Problem 21 - Precalculus: Functions and Graphs

Question

Find an equation of the parabola that satisfies the given conditions. Focus \(F(2,0), \quad\) directrix \(x=-2\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y^2 = 8x\).

Step 1: Understanding the Parabola

The parabola's focus is at \((2, 0)\) and its directrix is the vertical line \(x = -2\). The parabola will open horizontally because the directrix is vertical. The vertex will be the midpoint between the focus and the directrix.

Step 2: Find the Vertex

The vertex is the midpoint between the focus \((2,0)\) and the point on the directrix that is closest to the focus, which is \((-2,0)\). Find the midpoint:\[ x = \frac{2 + (-2)}{2} = 0, \quad y = \frac{0 + 0}{2} = 0 \]So, the vertex is \((0,0)\).

Step 3: Determine the Distance (p)

The distance \(p\) from the vertex to both the focus \((2,0)\) and the directrix \(x = -2\) is 2 units.

Step 4: Write the Parabola Equation

Since the parabola opens horizontally with vertex \((0, 0)\) and \(p = 2\), the equation for a horizontally opening parabola is of the form:\[(y - k)^2 = 4p(x - h)\]Substitute \(h = 0\), \(k = 0\), and \(p = 2\):\[y^2 = 4(2)(x) = 8x\]Thus, the equation of the parabola is \(y^2 = 8x\).

Key Concepts

Understanding Focus and Directrix

In the study of parabolas, the concepts of focus and directrix are pivotal. Let's dive into what they mean and how they impact the equation of a parabola.
To start, a **focus** is a fixed point used to define the parabola. The geometric rule is that the distance from any point on the parabola to the focus is equal to the distance of that point to the directrix.
The **directrix** is a fixed line. In this context, with a vertical directrix at \(x = -2\), the parabola will open **horizontally**. To visualize, imagine all the points on the curve maintain equal distances to both the directrix and the focus \((2,0)\).
This crucial configuration guides the parabola's shape and orientation, leading to the vertex lying precisely halfway between the focus and the directrix.

Defining the Vertex of a Parabola

The vertex of a parabola is the point where the curve turns or changes direction. It's a key component, as it helps in determining the parabola's position and orientation.
For a parabola, the vertex is centrally located between the focus and the directrix. In our exercise, with a focus at \((2, 0)\) and a directrix along \(x = -2\), the midpoint becomes crucial.
The midpoint between these two, calculated as

• \(x = \frac{2 + (-2)}{2} = 0\)
• \(y = \frac{0 + 0}{2} = 0\)

Places the vertex of the parabola at \((0, 0)\). The vertex serves both as a symmetry axis and the chief reference point in the standard form of the parabola equation.

Exploring the Parabola Opening Direction

The direction in which a parabola opens is a fundamental aspect determined primarily by its focus and directrix.
In this instance, since the directrix is vertical \((x = -2)\), we anticipate the parabola to open horizontally. More precisely, as the focus is to the right of the directrix, the parabola will open towards the focus, which is the right direction.
This understanding is consistent with the formula structure of horizontally opening parabolas, typically denoted in the form

• \((y - k)^2 = 4p(x - h)\)

In our problem, since the vertex is \((0,0)\) and \(p = 2\), the resulting equation becomes \(y^2 = 8x\). This captures the rightward opening, seen in the calculated equation, which assists in visualizing the curve's geometry.

Understanding the Standard Form of Parabola Equation

The standard form of a parabola equation depends on its orientation and location. This 'standard form' provides a structured format to write the parabola's equation based on its focus, vertex, and opening direction.
For a parabola opening either right or left, the equation is of the form:

• \((y - k)^2 = 4p(x - h)\)

Where:

• \((h, k)\) represents the **vertex**
• \(p\) is the distance from the vertex to the focus or directrix

By applying these to our specific example: with vertex \((0,0)\) and \(p = 2\), our equation molds into \(y^2 = 8x\). This equation relays all pertinent information about the parabola, embodying its relationship with the focus and directrix, cementing its structure within this essential standard form.