Finding The Equation, Focus, And Directrix Of A Parabola

Hey guys! Let's dive into the fascinating world of parabolas. Today, we're tackling a specific problem: finding the equation of a parabola given certain key features. We'll break it down step-by-step, so you'll not only understand the mechanics but also the why behind each step. We'll be focusing on parabolas with a vertical axis of symmetry. These parabolas open either upwards or downwards, making our lives a little easier.

Problem Statement: Our Parabola's Particulars

So, here's the scenario: We need to find the equation of a parabola that has its vertex at the point (4, -8). That's our parabola's turning point. We also know it has a vertical axis of symmetry. This means the parabola is symmetrical around a vertical line passing through the vertex. Crucially, we're given that the parabola has a y-intercept of 24. This means the parabola crosses the y-axis at the point (0, 24). Finally, our mission doesn't end with the equation; we also need to locate the focus and the directrix of this parabola. The focus is a special point inside the curve of the parabola, and the directrix is a special line outside the curve. They're both fundamental to understanding a parabola's shape and properties. Think of it like finding the treasure (the equation), but also the landmarks that lead to it (focus and directrix). Trust me, it's not as daunting as it sounds. We'll take it slow and make sure every step is crystal clear. So, grab your pencils and let's get started!

1. The Standard Equation: Our Starting Point

When dealing with parabolas that have a vertical axis of symmetry, the standard form equation is our best friend. It looks like this: (x - h)^2 = 4p(y - k). Now, let's break down what each of these variables represents. The dynamic duo (h, k) represents the coordinates of the vertex of the parabola. Remember, the vertex is that turning point we talked about earlier. The variable p is a crucial parameter. It determines the distance between the vertex and the focus, as well as the distance between the vertex and the directrix. The sign of p also tells us whether the parabola opens upwards (if p is positive) or downwards (if p is negative). Think of this standard form as a template. We're going to plug in the information we have to customize it for our specific parabola. This standard equation is not just a random formula; it's a powerful tool derived from the very definition of a parabola. A parabola is the set of all points that are equidistant to the focus (a point) and the directrix (a line). This beautiful geometric property is encapsulated in this neat algebraic equation. By understanding the origin of the standard form, we gain a deeper appreciation for its structure and how it relates to the parabola's shape. Remember, understanding is always better than just memorizing. With the standard equation in our toolkit, we're ready to start plugging in the information we have and unlocking the secrets of our parabola.

2. Plugging in the Vertex: Filling in the Blanks

Great! We've got our standard equation ready. Now, let's use the information we were given. We know the vertex of our parabola is located at the point (4, -8). Remember, in our standard equation, the vertex is represented by (h, k). So, we can directly substitute these values into our equation. This gives us h = 4 and k = -8. Let's plug these values into our standard form equation: (x - h)^2 = 4p(y - k) becomes (x - 4)^2 = 4p(y - (-8)) which simplifies to (x - 4)^2 = 4p(y + 8). See how we're making progress? We've already filled in two of the key pieces of the puzzle. Notice that the equation now looks more specific to our parabola. The (x - 4) part tells us the parabola is shifted horizontally by 4 units, and the (y + 8) part tells us it's shifted vertically by 8 units. These shifts are what give our parabola its specific location in the coordinate plane. The only unknown we have left is p. This parameter, as we discussed earlier, is crucial for determining the shape and orientation of the parabola. Finding p is our next big step. Don't worry, we have enough information to do it! We'll use the other piece of information we were given: the y-intercept. By strategically using the y-intercept, we'll be able to crack the code and find the value of p. So, stay tuned, and let's move on to the next step!

3. Using the Y-Intercept: Solving for 'p'

Okay, time to use the y-intercept. We know the parabola passes through the point (0, 24). This is super helpful because any point on the parabola's curve must satisfy the parabola's equation. So, we can substitute x = 0 and y = 24 into the equation we derived in the last step: (x - 4)^2 = 4p(y + 8). Substituting these values, we get: (0 - 4)^2 = 4p(24 + 8). Now, let's simplify this equation and solve for 'p'. First, we have (-4)^2 = 16 on the left side. On the right side, we have 4p(32), which simplifies to 128p. So, our equation becomes: 16 = 128p. To isolate 'p', we divide both sides of the equation by 128: p = 16 / 128. Simplifying the fraction, we find that p = 1/8. Fantastic! We've found the value of p. This is a major breakthrough because 'p' holds the key to understanding the parabola's shape and position. A positive value for 'p' (which we have here) tells us that the parabola opens upwards. The magnitude of 'p' (1/8 in our case) tells us how "wide" or "narrow" the parabola is. A smaller value of 'p' means a wider parabola, and a larger value means a narrower parabola. Now that we know 'p', we can plug it back into our equation to get the complete equation of the parabola. We're almost there! Let's move on to the next step and write out the final equation.

4. The Grand Finale: The Equation of the Parabola

Alright, we've done the hard work. We've found the value of 'p', which was the last piece of the puzzle. Now, we can write out the complete equation of our parabola. We had the equation: (x - 4)^2 = 4p(y + 8). We now know that p = 1/8. So, let's substitute that value into the equation: (x - 4)^2 = 4 * (1/8) * (y + 8). Simplifying, we get: (x - 4)^2 = (1/2)(y + 8). This is the equation of our parabola in standard form! We've successfully found the equation that describes this parabola with a vertex at (4, -8), a vertical axis of symmetry, and a y-intercept of 24. But our journey doesn't end here. Remember, we also need to find the focus and the directrix. The equation we've found is like the map to the treasure, but the focus and directrix are like the hidden landmarks along the way. To find them, we'll use the value of 'p' and our knowledge of the parabola's properties. The focus and directrix are intimately related to the value of 'p'. They are located a distance of |p| away from the vertex, one inside the curve of the parabola (the focus) and the other outside the curve (the directrix). So, let's move on to the next step and pinpoint these important features of our parabola.

5. Locating the Focus: Inside the Curve

Time to find the focus! Remember, the focus is a special point located inside the curve of the parabola. For a parabola with a vertical axis of symmetry that opens upwards (like ours, since p is positive), the focus is located 'p' units above the vertex. We know our vertex is at (4, -8) and p = 1/8. So, to find the focus, we need to add 'p' to the y-coordinate of the vertex. The x-coordinate of the focus will be the same as the x-coordinate of the vertex. Therefore, the focus will be at the point (4, -8 + 1/8). Let's simplify that y-coordinate: -8 + 1/8 = -64/8 + 1/8 = -63/8. So, the focus of our parabola is located at the point (4, -63/8). We've found the focus! This point has a special property: it's the point that "reflects" all incoming rays parallel to the axis of symmetry towards the focus. This is why parabolas are used in satellite dishes and other applications where focusing signals is important. Now that we've found the focus, let's find the directrix. The directrix is like the focus's counterpart, located on the opposite side of the vertex. It's a line, not a point, and it plays a crucial role in the parabola's definition. So, let's move on to the next step and find the directrix of our parabola.

6. Finding the Directrix: The Guiding Line

Now, let's find the directrix. The directrix is a line located outside the curve of the parabola. For a parabola with a vertical axis of symmetry that opens upwards, the directrix is a horizontal line located 'p' units below the vertex. We know our vertex is at (4, -8) and p = 1/8. So, to find the equation of the directrix, we need to subtract 'p' from the y-coordinate of the vertex. The directrix is a horizontal line, so its equation will be of the form y = constant. Therefore, the directrix will be the line y = -8 - 1/8. Let's simplify that: -8 - 1/8 = -64/8 - 1/8 = -65/8. So, the equation of the directrix of our parabola is y = -65/8. We've found the directrix! This line has a special relationship with the focus. As we mentioned earlier, a parabola is defined as the set of all points that are equidistant to the focus and the directrix. This fundamental property ties the focus and directrix together and gives the parabola its unique shape. Congratulations! We've successfully found the equation of the parabola, located the focus, and found the equation of the directrix. We've tackled a complete parabola problem from start to finish. You've now got the tools and the knowledge to solve similar problems. Remember, the key is to break down the problem into smaller steps, understand the concepts behind the formulas, and practice, practice, practice! Keep up the great work!

Conclusion: Parabola Power!

Wow, guys! We really dove deep into parabolas today. We started with the basics – the standard equation – and used the given information (vertex and y-intercept) to find the specific equation for our parabola: (x - 4)^2 = (1/2)(y + 8). Then, we went even further, pinpointing the focus at (4, -63/8) and the directrix with the equation y = -65/8. Finding the equation of a parabola isn't just about plugging numbers into a formula. It's about understanding the relationship between the vertex, focus, directrix, and the value of 'p'. By visualizing these elements and how they interact, you gain a deeper understanding of the parabola's unique shape and properties. Remember, parabolas aren't just abstract mathematical concepts. They're all around us! From the curves of satellite dishes to the trajectory of a baseball, parabolas play a vital role in our world. By mastering these concepts, you're not just acing your math class; you're unlocking a deeper understanding of the world around you. So, keep exploring, keep questioning, and keep practicing. You've got the power of parabolas on your side!