Problem 28 - Precalculus with Limits

Question

In Exercises 19-32, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x = 3\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x = -12y\).

Step 1: Determine the Orientation of the Parabola

The equation of the directrix is given as \(x = 3\), this indicates that the parabola opens towards the left (if 'p' is negative) or to the right (if 'p' is positive). Since the directrix is to the right of the origin, the parabola opens to the left, implying that 'p' is negative.

Step 2: Determine the value of 'p'

The distance from the vertex (0, 0) to the directrix \(x = 3\) is 'p'. Since the parabola opens to the left, 'p' will be negative. Therefore, the distance value, which is 3, will have a negative sign, i.e., \(p = -3\).

Step 3: Formulate the equation of the parabola

Given that 'p' for parabola opening to the left or right is represented in the standard equation \(x = 4py\), substituting for 'p', we get \(x = 4(-3)y\) or \(x = -12y\).

Key Concepts

Directrix

In the context of parabolas, the directrix is a fixed line that, together with the focus, helps define the shape and orientation of the parabola. The directrix is perpendicular to the axis of symmetry of the parabola.

• For a parabola whose vertex is at the origin, the location of the directrix can help us determine in which direction the parabola opens.
• If the directrix is parallel to the y-axis, the parabola opens horizontally. If it is parallel to the x-axis, the parabola opens vertically.

For the given exercise, the directrix is at \(x = 3\), which means it is vertical. This reveals that our parabola opens left or right. Because the directrix is on the right of the origin \(at \(x = 0\)\), it indicates that the parabola opens to the left.

Vertex at the Origin

The vertex of a parabola is the point where it changes direction, and in this exercise, the vertex is at the origin (0, 0). This simplifies finding the equation of the parabola, as it streamlines the formula to include the directrix and focus only.

• With the vertex at the origin, the symmetry of the parabola is consistent, making it easier to predict its behavior across the coordinate system.
• For our horizontally opening parabola, the standard equation becomes of the form \(x = 4py\) or \(y = 4px\).

The vertex at the origin ensures that calculations are straightforward since the origin (0, 0) coordinates do not modify the equation's terms.

Standard Form of Equation

The standard form of a parabola's equation depends on its orientation. For parabolas oriented left or right, like in our case, the standard form is \(x = 4py\).

• In this equation, \(p\) represents the distance from the vertex to the focus or, equivalently (with an opposite sign), from the vertex to the directrix.
• A negative \(p\) indicates the parabola opens leftwards, while a positive \(p\) signals a rightward opening.

In this scenario, where \(x = 3\) is the directrix to the right, \(p = -3\) for a leftward opening. Thus, inserting \(p\) into the standard form, we derive the equation \(x = -12y\). This fully describes our leftward-opening parabola with its vertex at the origin.