Finding The Vertex And Focus Of The Parabola X^2+6x-8y+1=0 A Comprehensive Guide

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Hey guys! Ever stumbled upon a parabola and felt a little lost? Don't worry, you're not alone! Parabolas might seem intimidating at first, but with a few simple steps, you can easily find their vertex and focus. These two points are super important for understanding the shape and properties of a parabola. In this article, we'll break down the process using a real example, making it crystal clear for you. So, grab your pencils, and let's dive into the fascinating world of parabolas!

Understanding the Parabola Equation

Before we jump into the calculations, let's quickly review the standard form of a parabola equation. This will give us a solid foundation for tackling the problem. The equation we're dealing with is:

x² + 6x - 8y + 1 = 0

This equation represents a parabola that opens either upwards or downwards. Why? Because the x term is squared, and the y term is linear. If the y term were squared instead, the parabola would open to the left or right. Now, to find the vertex and focus, we need to rewrite this equation in a more convenient form, called the standard form. The standard form for a parabola opening upwards or downwards looks like this:

(x - h)² = 4p(y - k)

Where:

  • (h, k) represents the vertex of the parabola – the point where the parabola changes direction. Think of it as the parabola's "corner."
  • p is the distance from the vertex to the focus and from the vertex to the directrix (a line that defines the parabola's shape). This p value is crucial for locating the focus.

Our goal is to manipulate the given equation (x² + 6x - 8y + 1 = 0) to match this standard form. This involves a technique called completing the square, which we'll explore in the next section.

Completing the Square: The Key to Unlocking the Vertex

The secret to transforming our equation into standard form lies in a technique called completing the square. This might sound a bit scary, but it's actually a pretty straightforward process. Here’s the idea: we want to rewrite the x terms in our equation (x² + 6x - 8y + 1 = 0) as a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x + a)² or (x - a)².

Let's focus on the x terms: x² + 6x. To complete the square, we need to add a constant term that will make this expression a perfect square. Here's how we find that constant:

  1. Take half of the coefficient of the x term (which is 6). Half of 6 is 3.
  2. Square the result from step 1. 3 squared is 9.

So, the constant we need to add is 9. But remember, in an equation, we can't just add a number to one side without adding it to the other! So, we'll add 9 to both sides of our equation, but with a little twist to keep things organized. Let's rewrite the original equation, group the x terms, and then add 9 strategically:

(x² + 6x        ) - 8y + 1 = 0
(x² + 6x + 9) - 8y + 1 = 9

Notice that we added 9 inside the parentheses on the left side. This allows us to complete the square. Now, the expression inside the parentheses, x² + 6x + 9, is a perfect square trinomial! We can factor it as (x + 3)². This is a huge step forward. Our equation now looks like this:

(x + 3)² - 8y + 1 = 9

Next, we want to isolate the squared term on one side of the equation. To do this, we'll add 8y and subtract 1 from both sides:

(x + 3)² = 8y - 1 + 9
(x + 3)² = 8y + 8

We're almost there! Now, we need to factor out the coefficient of the y term on the right side. This will help us clearly identify the value of p in the standard form equation:

(x + 3)² = 8(y + 1)

Now, let's compare this to the standard form (x - h)² = 4p(y - k). We can see that our equation is now in the standard form! We've successfully completed the square, and we're ready to extract the vertex and focus information.

Identifying the Vertex: The Parabola's Turning Point

With our equation now in the standard form, (x + 3)² = 8(y + 1), finding the vertex is a breeze! Remember, the vertex is represented by the coordinates (h, k) in the standard form equation (x - h)² = 4p(y - k).

Let's carefully compare our equation to the standard form. Notice that we have (x + 3)², which can be rewritten as (x - (-3))². This means that h is -3. Similarly, we have (y + 1), which can be rewritten as (y - (-1)). This means that k is -1.

Therefore, the vertex of our parabola is (-3, -1).

The vertex is a crucial point on the parabola. It's the turning point, the place where the parabola changes direction. It's also the point of symmetry for the parabola. Knowing the vertex gives us a good starting point for sketching the parabola's graph.

But we're not done yet! We still need to find the focus. To do that, we need to determine the value of p, which we'll tackle in the next section.

Finding the Focus: The Heart of the Parabola

Now that we've found the vertex, let's hunt down the focus! The focus is another key point associated with a parabola. It's a fixed point inside the curve of the parabola, and it plays a vital role in defining the parabola's shape. Remember, the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix (a line outside the parabola).

To find the focus, we need to determine the value of p in our standard form equation, (x + 3)² = 8(y + 1). Recall that the standard form equation is (x - h)² = 4p(y - k). By comparing our equation to the standard form, we can see that 4p = 8.

To solve for p, we simply divide both sides of the equation by 4:

4p = 8
p = 8 / 4
p = 2

So, p = 2. This means the distance between the vertex and the focus is 2 units. Now, we need to figure out in which direction the focus lies relative to the vertex. Since our parabola equation is in the form (x - h)² = 4p(y - k), and p is positive (p = 2), the parabola opens upwards. This means the focus will be located above the vertex.

To find the coordinates of the focus, we start at the vertex (-3, -1) and move p units (which is 2 units) upwards along the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex, and its equation is x = -3 in this case.

So, to find the focus, we keep the x-coordinate the same (-3) and add p to the y-coordinate:

Focus: (-3, -1 + 2) = (-3, 1)

And there you have it! We've successfully found the focus of our parabola. It's located at the point (-3, 1).

Putting It All Together: Vertex and Focus Revealed

Let's recap what we've accomplished. We started with the equation of a parabola:

x² + 6x - 8y + 1 = 0

Through the power of completing the square, we transformed this equation into the standard form:

(x + 3)² = 8(y + 1)

From the standard form, we were able to easily identify the vertex as (-3, -1) and, after calculating p, the focus as (-3, 1).

These two points, the vertex and the focus, are key characteristics of the parabola. They help us understand its shape, position, and orientation in the coordinate plane. By mastering the technique of completing the square and understanding the standard form of a parabola equation, you can confidently tackle any parabola problem that comes your way!

So, next time you encounter a parabola, remember the steps we've covered in this article. You'll be able to find the vertex and focus like a pro! Keep practicing, and you'll become a parabola master in no time. Cheers, guys!