Parabola Equation: Focus (4,0), Directrix X=-4

by Andrew McMorgan 47 views

Alright, guys, let's dive into the fascinating world of parabolas! Today, we're tackling a classic problem: finding the equation of a parabola given its focus and directrix. Specifically, we're looking at a parabola with its focus chilling at the point (4, 0) and its directrix hanging out at the line x = -4. Sounds like a plan? Let's break it down step by step. Understanding parabolas is super important because they show up everywhere – from satellite dishes to the trajectory of a baseball. So, stick with me, and we'll nail this.

Understanding the Parabola

Before we jump into the nitty-gritty, let's refresh our understanding of what a parabola actually is. A parabola is defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). Think of it like this: if you pick any point on the parabola, its distance to the focus will be exactly the same as its distance to the directrix. This property is key to finding the equation.

  • Focus: The focus is a fixed point inside the curve of the parabola. It plays a crucial role in defining the shape and orientation of the parabola.
  • Directrix: The directrix is a fixed line outside the curve of the parabola. It's always perpendicular to the axis of symmetry of the parabola.
  • Vertex: The vertex is the point on the parabola that is closest to both the focus and the directrix. It's essentially the "tip" of the parabola. The vertex is exactly halfway between the focus and the directrix.
  • Axis of Symmetry: This is the line that passes through the focus and the vertex, dividing the parabola into two symmetrical halves.

In our case, the focus is at (4, 0) and the directrix is x = -4. This tells us a few things right off the bat. Since the directrix is a vertical line, we know that the parabola opens either to the right or to the left. Because the focus is to the right of the directrix, the parabola must open to the right. This is an important observation that will help us narrow down our options later.

Finding the Vertex

The vertex is the midpoint between the focus and the directrix. Since the focus is at (4, 0) and the directrix is x = -4, the vertex will lie on the x-axis. To find the x-coordinate of the vertex, we simply take the average of the x-coordinate of the focus and the x-value of the directrix:

Vertex x-coordinate = (4 + (-4)) / 2 = 0

So, the vertex is at the point (0, 0). This is super convenient because it means our parabola is centered at the origin. Knowing the vertex is at the origin simplifies the equation quite a bit. If the vertex was somewhere else, we'd have to deal with more complicated translations, but we're in luck today!

The Standard Equation

For a parabola that opens to the right and has its vertex at the origin, the standard equation is:

y² = 4px

Where 'p' is the distance from the vertex to the focus (or from the vertex to the directrix – they're the same!). This distance is crucial because it determines how "wide" or "narrow" the parabola is. A larger value of 'p' means a wider parabola, while a smaller value means a narrower one.

In our problem, the vertex is at (0, 0) and the focus is at (4, 0). So, the distance 'p' between the vertex and the focus is simply:

p = 4 - 0 = 4

Now we have everything we need to write the equation of the parabola! Let's plug the value of 'p' into the standard equation.

Plugging in the Value of 'p'

We know that y² = 4px, and we've found that p = 4. So, we just substitute the value of p into the equation:

y² = 4 * 4 * x y² = 16x

And there you have it! The equation of the parabola with focus at (4, 0) and directrix at x = -4 is y² = 16x. This equation perfectly describes the relationship between the x and y coordinates of every point on the parabola. It tells us how the parabola opens to the right, how wide it is, and where it's located in the coordinate plane.

Why the Other Options Are Wrong

Let's quickly look at why the other answer choices are incorrect:

  • A. y² = -x: This parabola opens to the left, because of the negative sign. Our parabola opens to the right, so this is incorrect.
  • B. y² = x: This parabola opens to the right, which is good, but the value of 'p' would be 1/4, meaning the focus would be at (1/4, 0). That's not our focus, so this is wrong.
  • C. y² = -16x: This parabola also opens to the left, due to the negative sign. Again, this is not what we want.

Only option D, y² = 16x, matches the conditions given in the problem. This is why it's the correct answer. Understanding why the other options are wrong is just as important as knowing why the correct answer is right. It reinforces your understanding of the concepts and helps you avoid common mistakes.

Final Answer

So, after carefully analyzing the focus, directrix, and the definition of a parabola, we've confidently arrived at the answer:

D. y² = 16x

Guys, I hope this explanation helped you understand how to find the equation of a parabola given its focus and directrix. Remember the key steps: identify the focus and directrix, find the vertex, determine the value of 'p', and plug it into the standard equation. Keep practicing, and you'll become a parabola pro in no time!