Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Find an equation of a parabola that satisfies the given conditions. Focus \((2,1) ;\) directrix \(x=-1\)

Step-by-Step Solution

Answer

The equation of the parabola is \\( (y - 1)^2 = 6x - 3 \\\).

Step 1: Identify the Vertex

The vertex of the parabola is the midpoint between the focus and the directrix. The given focus is \( (2,1) \) and the directrix is the line \( x = -1 \). The midpoint is calculated as \( ((2 + (-1))/2 , 1) = (0.5, 1) \). Therefore, the vertex of the parabola is \( (0.5, 1) \).

Step 2: Determine the Orientation

The equation of the given directrix is \( x = -1 \), which is a vertical line. This indicates that the parabola opens horizontally, specifically to the right, since the focus has a greater x-value than the directrix.

Step 3: Use the Parabola Formula

The standard form of a horizontally opening parabola with vertex \( (h,k) \) is \( (y - k)^2 = 4p(x - h) \). Here, \( p \) is the distance from the vertex to the focus. In this case, the vertex \( (0.5, 1) \) and the focus \( (2, 1) \) are 1.5 units apart horizontally, so \( p = 1.5 \). The vertex form of the equation becomes \( (y - 1)^2 = 6(x - 0.5) \).

Step 4: Simplify the Equation

Expand \( 6(x - 0.5) \): \( 6x - 3 \). Thus, the equation of the parabola in a more standard form is \( (y - 1)^2 = 6x - 3 \).

Key Concepts

Focus and Directrix

In the world of parabolas, the focus and directrix form the backbone of its shape and orientation. The **focus** is a point inside the parabola, and the parabola curves around it. Meanwhile, the **directrix** is a straight line outside of the curve. Together, they work to define a parabola's precise shape.

Here's how they relate to a point on the parabola:

• The distance from the focus to any point on the parabola is equal to the distance from that point to the directrix. This balancing feature is essential for understanding the properties of a parabola.
• The focus and directrix help determine whether a parabola opens up, down, left, or right. When the directrix is a vertical line like in our exercise, the parabola opens sideways.

Understanding the focus and directrix empowers you with the core concepts needed to analyze any parabola's structure and orientation.

Vertex Form

The vertex form of a parabola offers a clear and simplified approach to writing the equation of the curve. It's particularly user-friendly because it directly reveals the vertex location, which is the parabola's peak or lowest point.

The vertex form of a parabola's equation is \((y - k)^2 = 4p(x - h)\) for horizontally opening parabolas and \((x - h)^2 = 4p(y - k)\) for vertically opening ones. Here:

• \((h, k)\) represents the vertex position.
• \(p\) indicates the distance from the vertex to the focus, driving the "width" of the parabola.

In the exercise, we found the vertex by considering the midpoint between the focus and directrix, yielding \((0.5, 1)\). This form simplifies computations, giving immediate insights into the parabola's orientation and opening direction.

Equation of a Parabola

The equation of a parabola is a powerful expression capturing all characteristics of its curve. It binds together concepts like the focus, directrix, and vertex to present a mathematical description of the parabola.

The standard equation for a horizontally opening parabola, as seen in the exercise, is \((y - k)^2 = 4p(x - h)\). To derive this equation:

• Identify the vertex \((h, k)\), which for us, is \((0.5, 1)\).
• Measure the distance \(p\) from the vertex to the focus, which was 1.5 units in our example.

This results in the equation \((y - 1)^2 = 6(x - 0.5)\), highlighting the orientation and behavior of the parabola. Expanding this further into its standard form, \((y - 1)^2 = 6x - 3\), you get a complete picture of how this mathematical curve is constructed from its foundational elements.