Problem 37 - College Algebra with Modeling and Visualization

Question

Find an equation of the parabola with vertex \((0,0)\) that satisfies the given conditions. $$ \text { Directrix } x=\frac{1}{4} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Form of the Parabola Equation

Based on the given vertex \((0,0)\), the general equation of a parabola can be of the form \(y^2 = 4px\). We need to find \(p\) such that the directrix is as given.

Step 2: Understand Directrix to Equation Relationship

The directrix of a parabola with equation \(y^2 = 4px\) is \(x = -p\). The directrix given is \(x = \frac{1}{4}\). So, we set \(-p = \frac{1}{4}\).

Step 3: Solve for p

Set \(-p = \frac{1}{4}\) and solve for \(p\): \[-p = \frac{1}{4}\]\[p = -\frac{1}{4}\].

Step 4: Write the Equation with Found p Value

Plug the value of \(p\) into the general parabola equation. The equation becomes:\[y^2 = 4(-\frac{1}{4})x\]Which simplifies to:\[y^2 = -x\].

Key Concepts

Directrix

The directrix is an essential component in defining the shape and orientation of a parabola. In simple words, it is a straight line that helps in constructing the parabola and is used to determine its geometric properties. The distance between any point on the parabola and the focus is equal to the perpendicular distance from the same point to the directrix. This property is key to understanding the geometry of parabolas.
For a parabola with the vertex at the origin, when the focus lies on the positive x-axis, the directrix will be a vertical line on the opposite side.

• If the parabola opens to the right, the directrix is a line to the left of the vertex.
• If it opens to the left, the directrix will be to the right.

In this case, since the directrix is given as \(x = \frac{1}{4}\), this suggests that the parabola opens to the left, because the directrix must lie in the opposite direction of the opening of the parabola, relative to the origin. Understanding the directrix helps in setting the equation in its correct algebraic form, leading us to find the correct parameter \(p\).

Vertex

The vertex of a parabola is one of its most important points as it acts as the origin of symmetry for the parabola. In mathematical terms, it's the point from which the parabola takes its unique shape. A given vertex allows for the precise configuration of the parabola's focus, axis, and directrix.
The vertex can be thought of as:

• The minimum or maximum point of a parabola that opens upwards or downwards, respectively.
• The turning point in parabolas that open left or right.

In the original exercise, the vertex is specifically located at \((0, 0)\). This means that it serves as the reference point from which all other characteristics and dimensions of the parabola are established. From here, the underlying structure of the parabola, including parameters like the orientation and position of the directrix, can be logically inferred.

General Parabola Formula

The general formula for a parabola shows us how to express its geometric properties through an algebraic equation. This relationship is vital for plotting and understanding parabolas. For a parabola with a vertical axis of symmetry, the general form is \(y = ax^2 + bx + c\). However, for parabolas that open either horizontally or vertically with the vertex at the origin, a simpler form is used to express them.

• Horizontal Parabolas: \(y^2 = 4px\)
• Vertical Parabolas: \(x^2 = 4py\)

For the exercise given, since the equation involves \(y^2\) and not \(x^2\), it's evident that the parabola opens horizontally.We can identify the coefficient \(p\) by relating it to the distance from the vertex to the focus or directrix. In this exercise, we determined that \(p = -\frac{1}{4}\).The full understanding of this formula was crucial in deriving the specific parabola equation of \(y^2 = -x\), which matches the given condition of the vertex and directrix.