Parabola Equation In Vertex Form Focus (2,-4) And Directrix Y=-6

Jul 10, 2025 by ADMIN 65 views

Unveiling the Equation of a Parabola in Vertex Form: A Comprehensive Guide

Embark on a journey into the fascinating world of parabolas, those elegant curves that grace the realm of mathematics and find applications in various aspects of our lives. In this comprehensive guide, we will delve into the intricacies of parabolas, focusing on deriving their equations in vertex form. Our specific quest is to determine the equation of a parabola given its focus and directrix. Let's unravel the secrets of these captivating curves!

Understanding the Parabola: A Geometric Definition

At its heart, a parabola is defined geometrically as the set of all points that are equidistant to a fixed point, known as the focus, and a fixed line, known as the directrix. The focus resides inside the curve of the parabola, while the directrix lies outside. The line passing through the focus and perpendicular to the directrix is called the axis of symmetry, which divides the parabola into two symmetrical halves. The point where the parabola intersects its axis of symmetry is the vertex, which represents the turning point of the curve.

To truly grasp the essence of a parabola, envision a point tracing a path such that its distance to the focus remains perpetually equal to its distance to the directrix. This elegant dance between point, focus, and directrix gives rise to the parabolic form. The vertex form of a parabola's equation provides a concise and insightful way to express this geometric relationship algebraically.

The Vertex Form: A Gateway to Parabola Equations

The vertex form of a parabola's equation serves as a powerful tool for unlocking the secrets of these curves. It elegantly encapsulates the key parameters that define a parabola: the vertex and the distance between the vertex and the focus. The general vertex form equation depends on whether the parabola opens vertically or horizontally:

For a parabola opening vertically (upwards or downwards), the vertex form is:
```
(x - h)^2 = 4p(y - k)
```
For a parabola opening horizontally (leftwards or rightwards), the vertex form is:
```
(y - k)^2 = 4p(x - h)
```

Here, (h, k) represents the coordinates of the vertex, and 'p' denotes the directed distance from the vertex to the focus. The sign of 'p' dictates the direction in which the parabola opens. If 'p' is positive, the parabola opens upwards (vertical) or rightwards (horizontal). Conversely, if 'p' is negative, the parabola opens downwards (vertical) or leftwards (horizontal).

Unveiling the Equation: Focus and Directrix as Guiding Stars

Now, let's embark on our primary quest: determining the equation of a parabola in vertex form given its focus and directrix. We are presented with a parabola whose focus resides at the point (2, -4) and whose directrix is the line y = -6. Our mission is to translate these geometric clues into the concise language of an equation.

The first step involves locating the vertex, the pivotal point that anchors our parabola. Recall that the vertex lies midway between the focus and the directrix. Geometrically, this means the vertex is the midpoint of the perpendicular segment connecting the focus and the directrix. Since the directrix is a horizontal line (y = -6), the perpendicular segment will be vertical. The x-coordinate of the vertex will match that of the focus, which is 2. To find the y-coordinate, we compute the average of the y-coordinates of the focus and the point where the perpendicular segment intersects the directrix:

Vertex y-coordinate = (-4 + (-6)) / 2 = -5

Thus, the vertex of our parabola gracefully resides at the point (2, -5).

Next, we must determine the directed distance, 'p', between the vertex and the focus. This distance plays a crucial role in shaping the parabola's equation. Since the focus lies above the vertex (the y-coordinate of the focus, -4, is greater than the y-coordinate of the vertex, -5), we know that the parabola opens upwards. The directed distance 'p' is simply the difference between the y-coordinates of the focus and the vertex:

p = -4 - (-5) = 1

The fact that 'p' is positive reaffirms our earlier observation that the parabola opens upwards.

Now, armed with the vertex (h, k) = (2, -5) and the directed distance p = 1, we are ready to invoke the vertex form equation for a parabola opening vertically:

(x - h)^2 = 4p(y - k)

Substituting our values, we get:

(x - 2)^2 = 4 * 1 * (y - (-5))

Simplifying, we arrive at the equation of our parabola in vertex form:

(x - 2)^2 = 4(y + 5)

Deciphering the Answer Choices: A Triumphant Match

Our derived equation, (x - 2)^2 = 4(y + 5), resonates harmoniously with answer choice D. Therefore, we confidently declare that option D is the correct equation of the parabola in vertex form.

Delving Deeper: Alternative Approaches and Key Insights

While we have successfully navigated our parabolic quest using the vertex form, let's briefly explore alternative approaches and glean some key insights.

The Geometric Definition Unveiled

Recall that a parabola is defined as the locus of points equidistant to the focus and the directrix. We could leverage this fundamental definition to derive the equation. Let (x, y) be any point on the parabola. The distance between (x, y) and the focus (2, -4) is:

√((x - 2)^2 + (y + 4)^2)

The distance between (x, y) and the directrix y = -6 is simply |y + 6|.

Equating these distances, we get:

√((x - 2)^2 + (y + 4)^2) = |y + 6|

Squaring both sides and simplifying, we would eventually arrive at the same equation, (x - 2)^2 = 4(y + 5).

Key Insights: A Parabola's Personality

The vertex form equation provides a clear snapshot of a parabola's key features: its vertex and the direction in which it opens.
The directed distance 'p' plays a crucial role in determining the parabola's shape. A larger absolute value of 'p' implies a wider parabola, while a smaller value indicates a narrower curve.
The geometric definition of a parabola, rooted in the equidistance property, offers a powerful alternative approach to deriving its equation.

Conclusion: Mastering the Parabola's Equation

In this comprehensive guide, we have successfully unraveled the equation of a parabola in vertex form, given its focus and directrix. We traversed the geometric landscape of parabolas, understood the significance of the vertex form, and applied our knowledge to a specific problem. By mastering the techniques and insights presented here, you are well-equipped to tackle a wide range of parabolic challenges. So, embrace the elegance of parabolas, and let their curves guide you through the fascinating world of mathematics!