Vertex & Focus: Parabola Y^2 - 12y + 4x + 4 = 0

by Alex Johnson 48 views

Understanding parabolas involves identifying key features such as the vertex and focus. This article will guide you through the process of finding these elements for the parabola defined by the equation y² - 12y + 4x + 4 = 0. We'll break down each step, making it easy to follow along and apply to similar problems. Let's dive in and unlock the secrets of this parabolic equation!

1. Understanding the Standard Forms of a Parabola

To effectively find the vertex and focus, it's crucial to understand the standard forms of a parabola equation. Parabolas can open either horizontally or vertically, and each orientation has its corresponding standard form. Knowing these forms allows us to manipulate the given equation into a recognizable structure, making it easier to extract the vertex and focus information. Let's explore these standard forms in detail to lay a solid foundation for solving the problem.

  • Horizontal Parabolas: Parabolas that open to the left or right have the standard form:

    (y - k)² = 4p(x - h)

    Where:

    • (h, k) represents the vertex of the parabola.
    • p is the distance from the vertex to the focus and from the vertex to the directrix.
    • If p > 0, the parabola opens to the right.
    • If p < 0, the parabola opens to the left.

  • Vertical Parabolas: Parabolas that open upwards or downwards have the standard form:

    (x - h)² = 4p(y - k)

    Where:

    • (h, k) represents the vertex of the parabola.
    • p is the distance from the vertex to the focus and from the vertex to the directrix.
    • If p > 0, the parabola opens upwards.
    • If p < 0, the parabola opens downwards.

Recognizing the standard forms is the first step in deciphering the given equation. By comparing the given equation to these standard forms, we can determine the orientation of the parabola (horizontal or vertical) and then proceed to identify the vertex and the value of p, which is essential for locating the focus. In the next section, we'll apply this knowledge to our specific equation, y² - 12y + 4x + 4 = 0, and transform it into a standard form to make finding the vertex and focus a straightforward process.

2. Transforming the Equation into Standard Form

Now, let's take the given equation, y² - 12y + 4x + 4 = 0, and manipulate it into the standard form of a parabola. This involves a process called completing the square. Completing the square allows us to rewrite the equation in a way that clearly reveals the vertex and the p value, which are crucial for determining the focus. By following this step-by-step transformation, we'll bring the equation into a form that directly matches one of the standard parabola equations discussed earlier.

  1. Isolate the y-terms: First, we'll group the y terms together and move the x term and the constant to the other side of the equation:

    y² - 12y = -4x - 4

  2. Complete the square for the y-terms: To complete the square, we need to add and subtract the square of half the coefficient of the y term. The coefficient of our y term is -12, half of which is -6, and the square of -6 is 36. So, we add 36 to both sides of the equation:

    y² - 12y + 36 = -4x - 4 + 36

  3. Rewrite as a squared term: Now, the left side of the equation can be written as a perfect square:

    (y - 6)² = -4x + 32

  4. Factor out the coefficient of x: To get the equation into the standard form, we need to factor out the coefficient of the x term, which is -4, from the right side of the equation:

    (y - 6)² = -4(x - 8)

Now our equation is in the standard form: (y - 6)² = -4(x - 8). By comparing this to the standard form (y - k)² = 4p(x - h), we can easily identify the vertex and the value of p. This transformation is a critical step in solving the problem, as it sets the stage for a straightforward determination of the parabola's key features. In the next section, we will use this standard form to pinpoint the vertex and calculate the focal distance p, leading us closer to finding the focus.

3. Identifying the Vertex

With the equation now in the standard form (y - 6)² = -4(x - 8), identifying the vertex is a breeze. Recall that the standard form for a horizontal parabola is (y - k)² = 4p(x - h), where (h, k) represents the vertex. By directly comparing our transformed equation with this standard form, we can extract the values of h and k and thus determine the vertex coordinates.

In our equation, we have:

  • (y - 6)², which corresponds to (y - k)², so k = 6.
  • (x - 8), which corresponds to (x - h), so h = 8.

Therefore, the vertex of the parabola is at the point (h, k) = (8, 6). This point serves as the central reference for the parabola, and knowing its coordinates is essential for locating other key features, such as the focus and directrix. The vertex is, in essence, the "corner" of the parabola, and its position dictates the overall placement of the curve in the coordinate plane. In the following section, we will leverage the standard form equation further to calculate the focal distance p and, ultimately, determine the coordinates of the focus.

4. Calculating the Focal Distance (p)

Next, we need to calculate the focal distance, p, which is the distance between the vertex and the focus, and also between the vertex and the directrix. This value is crucial for locating the focus, which is one of the key elements we're aiming to find. The value of p is directly related to the coefficient in the standard form equation, and by carefully examining our equation, we can easily determine its value. Understanding the sign of p will also tell us the direction in which the parabola opens, which is essential for correctly positioning the focus.

From the standard form equation, (y - 6)² = -4(x - 8), we can see that the term 4p corresponds to -4. Therefore, we can set up the equation:

4p = -4

Dividing both sides by 4, we get:

p = -1

The fact that p is -1 tells us two important things:

  1. The distance between the vertex and the focus is 1 unit.
  2. Since p is negative, the parabola opens to the left. This is because for parabolas in the form (y - k)² = 4p(x - h), a negative p value indicates a leftward-opening parabola.

Now that we've calculated p and know the direction the parabola opens, we have all the necessary information to pinpoint the focus. In the next section, we will use the vertex coordinates and the value of p to determine the exact location of the focus, completing our solution.

5. Determining the Focus

With the vertex at (8, 6) and the focal distance p = -1, we can now determine the focus of the parabola. Since the parabola opens to the left (because p is negative), the focus will be located to the left of the vertex. The distance between the vertex and the focus is the absolute value of p, which is 1 in this case. To find the coordinates of the focus, we simply adjust the x-coordinate of the vertex by the value of p.

The x-coordinate of the vertex is 8. Since the parabola opens to the left, we subtract the absolute value of p (which is 1) from the x-coordinate:

x_focus = 8 + p = 8 + (-1) = 7

The y-coordinate of the focus remains the same as the y-coordinate of the vertex, which is 6.

Therefore, the focus of the parabola is at the point (7, 6). This point represents the heart of the parabola, and its location is directly influenced by the vertex and the focal distance. By systematically applying the standard form equation and the properties of parabolas, we've successfully navigated the process of finding the focus.

Conclusion

In this article, we successfully found the vertex and focus of the parabola defined by the equation y² - 12y + 4x + 4 = 0. By transforming the equation into standard form, we identified the vertex as (8, 6) and calculated the focal distance p to be -1. This allowed us to determine that the parabola opens to the left and that the focus is located at (7, 6). Understanding these key features provides a comprehensive understanding of the parabola's shape and position.

For further exploration of parabolas and conic sections, you can visit Khan Academy's section on Conic Sections.