Transforming Quadratic Functions To Vertex Form And Finding The Vertex For F(x)=2x^2-16x+34

Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically how to transform them into the vertex form. This form, ${f(x) = a(x - h)^2 + k}$, is super useful because it instantly reveals the vertex of the parabola, which is a key feature of any quadratic graph. We'll be tackling the function ${f(x) = 2x^2 - 16x + 34}$ and breaking down each step so you can master this skill too. So, let's get started!

Understanding the Vertex Form

Before we jump into the nitty-gritty, let's make sure we're all on the same page about the vertex form. The vertex form of a quadratic function is expressed as ${f(x) = a(x - h)^2 + k}$. In this form:

• ${a}$ determines the direction and steepness of the parabola. If ${a}$ is positive, the parabola opens upwards, and if it's negative, it opens downwards. The larger the absolute value of ${a}$, the steeper the parabola.
• ${(h, k)}$ represents the vertex of the parabola. The vertex is the point where the parabola changes direction – it's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards).
• ${h}$ is the x-coordinate of the vertex, and ${k}$ is the y-coordinate of the vertex. Knowing these coordinates is crucial for graphing and understanding the behavior of the quadratic function.

Why is the vertex form so important? Well, it makes it incredibly easy to identify the vertex of the parabola. This is super helpful for various applications, such as finding the maximum height of a projectile, determining the minimum cost in a business model, or optimizing any scenario that can be modeled by a quadratic function. So, understanding how to convert a quadratic function into vertex form is a powerful tool in your mathematical arsenal.

Now, you might be wondering, "How do we actually get there?" The main technique we'll use is called completing the square. It might sound intimidating, but don't worry, we'll break it down into simple steps. Completing the square allows us to rewrite the quadratic expression in a way that highlights the squared term, which is what we need for the vertex form. Think of it as rearranging the furniture in a room to make the most important piece – the vertex – stand out.

Step-by-Step Conversion of ${f(x) = 2x^2 - 16x + 34}$ into Vertex Form

Alright, let's get our hands dirty and transform ${f(x) = 2x^2 - 16x + 34}$ into vertex form. This might seem like a daunting task at first, but by breaking it down into manageable steps, you'll see it's totally doable. We're going to use the method of completing the square, which is a fundamental technique for rewriting quadratic expressions.

Step 1: Factor out the Leading Coefficient

The first thing we need to do is factor out the leading coefficient (the number in front of the ${x^2}$ term) from the first two terms of the quadratic expression. In our case, the leading coefficient is 2. Factoring it out, we get:

${f(x) = 2(x^2 - 8x) + 34}$

Notice that we're only factoring out the 2 from the terms with ${x^2}$ and ${x}$. The constant term, 34, stays outside the parentheses for now. This step is crucial because it sets us up to create a perfect square trinomial inside the parentheses. Think of it like prepping the ingredients before you start cooking – we need to get everything in the right form before we can transform it.

Step 2: Completing the Square

This is where the magic happens! We need to figure out what constant to add inside the parentheses to make the expression a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form ${(x + a)^2}$ or ${(x - a)^2}$. To find this constant, we take half of the coefficient of the ${x}$ term (inside the parentheses), square it, and add it inside the parentheses.

In our case, the coefficient of the ${x}$ term inside the parentheses is -8. Half of -8 is -4, and squaring -4 gives us 16. So, we need to add 16 inside the parentheses. But here's the catch: we're not just adding 16 to the function; we're adding 16 inside the parentheses, which are being multiplied by 2. So, we're actually adding 2 * 16 = 32 to the function. To keep the function balanced, we need to subtract 32 outside the parentheses as well. This is a crucial step to maintain the equality of the expression.

Let's break that down in the equation:

${f(x) = 2(x^2 - 8x + 16) + 34 - 32}$

Now, the expression inside the parentheses, ${x^2 - 8x + 16}$, is a perfect square trinomial. It can be factored as ${(x - 4)^2}$. This is the heart of the completing the square method – we've transformed a regular quadratic expression into a perfect square, which is essential for getting to the vertex form. Think of it as finding the missing piece of a puzzle that makes the whole picture complete.

Step 3: Rewrite as Vertex Form

Now that we've completed the square, we can rewrite the function in vertex form. We'll replace the perfect square trinomial with its factored form and simplify the constant terms:

${f(x) = 2(x - 4)^2 + 34 - 32}$ ${f(x) = 2(x - 4)^2 + 2}$

And there you have it! We've successfully transformed the original quadratic function into vertex form. Notice how the structure of the equation now clearly reveals the vertex and the direction of the parabola. This is the power of the vertex form – it provides a clear snapshot of the key features of the quadratic function.

Identifying the Vertex

Now that we have the function in vertex form, ${f(x) = 2(x - 4)^2 + 2}$, identifying the vertex is a piece of cake! Remember, the vertex form is ${f(x) = a(x - h)^2 + k}$, where ${(h, k)}$ is the vertex. By comparing our equation with the vertex form, we can directly read off the coordinates of the vertex.

In our case, ${h = 4}$ and ${k = 2}$. So, the vertex of the parabola is ${(4, 2)}$. This means the parabola reaches its minimum (or maximum, depending on the sign of ${a}$) at the point (4, 2). Since ${a = 2}$, which is positive, the parabola opens upwards, and the vertex represents the minimum point of the function.

The vertex is a crucial point on the graph of the quadratic function. It's the point where the parabola changes direction, and it provides valuable information about the function's behavior. For example, if we were modeling the height of a ball thrown in the air, the vertex would represent the maximum height the ball reaches. Or, if we were modeling the profit of a business, the vertex could represent the point of maximum profit. Understanding how to find and interpret the vertex is therefore essential for applying quadratic functions to real-world problems.

Putting It All Together

Let's recap what we've done. We started with the quadratic function ${f(x) = 2x^2 - 16x + 34}$ and our mission was to rewrite it in vertex form and identify the vertex. We successfully navigated the steps of completing the square:

1. Factored out the leading coefficient from the ${x^2}$ and ${x}$ terms.
2. Completed the square by adding and subtracting the appropriate constant.
3. Rewrote the function in vertex form: ${f(x) = 2(x - 4)^2 + 2}$.
4. Identified the vertex as ${(4, 2)}$.

By following these steps, we not only transformed the function into vertex form but also gained a deeper understanding of its properties. We now know that the parabola opens upwards (because ${a = 2}$ is positive) and that its minimum point is at ${(4, 2)}$. This kind of information is invaluable for graphing the function, solving related problems, and applying quadratic functions in various contexts.

Practice Makes Perfect

Now that you've seen how to convert a quadratic function into vertex form, it's time to put your knowledge to the test. The best way to master this skill is to practice, practice, practice! Try converting other quadratic functions into vertex form and identifying their vertices. You can find plenty of examples in textbooks, online resources, or even create your own. The more you practice, the more comfortable and confident you'll become with this technique.

Remember, completing the square can seem tricky at first, but with each problem you solve, you'll develop a better understanding of the process. Don't be afraid to make mistakes – they're a natural part of learning. And if you get stuck, don't hesitate to review the steps we've covered or seek help from a teacher, tutor, or online community.

The ability to rewrite quadratic functions in vertex form is a powerful tool in your mathematical toolkit. It not only helps you understand the behavior of quadratic functions but also provides a foundation for solving a wide range of problems. So, keep practicing, keep exploring, and keep having fun with math!

Conclusion

In this comprehensive guide, we've unraveled the mystery of converting quadratic functions into vertex form. We took the function ${f(x) = 2x^2 - 16x + 34}$ and, through the magic of completing the square, transformed it into ${f(x) = 2(x - 4)^2 + 2}$. We then confidently identified the vertex as ${(4, 2)}$.

By mastering this technique, you've unlocked a powerful way to analyze and understand quadratic functions. The vertex form provides a clear picture of the parabola's vertex, direction, and overall behavior. This knowledge is not only essential for academic success but also for applying quadratic functions in real-world scenarios.

So, keep honing your skills, embrace the challenges, and remember that with practice and perseverance, you can conquer any mathematical hurdle. Happy calculating, guys!