Problem 40 - 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 Orientation

Since the directrix is a vertical line, the parabola opens either to the right or to the left. The directrix is given as \(x = -\frac{1}{8}\), which means the parabola opens to the right since the vertex is at the origin \((0,0)\) and the directrix is on the negative x-axis side.

Step 2: Using the Parabola Formula

The standard form equation for a parabola that opens sideways (horizontal orientation) is given by \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix). Since the vertex is at the origin, we need to find \(p\) based on its directrix.

Step 3: Calculating Parameter \(p\)

The distance \(p\) is the absolute value of the x-coordinate of the directrix, since the vertex \((0,0)\) is the midpoint between the focus and the directrix. Hence, \(p = \frac{1}{8}\). Because the directrix is to the left, the focus must be to the right and \(p\) remains positive.

Step 4: Writing Equation of the Parabola

Substitute \(p = \frac{1}{8}\) into the parabola equation \(y^2 = 4px\). This gives us \[ y^2 = 4\left(\frac{1}{8}\right)x \]. Simplifying this, we have \[ y^2 = \frac{1}{2}x \].

Key Concepts

Vertex Form of a Parabola

A parabola is a curve where any point is equidistant from a fixed point, known as the focus, and a fixed straight line, known as the directrix. The **vertex form** of a parabola provides a straightforward way to express this relationship. The general formula for a parabola in vertex form is \[ y = a(x-h)^2 + k \] where

• **\((h, k)\)** represents the coordinates of the vertex
• **\(a\)** determines the direction and width of the parabola

For the exercise above, the vertex is given at the origin, meaning it is at point \((0, 0)\). Hence, our vertex form simplifies. When a parabola opens sideways, as in this exercise, one might see the formula written differently:\[ x = a(y-k)^2 + h \]This shows the sideways opening effect, with transformations around the y-axis.
Exploring vertex forms further reveals how transformations such as shifts and stretches affect the graph. Understanding this allows students to predict how the equation of a parabola changes with different vertex positions and opening directions.

Parabola Directrix

The **directrix** is a critical element in defining a parabola. It is the line from which distances are measured to form a parabolic shape. In simple terms, a parabola is a set of all points that are equidistant from the focus and the directrix.
For a horizontally opening parabola, as stated in our problem, the directrix is vertical. The directrix also indicates the orientation of the parabola. In our given exercise, the directrix is at \(x = -\frac{1}{8}\), suggesting the parabola opens to the right.
To determine a parabola's equation, knowing the position of the directrix is essential. It helps calculate the parameter **\(p\)**, which is the distance from the vertex to the directrix, and is also used to find the distance to the focus. In this problem, with a vertex at the origin, \(p\) is the absolute distance to
\(-\frac{1}{8}\), which becomes \(\frac{1}{8}\). By understanding the role of the directrix, students can better grasp how it determines the shape and position of a parabola.

Parabola Focus

The **focus** of a parabola is a fixed point used in its definition and provides a reference for shaping the parabola. Together with the directrix, it helps plan the parabola’s path. The focus is crucial in determining both the position and the form of the parabola.
For parabolas opening horizontally, the focus can be found along the same horizontal line as the vertex. In our exercise, since the directrix is given and the vertex is at \((0,0)\), the focus can be derived by splitting the distance of \(\frac{1}{8}\) equally from the vertex to the focus and the directrix. This means the focus is positioned at \((\frac{1}{8}, 0)\).
In visualizing parabola equations, the focus affects the curvature significantly. The shorter the distance between vertex and focus, the tighter the curve. Understanding the concept of focus will help students grasp why certain parabolas are broad while others are narrow, depending on their focal distance.