2(x - 3)² - 11, we can see that h = 3 and k = -11. Therefore, the vertex of the quadratic function f(x) = 2x² - 12x + 7 is (3, -11). This matches the result we obtained using the vertex formula, which is a great confirmation that we've done the process correctly! Completing the square not only gives us the vertex but also transforms the function into a form that reveals other important information, such as the parabola's axis of symmetry and the minimum or maximum value of the function.

Comparing the Methods and Choosing the Right One

We've explored two distinct methods for finding the vertex of a quadratic function: the vertex formula and completing the square. Both methods are valid and will lead you to the correct answer, but they have different strengths and weaknesses. Understanding these differences can help you choose the most efficient method for a given problem. So, let's break down the comparison and discuss when to use each method.

The vertex formula (h = -b / 2a, k = f(h)) is generally the quicker and more direct method, especially when you just need to find the vertex and don't require the vertex form of the equation. It involves a straightforward calculation using the coefficients of the quadratic function, making it less prone to errors if you're comfortable with algebraic manipulations. If time is of the essence, or if you're dealing with a complex quadratic function, the vertex formula might be your go-to choice. However, the vertex formula doesn't provide the function in vertex form, which can be useful for other purposes, such as graphing the parabola or analyzing its properties.

On the other hand, completing the square is a more involved method, but it offers a deeper understanding of the quadratic function and its transformations. It's particularly useful when you need the vertex form of the equation, f(x) = a(x - h)² + k, as this form directly reveals the vertex (h, k) and the vertical stretch factor (a). Completing the square is also a valuable algebraic technique in its own right, with applications beyond finding vertices, such as solving quadratic equations and simplifying expressions. While it takes more steps and can be more challenging to master, the benefits of completing the square extend beyond just finding the vertex.

So, how do you choose the right method? If you only need the vertex coordinates and speed is a priority, the vertex formula is likely the better choice. If you need the vertex form of the equation or want a more thorough understanding of the quadratic function's structure, completing the square is the way to go. Also, consider your personal strengths and preferences. If you're comfortable with algebraic manipulations and memorizing formulas, the vertex formula might feel more natural. If you prefer a step-by-step process that reveals the underlying structure of the function, completing the square might be a better fit.

Ultimately, the best approach is to master both methods. Knowing both techniques gives you flexibility and a deeper understanding of quadratic functions. Practice both methods with various examples, and you'll develop a sense of which method is most appropriate for different situations. In our example, f(x) = 2x² - 12x + 7, both methods led us to the same vertex (3, -11), reinforcing the validity of both approaches. Now that we've explored the methods and their comparisons, let's summarize our findings and reinforce the key takeaways.

Conclusion

Alright, guys! We've journeyed through the world of quadratic functions and conquered the challenge of finding the vertex. We started with understanding what quadratic functions and the vertex are, emphasizing the importance of the vertex as the turning point of the parabola and its connection to the function's minimum or maximum value. Then, we dived into two powerful methods for finding the vertex: using the vertex formula and completing the square.

The vertex formula, with its direct calculation (h = -b / 2a, k = f(h)), offers a quick and efficient way to determine the vertex coordinates. It's a great tool when you need the vertex fast and don't require the vertex form of the equation. We applied the formula to our example, f(x) = 2x² - 12x + 7, and effortlessly found the x-coordinate (h = 3) and y-coordinate (k = -11), leading us to the vertex (3, -11).

Completing the square, while more involved, provides a deeper understanding of quadratic functions and transforms the function into vertex form, f(x) = a(x - h)² + k. This form directly reveals the vertex (h, k) and offers insights into the parabola's shape and position. We meticulously walked through the steps of completing the square for f(x) = 2x² - 12x + 7, ultimately arriving at the vertex form f(x) = 2(x - 3)² - 11 and confirming our vertex as (3, -11).

We then compared the two methods, highlighting their strengths and weaknesses. The vertex formula excels in speed and directness, while completing the square offers a deeper understanding and provides the vertex form. The choice between the methods depends on the specific problem, your personal preferences, and the desired level of understanding.

In conclusion, finding the vertex of a quadratic function is a fundamental skill with wide-ranging applications. By mastering both the vertex formula and completing the square, you'll be well-equipped to tackle quadratic function problems with confidence. Remember, practice makes perfect! So, grab some more quadratic functions and put these methods to the test. Keep exploring, keep learning, and keep having fun with math! You've got this!