Parabola Unveiled: Vertex, Focus, Directrix & Sketching

Hey everyone, let's dive into the fascinating world of parabolas! Today, we're going to break down how to find the vertex, focus, and directrix of a given parabola, and then we'll learn how to sketch the parabola, marking the focus and directrix for a complete visual understanding. Specifically, we'll be tackling the equation: y² + 2y + 12x - 23 = 0. So, grab your pencils and let's get started. Understanding these elements is key to grasping the nature of these curves, which are fundamental in various fields, from physics to architecture. By the end of this, you’ll be able to not only solve for these features but also visualize the parabola's shape and orientation, which is super cool.

Step-by-Step Guide to Solving for Parabola Features

Okay, guys, first things first, let's get that equation into a more manageable form. Our goal is to rewrite the equation into the standard form of a horizontal parabola, which looks like this: (y - k)² = 4p(x - h). In this form, (h, k) represents the vertex, and p is the distance from the vertex to both the focus and the directrix. Let's start with the original equation: y² + 2y + 12x - 23 = 0. Our first move is to complete the square for the y terms. We have y² + 2y. To complete the square, take half of the coefficient of the y term (which is 2), square it (1), and add it to both sides of the equation. This gives us: y² + 2y + 1. But, we can't just change the equation, so we need to balance it out. Since we added 1 to the y side, we need to balance the equation by also adding 1 to the x side. Thus, the equation becomes y² + 2y + 1 = -12x + 23 + 1. Now we can rewrite the left side as a perfect square: (y + 1)² = -12x + 24. Further simplification involves factoring out a -12 from the right side: (y + 1)² = -12(x - 2). Now, we have our equation in the standard form: (y - (-1))² = 4(-3)(x - 2). This form makes it super easy to identify the vertex and the value of p. The vertex, (h, k), is clearly (2, -1). And, the value of p is -3. Keep in mind that p tells us the distance and the direction of the focus and directrix from the vertex. Because p is negative, this parabola opens towards the negative x-direction (to the left).

Finding the Vertex

As we derived above, the vertex, the turning point of our parabola, is (2, -1). This point is super important because it's the anchor from which we'll find the focus and the directrix. Think of the vertex as the central reference point, the balancing point of the curve. It's the point where the parabola changes direction, the point closest to the directrix. Knowing the vertex is always the first step in understanding the parabola's shape and position.

Determining the Focus

Alright, let's figure out the focus. The focus is a point inside the parabola, and it plays a critical role in defining its shape. Given that p = -3, the focus is p units away from the vertex. Since our parabola opens horizontally, we'll move p units along the x-axis from the vertex. Our vertex is at (2, -1), and since p is -3, we subtract 3 from the x-coordinate of the vertex. Therefore, the focus is at (2 - 3, -1), which simplifies to (-1, -1). So, the focus is located at the point (-1, -1), inside the curve of the parabola. All points on the parabola are equidistant from the focus and the directrix. The focus is a critical element in understanding the parabola's reflective properties.

Calculating the Directrix

The directrix is a line outside the parabola. Every point on the parabola is equidistant from the focus and the directrix. Since p is -3, the directrix is p units away from the vertex in the opposite direction from the focus. Because the parabola opens to the left, the directrix will be a vertical line to the right of the vertex. The vertex is at x = 2, and p is -3, so we add 3 to the x-coordinate of the vertex to find the directrix. This places the directrix at the line x = 2 - (-3) = 5. Hence, the directrix is the vertical line x = 5. The directrix is crucial for understanding the parabola's geometric definition. It's the straight line that, together with the focus, defines the shape of the curve.

Sketching the Parabola

Now, let's bring it all together with a sketch! Sketching helps you visualize what you've calculated and understand the spatial relationship between the vertex, focus, and directrix. First, plot the vertex at (2, -1). Next, mark the focus at (-1, -1). Then, draw the directrix as a vertical line at x = 5. Since p is negative, you know the parabola opens to the left, towards the focus. The vertex is the turning point, and the curve extends from there, getting wider as it goes. Remember, the parabola's curve hugs the focus, with every point on the curve the same distance from the focus as it is from the directrix. The sketch should clearly show the vertex, the focus, the directrix, and the direction in which the parabola opens. Make sure your sketch is clean and clearly labeled, so it makes sense to anyone looking at it.

Drawing the Parabola

To draw the parabola, start at the vertex, and make sure that the curve bends around the focus. The parabola should get wider as it moves away from the vertex. Be precise, so the parabola's shape and position reflect your calculations. Remember that the directrix is a reference line; the curve should never touch or cross it. Make sure the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. It's all about precision, so take your time and make your sketch clear and accurate. You can also use a few additional points on the curve to make the sketch more precise. For example, knowing the points where the parabola intersects the horizontal line through the focus can help make the curve more accurate.

Labeling Key Elements

Make sure to label your sketch properly. Label the vertex with its coordinates (2, -1). Clearly mark the focus and label it with its coordinates (-1, -1). Label the directrix as the line x = 5. Labeling is essential, so anyone can immediately understand the key features of the parabola. Don't forget to include the axis of symmetry, which, for this parabola, is the horizontal line y = -1. The axis of symmetry runs through the vertex and the focus, dividing the parabola into two symmetrical halves. Properly labeling the components of your sketch is important to make it easy to understand the relationship between the key elements and the overall shape.

Conclusion

And there you have it, guys! We've successfully found the vertex, focus, and directrix of the parabola defined by the given equation and sketched it out, too. Remember, the most important thing is to understand the concepts behind these calculations and how they relate visually. So, keep practicing, and you'll become a pro at these problems! Hopefully, you found this tutorial helpful and now have a better grasp of parabolas. Remember to use the standard form to make these problems much easier. Happy calculating!