Finding The Quadratic Function Of A Parabola Given Vertex And Directrix

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In the realm of conic sections, the parabola holds a special place, characterized by its unique reflective property and its ubiquitous presence in various scientific and engineering applications. Understanding the properties of a parabola, such as its vertex and directrix, is crucial for determining its quadratic function, which provides a mathematical representation of its shape and position. In this comprehensive exploration, we will delve into the process of finding the quadratic function of a parabola when its vertex and directrix are known, providing a step-by-step guide and illustrative examples to solidify your understanding.

Understanding the Key Concepts

Before embarking on the journey of determining the quadratic function, let's first establish a firm grasp of the fundamental concepts that underpin the parabola. A parabola is defined as the set of all points that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix. The vertex of a parabola is the point where the parabola changes direction, and it lies exactly midway between the focus and the directrix.

The quadratic function of a parabola is a mathematical expression that describes the relationship between the x and y coordinates of points on the parabola. The general form of a quadratic function is given by:

y = a(x - h)^2 + k

where:

  • (h, k) represents the coordinates of the vertex of the parabola.
  • a is a constant that determines the parabola's direction and width.

The directrix of a parabola is a line that lies outside the parabola, and it plays a crucial role in defining the parabola's shape. The distance between any point on the parabola and the focus is equal to the distance between that point and the directrix.

Determining the Quadratic Function: A Step-by-Step Guide

Now that we have a solid understanding of the key concepts, let's proceed with the step-by-step guide to finding the quadratic function of a parabola when its vertex and directrix are given.

Step 1: Identify the Vertex and Directrix

The first step is to identify the coordinates of the vertex, denoted as (h, k), and the equation of the directrix. These values will be provided in the problem statement.

For instance, let's consider the example where the vertex of the parabola is at (-3, 1) and the directrix is given by the equation y = 6.

Step 2: Determine the Value of 'a'

The constant a in the quadratic function determines the parabola's direction and width. To find the value of a, we utilize the relationship between the vertex, directrix, and the focus of the parabola.

The distance between the vertex and the directrix is equal to the absolute value of 1/(4a). In our example, the vertex is at (-3, 1) and the directrix is y = 6. The distance between the vertex and the directrix is |6 - 1| = 5. Therefore, we have:

|1/(4a)| = 5

Solving for a, we get two possible values:

a = 1/(4 * 5) = 1/20

or

a = -1/(4 * 5) = -1/20

To determine the correct value of a, we consider the parabola's orientation. Since the directrix is above the vertex, the parabola opens downward, which means the value of a must be negative. Therefore, we choose a = -1/20.

Step 3: Substitute the Values into the Quadratic Function

Now that we have determined the values of h, k, and a, we can substitute them into the general form of the quadratic function:

y = a(x - h)^2 + k

Substituting h = -3, k = 1, and a = -1/20, we get:

y = (-1/20)(x - (-3))^2 + 1

Simplifying the equation, we obtain the quadratic function of the parabola:

y = (-1/20)(x + 3)^2 + 1

Illustrative Examples

To further solidify your understanding, let's consider a few more examples.

Example 1:

Find the quadratic function of the parabola with vertex at (2, -3) and directrix x = 5.

  • Step 1: Identify the vertex and directrix: (h, k) = (2, -3) and x = 5.
  • Step 2: Determine the value of a. The distance between the vertex and the directrix is |5 - 2| = 3. Since the directrix is to the right of the vertex, the parabola opens to the left, and a is negative. Therefore, |1/(4a)| = 3, which gives a = -1/12.
  • Step 3: Substitute the values into the quadratic function: y = (-1/12)(y - (-3))^2 + 2. Simplifying, we get x = (-1/12)(y + 3)^2 + 2.

Example 2:

Find the quadratic function of the parabola with vertex at (0, 0) and directrix y = -2.

  • Step 1: Identify the vertex and directrix: (h, k) = (0, 0) and y = -2.
  • Step 2: Determine the value of a. The distance between the vertex and the directrix is |-2 - 0| = 2. Since the directrix is below the vertex, the parabola opens upward, and a is positive. Therefore, |1/(4a)| = 2, which gives a = 1/8.
  • Step 3: Substitute the values into the quadratic function: y = (1/8)(x - 0)^2 + 0. Simplifying, we get y = (1/8)x^2.

Conclusion

In this comprehensive exploration, we have successfully navigated the process of finding the quadratic function of a parabola when its vertex and directrix are known. By understanding the fundamental concepts of parabolas, directrices, and quadratic functions, and by following the step-by-step guide outlined in this article, you can confidently determine the quadratic function for any parabola given its vertex and directrix. This knowledge empowers you to delve deeper into the fascinating world of conic sections and their applications in various fields of study.

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