Converting Quadratics To Vertex Form A Step By Step Guide

Understanding quadratic equations is fundamental in algebra, and one of the most useful forms for analyzing these equations is the vertex form. The vertex form of a quadratic equation not only reveals the vertex of the parabola but also simplifies graphing and understanding the equation’s properties. In this comprehensive guide, we will delve deep into the process of converting a quadratic equation from its standard form to vertex form. Specifically, we'll address the question of rewriting the equation $y = 6x^2 + 12x - 10$ in vertex form. We will explore the step-by-step method of completing the square, a powerful algebraic technique used to achieve this transformation. This detailed explanation will empower you to confidently convert any quadratic equation into vertex form, enhancing your problem-solving skills and deepening your understanding of quadratic functions.

What is Vertex Form?

Vertex form provides a clear representation of the parabola's vertex (h, k), which is crucial for understanding the graph's characteristics. The vertex form of a quadratic equation is expressed as:

$y = a(x - h)^2 + k$

where:

• (h, k) represents the vertex of the parabola.
• 'a' determines the direction and stretch of the parabola.

Why Convert to Vertex Form?

Converting to vertex form offers significant advantages, particularly when analyzing and graphing quadratic functions. The vertex form directly reveals the vertex (h, k), which is the minimum or maximum point of the parabola. This is extremely useful in various applications, such as optimization problems in physics and engineering. Furthermore, the vertex form simplifies the process of graphing the parabola, as you can easily plot the vertex and use the 'a' value to determine the shape and direction of the parabola. Understanding the vertex form enhances your ability to quickly grasp the key features of a quadratic function, making it an indispensable tool in algebra.

Step-by-Step Conversion: Completing the Square

To convert the given equation $y = 6x^2 + 12x - 10$ into vertex form, we will employ the method of completing the square. This technique allows us to rewrite the quadratic expression in a form that directly reveals the vertex of the parabola. The process involves several key steps, each designed to systematically transform the equation while preserving its original meaning. By carefully following these steps, we can successfully convert the equation into vertex form, gaining valuable insights into the function's properties and behavior. Let's walk through each step in detail to ensure a clear understanding of the method.

Step 1: Factor out the Coefficient of $x^2$

Our initial equation is $y = 6x^2 + 12x - 10$. The first step in completing the square is to factor out the coefficient of the $x^2$ term, which in this case is 6, from the first two terms of the equation. This isolates the $x^2$ and $x$ terms, making it easier to complete the square within the parentheses. Factoring out 6 from $6x^2 + 12x$ gives us $6(x^2 + 2x)$. Therefore, we rewrite the equation as:

$y = 6(x^2 + 2x) - 10$

This step sets the stage for completing the square by simplifying the quadratic expression inside the parentheses.

Step 2: Complete the Square Inside the Parentheses

To complete the square, we focus on the expression inside the parentheses, which is $x^2 + 2x$. We need to add a constant term that will make this expression a perfect square trinomial. A perfect square trinomial can be factored into the form $(x + n)^2$, where n is a constant. To find this constant, we take half of the coefficient of the x term (which is 2), square it, and add it inside the parentheses. Half of 2 is 1, and 1 squared is 1. So, we add 1 inside the parentheses:

$y = 6(x^2 + 2x + 1) - 10$

However, since we've added 1 inside the parentheses, which is being multiplied by 6, we've effectively added 6 to the right side of the equation. To maintain the equation's balance, we must subtract 6 outside the parentheses. This ensures that the overall value of the equation remains unchanged.

Step 3: Balance the Equation

As explained in the previous step, adding 1 inside the parentheses (which is multiplied by 6) means we've added 6 to the right side of the equation. To maintain the balance, we need to subtract 6 outside the parentheses. This ensures that the equation remains equivalent to the original equation. So, we subtract 6 from the constant term outside the parentheses:

$y = 6(x^2 + 2x + 1) - 10 - 6$

This step is crucial to ensure that we are only rewriting the equation and not changing its fundamental properties.

Step 4: Rewrite as a Squared Term

Now, we rewrite the expression inside the parentheses as a squared term. The expression $x^2 + 2x + 1$ is a perfect square trinomial and can be factored into $(x + 1)^2$. This is the essence of completing the square – transforming a quadratic expression into a squared term, which is a key component of the vertex form. Substituting this back into our equation, we get:

$y = 6(x + 1)^2 - 10 - 6$

This step simplifies the equation and brings us closer to the vertex form.

Step 5: Simplify the Constant Term

Finally, we simplify the constant term by combining the numbers outside the parentheses. We have $-10 - 6$, which simplifies to $-16$. This gives us the final form of the equation in vertex form:

$y = 6(x + 1)^2 - 16$

This step completes the conversion to vertex form, making it easy to identify the vertex of the parabola.

Identifying the Vertex

Now that we have the equation in vertex form, $y = 6(x + 1)^2 - 16$, we can easily identify the vertex of the parabola. Recall that the vertex form is $y = a(x - h)^2 + k$, where (h, k) is the vertex. Comparing our equation with the vertex form, we can see that:

• h = -1
• k = -16

Therefore, the vertex of the parabola is (-1, -16). This point represents the minimum value of the quadratic function, as the coefficient 'a' (which is 6) is positive, indicating that the parabola opens upwards.

Conclusion

In conclusion, by following the method of completing the square, we successfully converted the quadratic equation $y = 6x^2 + 12x - 10$ into vertex form, which is $y = 6(x + 1)^2 - 16$. This transformation allowed us to easily identify the vertex of the parabola as (-1, -16). Understanding how to convert quadratic equations into vertex form is a valuable skill in algebra, providing insights into the properties and behavior of quadratic functions. The correct answer to the question is:

D. $y = 6(x + 1)^2 - 16$

By mastering this technique, you can confidently analyze and graph quadratic functions, enhancing your problem-solving abilities in mathematics and related fields. The vertex form not only simplifies the identification of the vertex but also provides a clear representation of the parabola's key features, making it an essential tool in your mathematical toolkit.