First Step To Vertex Form Rewriting Y=-4x²+2x-7

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Hey guys! Today, we're diving deep into the fascinating world of quadratic equations, specifically how to transform them into the ever-so-useful vertex form. If you've ever wondered how to find the vertex of a parabola quickly or understand the maximum or minimum value of a quadratic function, then you're in the right place. We're going to break down the process step-by-step, making it super easy to grasp. Our focus? Rewriting the quadratic equation y = -4x² + 2x - 7 into the vertex form: y = a(x - h)² + k. Let's get started!

Understanding Vertex Form

Before we jump into the nitty-gritty, let's chat a bit about why vertex form is so awesome. The vertex form, y = a(x - h)² + k, is a fantastic way to represent a quadratic equation because it immediately reveals the vertex of the parabola, which is the point (h, k). The vertex is crucial because it represents the maximum or minimum point of the parabola, depending on whether the parabola opens upwards (a > 0) or downwards (a < 0). Knowing the vertex, we can quickly sketch the graph of the parabola and understand its key characteristics.

But that's not all! The 'a' in the vertex form tells us about the parabola's shape – specifically, whether it's wider or narrower compared to the standard parabola y = x². If |a| > 1, the parabola is narrower, and if |a| < 1, it's wider. So, by transforming a quadratic equation into vertex form, we unlock a treasure trove of information about its graph and behavior. This is super useful in many real-world applications, from physics problems involving projectile motion to optimizing business profits.

Think about it: if you're trying to figure out the maximum height a ball will reach when thrown or the minimum cost to produce a certain number of items, vertex form is your best friend. It cuts through the complexity and gives you the key information you need right at your fingertips. So, understanding how to get an equation into this form is a skill well worth mastering. Trust me, once you get the hang of it, you'll be amazed at how much easier quadratic equations become. It's like having a secret decoder ring for parabolas!

The Million-Dollar Question: What's the First Step?

Alright, let's tackle the question at hand: What's the very first move when rewriting y = -4x² + 2x - 7 into vertex form? We've got four options on the table, each with its own twist. To find the correct path, we need to understand the overall strategy for converting to vertex form. The main idea is to complete the square, which involves creating a perfect square trinomial within the equation. This might sound intimidating, but it's totally doable once we break it down.

So, let's look at our options:

  • A. 2 must be factored from 2x - 7: This option is a bit misleading. While you could factor 2 from 2x - 7, it's not the most strategic first step for completing the square. We're aiming to isolate the x² and x terms to create our perfect square, and this option doesn't quite get us there.
  • B. -4 must be factored from -4x² + 2x: Bingo! This is the winner. Factoring out the coefficient of the x² term (in this case, -4) from the x² and x terms is the crucial first step in completing the square. This sets us up to create that perfect square trinomial we talked about earlier. By factoring out -4, we're essentially normalizing the x² term, making the subsequent steps much smoother. This is like laying the foundation for a sturdy building – you've got to get it right from the start!
  • C. x must be factored from -4x² + 2x: While factoring out x is a valid algebraic manipulation, it doesn't directly lead us towards completing the square. It's more useful for finding the roots of the equation, but not for transforming it into vertex form. So, while it's a good technique to have in your toolbox, it's not the right one for this particular job.
  • D. -4 must be factored from -4x² - 7: This option misses the mark because it leaves out the 2x term, which is essential for completing the square. We need to work with the x² and x terms together to create our perfect square trinomial. Factoring -4 from just -4x² - 7 would leave us with a mixed bag that's not conducive to our goal.

So, the correct answer, without a doubt, is B. -4 must be factored from -4x² + 2x. This step is the key to unlocking the rest of the transformation process. It's like finding the right key to open a door – once you have it, the path forward becomes clear.

Why Factoring Out -4 is the Magic Move

Now that we've identified the first step, let's delve deeper into why factoring out -4 from -4x² + 2x is so important. This step is all about setting the stage for completing the square. When we factor out -4, we're essentially making the coefficient of the x² term equal to 1 inside the parentheses. This is crucial because completing the square works most smoothly when the leading coefficient (the coefficient of x²) is 1.

Think of it like this: we're preparing the ingredients for a perfect recipe. One of the key ingredients is having the x² term