Identifying The Graph Of -2x² - 4x - 2y² + 6y - 6 = 0 A Comprehensive Guide
Hey guys! Let's dive into this interesting math problem where we need to figure out what kind of graph the equation -2x² - 4x - 2y² + 6y - 6 = 0 represents. It might look intimidating at first, but don't worry, we'll break it down step by step. Our main goal here is to rewrite the equation in a standard form that we can easily recognize. This involves completing the square for both the x and y terms. So, grab your thinking caps, and let's get started!
Unpacking the Equation: Setting the Stage
Before we jump into any fancy techniques, let's take a good look at the equation: -2x² - 4x - 2y² + 6y - 6 = 0. Notice how both x and y are squared? This is a big clue that we might be dealing with a circle, an ellipse, or a hyperbola. But to be sure, we need to do some algebraic maneuvering. First things first, let’s simplify the equation by dividing everything by -2. This makes our lives a whole lot easier by getting rid of that pesky negative coefficient in front of the squared terms. So, after dividing, our equation transforms into: x² + 2x + y² - 3y + 3 = 0. Much cleaner, right? Now, the real fun begins – completing the square!
Completing the Square: The Key to Unlocking the Graph
Completing the square is a nifty algebraic technique that helps us rewrite quadratic expressions in a more manageable form. Think of it as turning a messy room into an organized one. We’re going to do this separately for both the x and y terms. For the x terms (x² + 2x), we need to add and subtract a value that will allow us to form a perfect square trinomial. Remember, a perfect square trinomial is something like (x + a)² or (x - a)². To find this magic value, we take half of the coefficient of the x term (which is 2), square it (1² = 1), and add and subtract it within the equation. So, we get: (x² + 2x + 1) - 1. Notice how (x² + 2x + 1) is just (x + 1)²? That's the beauty of completing the square! We do the exact same thing for the y terms (y² - 3y). Half of -3 is -3/2, and squaring that gives us 9/4. So, we add and subtract 9/4: (y² - 3y + 9/4) - 9/4. And just like before, (y² - 3y + 9/4) is (y - 3/2)². Now, let’s put it all together and see what our equation looks like now:
(x² + 2x + 1) - 1 + (y² - 3y + 9/4) - 9/4 + 3 = 0
We can rewrite this as:
(x + 1)² + (y - 3/2)² = 1/4
Spotting the Circle: Standard Form Reveals All
Take a good look at our transformed equation: (x + 1)² + (y - 3/2)² = 1/4. Does this look familiar? It should! This is the standard form of the equation of a circle: (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. Comparing our equation to the standard form, we can immediately see that the center of our circle is (-1, 3/2) and the radius squared (r²) is 1/4. Taking the square root of 1/4 gives us the radius, which is 1/2. So, there you have it! The equation -2x² - 4x - 2y² + 6y - 6 = 0 represents a circle with center (-1, 3/2) and radius 1/2. Woohoo! We cracked it!
Visualizing the Circle: A Graph Is Worth a Thousand Words
Now that we've determined that the equation represents a circle, let's take a moment to visualize it. Imagine a coordinate plane, the familiar x and y axes stretching out in all directions. Our circle has its center at the point (-1, 3/2). That means we move one unit to the left on the x-axis and 1.5 units up on the y-axis. That's the bullseye of our circle! Now, picture drawing a circle around this point with a radius of 1/2. That's a pretty small circle, just half a unit in every direction from the center. So, the circle will extend 1/2 unit to the left, right, up, and down from the point (-1, 3/2). Visualizing the graph can really solidify our understanding. We can see how the equation we started with actually translates into a concrete geometric shape. It's like magic, isn't it? The abstract algebra becomes a beautiful, tangible form. This is one of the coolest things about math – connecting equations to the real world, or in this case, to geometric shapes.
Beyond the Circle: A Glimpse into Conic Sections
Our journey with this equation doesn't just end with identifying it as a circle. This is actually part of a bigger family of curves known as conic sections. These curves are formed when a plane intersects a double cone. Imagine two cones stacked tip-to-tip, and then picture slicing through them with a flat plane. Depending on the angle and position of the plane, you can create different shapes: circles, ellipses, parabolas, and hyperbolas. We’ve encountered a circle today, but the other conic sections have their own unique equations and properties. Ellipses are like stretched-out circles, parabolas are the familiar U-shaped curves, and hyperbolas have two separate branches. Each of these shapes has its own standard form equation, just like the circle. Learning to recognize these standard forms is key to quickly identifying the type of conic section you're dealing with. The process of completing the square, which we used to find the standard form of the circle equation, is also incredibly useful for identifying other conic sections. It allows us to rewrite the equations in a way that makes the key features, like the center, radius, vertices, and axes, crystal clear. So, mastering this technique is a major win in your mathematical toolkit!
Wrapping Up: Why This Matters
So, why did we go through all this trouble to figure out which graph is represented by the equation -2x² - 4x - 2y² + 6y - 6 = 0? Well, understanding the connection between equations and their graphs is fundamental in mathematics and has wide-ranging applications in various fields. Whether it's physics, engineering, computer graphics, or even economics, the ability to visualize and interpret equations is a powerful tool. For example, in physics, understanding the equation of a projectile's trajectory (which is a parabola) can help us predict its path. In engineering, knowing the properties of ellipses is crucial for designing arches and bridges. And in computer graphics, conic sections are used to create smooth curves and shapes in images and animations. The more we understand these fundamental concepts, the better equipped we are to tackle real-world problems. Plus, it's just plain cool to see how abstract math concepts can translate into tangible shapes and phenomena. So, keep exploring, keep questioning, and keep connecting the dots – the world of mathematics is full of fascinating discoveries waiting to be made!
In a nutshell, we started with a seemingly complex equation, used the technique of completing the square to transform it into standard form, and then identified the graph as a circle. We also took a peek into the broader world of conic sections and discussed the real-world applications of this knowledge. Math might seem like a bunch of symbols and equations, but it's actually a powerful language for describing and understanding the world around us. Keep practicing, keep exploring, and you'll be amazed at what you can achieve!