Problem 41 - 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: \(x=\frac{1}{20}\)

Step-by-Step Solution

Answer

The equation is \(y^2 = -\frac{1}{5}x\).

Step 1: Understand the Problem

We need to find an equation for a parabola with its vertex at the origin. The directrix is given as a vertical line, which suggests the parabola opens horizontally.

Step 2: Recall the Standard Form

For a parabola that opens left or right, the standard form is \((y - k)^2 = 4p(x-h)\) where \((h, k)\) is the vertex. Since our vertex is at the origin, this simplifies to \(y^2 = 4px\).

Step 3: Identify the Parameters

The directrix is \(x = \frac{1}{20}\). For the standard form \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus and also from the vertex to the directrix, we have \(-p = -\frac{1}{20}\). This tells us the directrix is \(\frac{1}{20}\) units away from the vertex on the positive side of the x-axis.

Step 4: Solve for Parameter \(p\)

From the equation of the directrix \(x = \frac{1}{20}\), we know \(-p = -\frac{1}{20}\) and therefore \(p = -\frac{1}{20}\). The negative sign confirms that the parabola opens to the negative x-direction.

Step 5: Write the Parabola's Equation

Insert the value of \(p\) back into the simplified equation. We have \(y^2 = 4(-\frac{1}{20})x\). Simplifying it, we get \(y^2 = -\frac{1}{5}x\).

Step 6: Verify the Solution

Check that the characteristics of the derived equation conform to the given facts (vertex at origin, correct position of the directrix). The parabola's equation \(y^2 = -\frac{1}{5}x\) is aimed leftward due to the negative sign, matching the condition set by the directrix at \(x = \frac{1}{20}\).

Key Concepts

Directrix of a Parabola

The directrix of a parabola is an essential element that helps define its geometric properties. A directrix is a line which, along with the parabola’s focus, helps maintain the symmetrical, uniform shape of the parabola.
In the problem at hand, the directrix is a vertical line given by the equation \(x=\frac{1}{20}\). This means it is positioned at \(\frac{1}{20}\) units to the right of the vertex, which is located at the origin \((0,0)\). The parabola must be symmetric relative to this fixed line.
Key points to know about the directrix:

• It indicates the direction in which the parabola opens (opposite to where the directrix lies).
• It is one part of the relationship defining the curve, the other being the focus.
• Together with the vertex, it helps determine the orientation and steepness of the parabola.

Understanding the directrix’s role is a cornerstone in determining the equation of the parabola.

Equation of a Parabola

The equation of a parabola reveals the structure and orientation of the curve. For a parabola with a vertex at the origin, the form can be simplified based on its opening direction.
For parabolas that open horizontally (to the left or right), the formula is \((y-k)^2 = 4p(x-h)\), with \((h, k)\) being the vertex. Since the vertex is at \((0,0)\), our equation diminishes to \(y^2 = 4px\).
The parameter \(p\) in this formula represents the distance from the vertex to the focus (or equivalently, to the directrix on the opposite side).

• \(p > 0\) implies the parabola opens to the right.
• \(p < 0\) means it opens to the left.

Thus, knowing the coordinates of the directrix and the vertex, we derived the equation for the parabola: \(y^2 = -\frac{1}{5}x\). This equation confirms the shape opens leftward, aligning with all defined properties.

Parabola Opening Direction

Understanding the parabola's opening direction is crucial in graphing and analyzing its behavior. This direction is primarily dictated by the sign of the parameter \(p\) in the parabola’s equation.
Since our derived equation is \(y^2 = -\frac{1}{5}x\), the negative sign in front of \(\frac{1}{5}x\) indicates that the parabola opens to the left.
Here is how you identify and understand the opening direction:

• If \(p > 0\), the parabola opens to the right.
• If \(p < 0\), it opens to the left.
• The orientation (horizontal or vertical) is determined by whether \(y\) or \(x\) is squared in the equation.

In our problem, the vertex is at the origin, and the directrix, being on the positive side of the x-axis, leads to a negative \(p\). Thus, confirming the negative direction and finding the equation \(y^2 = -\frac{1}{5}x\). This understanding prevents any complications while working with various parabolic shapes.