Parabola Unveiled Find Vertex Focus And Directrix For Y^2-y-x+3=0

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of parabolas. We'll tackle a problem that might seem a bit intimidating at first, but trust me, we'll break it down step by step. Our mission? To find the vertex, focus, and directrix of the parabola defined by the equation y² - y - x + 3 = 0, and of course, we'll sketch its graph too. So, buckle up, and let's get started!

Watch and Learn

Before we jump into the nitty-gritty, it's always helpful to have a visual aid. I highly recommend watching a video that explains the concepts of parabolas, vertices, foci, and directrices. Visualizing these elements can make the problem-solving process much smoother. You can find tons of excellent videos online – just search for "parabola vertex focus directrix" on your favorite platform. Watching a video that illustrates how to find the parabola's vertex, focus, and directrix will provide a solid foundation for tackling our specific problem.

The Parabola Puzzle: y² - y - x + 3 = 0

Now, let's get our hands dirty with the equation y² - y - x + 3 = 0. This might not look like the standard form of a parabola equation, but don't worry, we'll transform it. Our goal is to rewrite it in a form that reveals the parabola's key features. Remember, the standard form of a parabola opening horizontally is (y - k)² = 4p(x - h), where (h, k) is the vertex and p is the distance between the vertex and the focus, as well as the vertex and the directrix. Understanding this standard form is crucial to unlocking the parabola's vertex, focus, and directrix.

Step 1: Completing the Square

The first step in our transformation journey is to complete the square for the y terms. This is a classic algebraic technique that helps us rewrite quadratic expressions in a more manageable form. Let's isolate the y terms: y² - y = x - 3. To complete the square, we need to add and subtract (1/2)² = 1/4 on the left side: y² - y + 1/4 - 1/4 = x - 3. Now, we can rewrite the left side as a perfect square: (y - 1/2)² - 1/4 = x - 3. Completing the square is a fundamental technique for rewriting the equation and making it easier to identify the vertex and other key features of the parabola.

Step 2: Isolating the Squared Term

Next, let's isolate the squared term by adding 1/4 to both sides of the equation: (y - 1/2)² = x - 3 + 1/4. Simplifying the right side, we get (y - 1/2)² = x - 11/4. This is starting to look more like the standard form we discussed earlier! By isolating the squared term, we are getting closer to the standard form of the parabola equation, which will help us easily identify the vertex, focus, and directrix.

Step 3: The Standard Form Reveal

Now, we have the equation in the form (y - 1/2)² = x - 11/4. Comparing this to the standard form (y - k)² = 4p(x - h), we can identify the following: h = 11/4, k = 1/2, and 4p = 1. From 4p = 1, we get p = 1/4. This is a major breakthrough! By rewriting the equation in standard form, we can directly read off the values of h, k, and p, which are essential for finding the vertex, focus, and directrix.

Unveiling the Vertex

The vertex of the parabola is the point (h, k). From our standard form equation, we know that h = 11/4 and k = 1/2. Therefore, the vertex of the parabola is (11/4, 1/2). Congratulations, we've found our first key element! The vertex is a crucial point on the parabola, as it represents the turning point of the curve. Its coordinates, (h, k), are directly obtained from the standard form of the equation.

Pinpointing the Focus

The focus of a parabola is a special point inside the curve. For a parabola that opens horizontally, the focus is located at (h + p, k). We already know that h = 11/4, k = 1/2, and p = 1/4. Plugging these values in, we get the focus as (11/4 + 1/4, 1/2) = (12/4, 1/2) = (3, 1/2). Awesome, we've located the focus! The focus is a key defining point of the parabola. Its location, relative to the vertex and the directrix, determines the shape and orientation of the parabola.

Locating the Directrix

The directrix is a line that lies outside the parabola. For a parabola that opens horizontally, the directrix is a vertical line with the equation x = h - p. Using our values, h = 11/4 and p = 1/4, the directrix is x = 11/4 - 1/4 = 10/4 = 5/2. So, the directrix is the vertical line x = 5/2. We've successfully found the directrix! The directrix is a line such that for any point on the parabola, the distance to the focus is equal to the distance to the directrix. This property is fundamental to the definition of a parabola.

Sketching the Graph

Now for the fun part: sketching the graph! We know the vertex is at (11/4, 1/2), the focus is at (3, 1/2), and the directrix is the line x = 5/2. Since p is positive, the parabola opens to the right. Plot the vertex, focus, and directrix. The parabola will curve away from the directrix and wrap around the focus. Draw a smooth curve that represents the parabola. Sketching the graph provides a visual representation of the parabola and helps to solidify our understanding of its properties.

Key Takeaways

Let's recap what we've learned. We started with the equation y² - y - x + 3 = 0 and, through the power of completing the square and understanding the standard form of a parabola equation, we successfully found the vertex, focus, and directrix. We also sketched the graph to visualize our results. This problem highlights the importance of algebraic manipulation and a solid understanding of conic sections. Finding the vertex, focus, and directrix of a parabola involves a combination of algebraic techniques and geometric understanding. By mastering these concepts, you'll be well-equipped to tackle a wide range of parabola-related problems.

So, guys, keep practicing, and you'll become parabola pros in no time! Remember, math is like a puzzle – each piece fits together to reveal a beautiful solution. And with parabolas, that solution is a stunning curve with fascinating properties.

Final Answer

• Vertex: (11/4, 1/2)
• Focus: (3, 1/2)
• Directrix: x = 5/2