Problem 36 - Precalculus Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(x=-\frac{1}{8}\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y^2 = \frac{1}{2}x\).

Step 1: Understanding the Parabola's Orientation

Since the directrix of the parabola is a vertical line, the parabola opens horizontally. This positioning means that the parabola has the form \(y^2 = 4px\), with its vertex at the origin \((0,0)\). Here, \(p\) represents the distance from the vertex to the focus or from the vertex to the directrix.

Step 2: Determine the Value of p

The equation of the directrix is given as \(x = -\frac{1}{8}\). The vertex is at \(x = 0\). Thus, the distance \(p\) is \(\frac{1}{8}\). Since the parabola opens to the right (directrix is to the left of the vertex), \(p\) is \(\frac{1}{8}\).

Step 3: Substitute p into the Standard Form Equation

With \(p = \frac{1}{8}\), substitute into the standard form equation of a horizontal parabola: \(y^2 = 4px\). This gives \(y^2 = 4\times\frac{1}{8}\times x\), which simplifies to \(y^2 = \frac{1}{2}x\).

Key Concepts

Vertex at Origin

In the context of parabolas, the term 'vertex' refers to the point at which the curve changes direction. The vertex is a critical point for any parabola, as it dictates its starting position and orientation. When the vertex is at the origin, this means that the parabola begins at the coordinate point

• (0,0)

This positioning simplifies the equation of the parabola, as it eliminates the need for additional terms accounting for vertex translation. The equation format depends on other properties, such as the direction the parabola opens in.
For instance, a parabola with its vertex at the origin that opens upwards or downwards follows the equation format:

• \( x^2 = 4py \)

Whereas a horizontal parabola fits the format:

• \( y^2 = 4px \)

Thus, knowing that a parabola originates from the vertex helps you to identify its equation quickly. Remember, when dealing with origin-based vertices, you're working with the simplest form possible, focusing solely on direction and scale.

Directrix of a Parabola

The directrix of a parabola is a specific line used to define and construct the parabola along with the focus. It's a line that is equally distant from all points on the curve to the parabola's focus, helping set the curve's width and direction. In this exercise, the given directrix is

• \( x = -\frac{1}{8} \)

A directrix provides valuable information about the properties of a parabola:

• If the directrix is vertical (like in this case), the parabola opens horizontally.
• If the directrix is horizontal, the parabola opens vertically.

The formula for the value of \( p \), the distance from the vertex to the directrix (or to the focus), is crucial here. When the directrix is given, the value of \( p \) can be calculated by the absolute distance between the vertex and the directrix's position. Here, it's simply

• \( p = \frac{1}{8} \)

Understanding the directrix not only helps in writing the equation but also in visualizing the overall shape of the parabola.

Horizontal Parabola

A horizontal parabola has a distinct orientation compared to its more commonly known vertical counterpart. In our case, the parabola opens horizontally due to the vertical orientation of the directrix. This leads to a characteristic equation format, aligned along the y-axis:

• \( y^2 = 4px \)

Here, the placement of \( y^2 \) at the beginning tells us that we're dealing with a horizontal parabola. The value of \( p \) is half the distance from the vertex to either the focus or the directrix. Knowing this distance allows you to calculate how "wide" or "narrow" the parabola will open. Since our given directrix is

• \( x = -\frac{1}{8} \)

Then \( p = \frac{1}{8} \). Substituting this into our standard horizontal parabola equation, we get:

• \( y^2 = \frac{1}{2}x \)

Visualizing horizontal parabolas can often be more tricky than their vertical counterparts, so it helps to plot key points or use graphing tools to gain a better understanding of the shape and spread.