Rewrite $y=-6x^2+3x+2$ In Vertex Form: A Guide

Hey guys! Let's dive into the fascinating world of quadratic equations, specifically focusing on how to rewrite the equation $y = -6x^2 + 3x + 2$ into its vertex form. This isn't just a mathematical exercise; it's a powerful technique that unlocks a wealth of information about the parabola represented by this equation, including its vertex, axis of symmetry, and maximum or minimum value. Think of it as giving our equation a makeover, revealing its most important features in a clear and concise way. So, grab your thinking caps, and let's embark on this mathematical adventure!

Understanding Vertex Form: The Key to Unlocking Parabolas

Before we tackle the specific equation, it's crucial to understand what vertex form actually is. 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. The vertex, as the name suggests, is the "turning point" of the parabola – its highest or lowest point. The coefficient a plays a crucial role too; it determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and also influences the "width" of the parabola. A larger absolute value of a means a narrower parabola, while a smaller absolute value results in a wider one.

Why is vertex form so useful? Well, it provides an immediate snapshot of the parabola's key characteristics. The vertex $(h, k)$ is directly visible, allowing us to quickly identify the parabola's maximum or minimum point. The axis of symmetry, which is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves, is simply the line $x = h$. Furthermore, knowing the vertex and the direction the parabola opens (determined by the sign of a) gives us a good sense of the parabola's overall shape and position on the coordinate plane.

In contrast, the standard form of a quadratic equation, which is $y = ax^2 + bx + c$, doesn't readily reveal the vertex. While it's useful for other purposes, such as easily identifying the y-intercept (which is c), extracting the vertex requires additional steps. This is where the technique of completing the square comes into play, allowing us to transform the standard form into the much more informative vertex form. So, now that we appreciate the power of vertex form, let's get our hands dirty and apply it to our specific equation.

Completing the Square: Our Transformation Tool

Alright, guys, here's where the magic happens! We're going to use a technique called completing the square to rewrite $y = -6x^2 + 3x + 2$ into vertex form. Completing the square might sound intimidating, but it's a systematic process that involves manipulating the quadratic expression to create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, like $(x + a)^2$ or $(x - a)^2$.

Here's a step-by-step breakdown of how we'll complete the square for our equation:

1. Factor out the coefficient of the $x^2$ term: In our case, the coefficient of $x^2$ is -6. We factor this out from the first two terms: $y = -6(x^2 - rac{1}{2}x) + 2$. Notice how we've carefully factored out the -6, leaving a space inside the parentheses where we'll create our perfect square trinomial. The +2 remains outside for now.

2. Complete the square inside the parentheses: This is the heart of the process. We take half of the coefficient of our x term (which is -1/2), square it, and add it inside the parentheses. Half of -1/2 is -1/4, and squaring that gives us 1/16. So, we add 1/16 inside the parentheses: $y = -6(x^2 - rac1}{2}x + rac{1}{16}) + 2$. Now, hold on! We can't just add 1/16 without compensating for it. Remember, this 1/16 is inside the parentheses, which are being multiplied by -6. So, we've effectively subtracted -6 * (1/16) = -3/8 from the expression. To balance this out, we need to add 3/8 outside the parentheses $y = -6(x^2 - rac{1{2}x + rac{1}{16}) + 2 + rac{3}{8}$.

3. Rewrite the perfect square trinomial as a squared binomial: The expression inside the parentheses is now a perfect square trinomial! It can be factored as $(x - rac1}{4})^2$. So, we have $y = -6(x - rac{1{4})^2 + 2 + rac{3}{8}$.

4. Simplify the constant term: We combine the constants outside the parentheses: $2 + rac{3}{8} = rac{16}{8} + rac{3}{8} = rac{19}{8}$. This gives us our final vertex form equation!

The Vertex Form Unveiled: $y = -6(x - rac{1}{4})^2 + rac{19}{8}$

Ta-da! We've successfully rewritten the equation into vertex form: $y = -6(x - rac{1}{4})^2 + rac{19}{8}$. Now, let's reap the rewards of our hard work and extract the valuable information this form provides.

By comparing our equation to the general vertex form $y = a(x - h)^2 + k$, we can immediately identify the vertex: $(h, k) = (rac{1}{4}, rac{19}{8})$. This tells us that the parabola's turning point is at the coordinates (1/4, 19/8). Since the coefficient a is -6 (which is negative), the parabola opens downwards, meaning the vertex represents the maximum point of the parabola. The maximum value of y is therefore 19/8.

The axis of symmetry is the vertical line that passes through the vertex. Its equation is simply $x = h$, which in our case is $x = rac{1}{4}$. This line divides the parabola into two perfectly symmetrical halves.

Furthermore, the coefficient a = -6 tells us that the parabola is narrower than the standard parabola $y = x^2$. The larger the absolute value of a, the steeper the parabola's sides.

So, by completing the square and arriving at the vertex form, we've gained a comprehensive understanding of the parabola's key features: its vertex, axis of symmetry, maximum value, and overall shape. This demonstrates the power and elegance of vertex form in analyzing quadratic equations.

Putting it All Together: Why Vertex Form Matters

Guys, we've journeyed through the process of rewriting a quadratic equation into vertex form, and we've seen the wealth of information it unlocks. But let's take a step back and appreciate the bigger picture: why does all of this matter in the real world?

Quadratic equations and parabolas pop up in various applications, from physics to engineering to economics. For instance, the trajectory of a projectile (like a ball thrown in the air) follows a parabolic path. Knowing the vertex of this parabola allows us to determine the maximum height the projectile reaches and the distance it travels. In engineering, parabolic shapes are used in bridge designs, satellite dishes, and even the reflectors in car headlights, all because of their unique focusing properties.

In economics, quadratic functions can model cost, revenue, and profit. Finding the vertex in these models helps businesses determine the optimal production level to maximize profit or minimize cost. The ability to quickly identify the maximum or minimum value of a quadratic function is a powerful tool in optimization problems.

Beyond these specific applications, understanding vertex form enhances our overall mathematical literacy. It reinforces the connection between algebraic representations (the equation) and geometric representations (the parabola). It deepens our understanding of transformations and how manipulating equations can reveal hidden properties. And, perhaps most importantly, it empowers us to approach mathematical challenges with confidence and a systematic problem-solving approach.

So, the next time you encounter a quadratic equation, remember the power of vertex form. It's not just a different way of writing the equation; it's a key that unlocks the secrets of the parabola and its many applications in the world around us. Keep exploring, keep learning, and keep those mathematical gears turning!