Vertex Form: Completing The Square For Y=x²-2x+7

Have you ever wondered how to transform a quadratic function into its vertex form? Understanding the vertex form can unlock valuable insights into the behavior of the parabola, revealing its vertex (the maximum or minimum point) and axis of symmetry. In this comprehensive guide, we'll walk through the process of completing the square to rewrite the quadratic function y = x² - 2x + 7 into vertex form. Let's dive in and make quadratic functions a breeze!

Understanding Vertex Form

Before we jump into the process, let's first understand what vertex form actually is. A quadratic function in vertex form looks like this:

y = a(x - h)² + k

Where:

• (h, k) represents the vertex of the parabola.
• a determines the direction the parabola opens (upward if a > 0, downward if a < 0) and its vertical stretch or compression.

The vertex form makes it incredibly easy to identify the vertex, which is a crucial point for understanding the graph of the quadratic function. Our goal is to transform the given equation, y = x² - 2x + 7, into this form.

Step-by-Step Guide to Completing the Square

Now, let's get our hands dirty with the actual process. We'll break it down into manageable steps, so you can follow along easily.

Step 1: Group the x terms

Our first step is to group the terms containing x together. This sets the stage for completing the square.

y = (x² - 2x) + 7

We've simply put parentheses around the x² and -2x terms. The + 7 remains outside the parentheses for now.

Step 2: Complete the Square

This is the heart of the process. To complete the square, we need to add and subtract a specific value inside the parentheses. This value is determined by taking half of the coefficient of our x term (which is -2 in this case), squaring it, and then adding and subtracting it. Let’s calculate:

1. Take half of the coefficient of the x term: -2 / 2 = -1
2. Square the result: (-1)² = 1

Now, we add and subtract this value (1) inside the parentheses:

y = (x² - 2x + 1 - 1) + 7

Notice that we've added and subtracted the same value, so we haven't changed the equation's overall value. However, we've strategically set up a perfect square trinomial.

Step 3: Factor the Perfect Square Trinomial

The expression inside the parentheses, x² - 2x + 1, is a perfect square trinomial. This means it can be factored into the square of a binomial. Specifically:

x² - 2x + 1 = (x - 1)²

Now our equation looks like this:

y = ((x - 1)2 - 1) + 7

Step 4: Simplify the Equation

Next, we need to simplify the equation by getting rid of the parentheses and combining the constant terms. First, distribute the implied 1 (in front of the parenthesis):

y = (x - 1)² - 1 + 7

Now, combine the constant terms -1 and +7:

y = (x - 1)² + 6

Step 5: Identify the Vertex

Congratulations! We've successfully rewritten the quadratic function in vertex form:

y = (x - 1)² + 6

Now, we can easily identify the vertex. Comparing our equation to the vertex form y = a(x - h)² + k, we see that:

• h = 1
• k = 6

Therefore, the vertex of the parabola is (1, 6).

Putting It All Together

Let’s recap the entire process to ensure we’ve got a firm grasp on it:

1. Group the x terms: y = (x² - 2x) + 7
2. Complete the Square: y = (x² - 2x + 1 - 1) + 7
3. Factor the Perfect Square Trinomial: y = ((x - 1)2 - 1) + 7
4. Simplify the Equation: y = (x - 1)² + 6
5. Identify the Vertex: Vertex is (1, 6)

By following these steps, you can confidently rewrite any quadratic function into vertex form. This skill is invaluable for graphing quadratic functions and understanding their key characteristics.

Why is Vertex Form Useful?

You might be wondering,