Problem 19 - College Algebra with Modeling and Visualization

Question

Graph the parabola. Label the vertex, focus, and directrix. $$ x=\frac{1}{8} y^{2} $$

Step-by-Step Solution

Answer

Vertex: (0,0), Focus: (2,0), Directrix: \(x = -2\).

Step 1: Identify the Form of the Equation

First, recognize that the equation \(x = \frac{1}{8} y^2\) depicts a parabola that opens sideways because the variable \(x\) is isolated. This is in the form \(x = a(y - k)^2 + h\), where \(h = 0\) and \(k = 0\).

Step 2: Determine the Vertex

Since the terms \(h\) and \(k\) are both zero, it means the vertex of the parabola is at the origin, \((0, 0)\).

Step 3: Calculate Value of 'a' and Determine the Focus and Directrix

The coefficient \(a = \frac{1}{8}\) gives the distance from the vertex to the focus or the directrix. The standard form equation for a parabola that opens horizontally is \(x = a(y - k)^2 + h\) with focus at \((h + \frac{1}{4a}, k)\). Thus, the focus is at \((\frac{1}{4 \times \frac{1}{8}}, 0) = (2, 0)\). The equation of the directrix is \(x = h - \frac{1}{4a} = 0 - 2 = -2\).

Step 4: Sketch the Graph

Draw a horizontal parabola with its vertex at the origin \((0,0)\), focus at \((2,0)\), and directrix at \(x = -2\). The parabola will open to the right since \(a > 0\). Label the points and the directrix line accordingly.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point because it serves as a turning point for the graph. In simpler terms, it's the point where the parabola changes direction. For equations like \(x = a(y - k)^2 + h\), the vertex is given by the coordinates \((h, k)\). This vertex represents the maximum or minimum point, depending on the parabola's orientation.

In our specific example, the equation is \(x = \frac{1}{8} y^2\). This means that the vertex is at \(0, 0\) since the equation is already very close to its simplest form where \(h = 0\) and \(k = 0\).

• This information becomes the starting point to sketch the parabola.
• It serves as a reference to determine other elements like the focus and directrix.

By finding the vertex, you can easily proceed to plot the rest of the parabola correctly.

Focus and Directrix of a Parabola

The concept of focus and directrix helps in understanding how a parabola is formed. At the heart of every parabola is a point known as the "focus," and a line called the "directrix." The parabola is essentially defined as the set of all points equidistant from the focus and directrix.

The location of the focus and the equation of the directrix depend on the coefficient \(a\) in the parabola's equation. For an equation like \(x = a(y-k)^2 + h\), the focus is \((h + \frac{1}{4a}, k)\) and the directrix is \(x = h - \frac{1}{4a}\).

• In our equation \(x = \frac{1}{8}y^2\), where \(a = \frac{1}{8}\), the focus calculates to \( (2, 0) \).
• The directrix is a vertical line \(x = -2\).

Understanding the relative positioning of these elements allows us to now plot the parabola precisely.

This interaction between focus and directrix ensures that the parabola maintains its curved shape.

Equation of a Parabola

The equation of a parabola is a representation of the set of points that form the characteristic "U" or "C" shape graph. For horizontal parabolas, the general form is \(x = a(y-k)^2 + h\), and for vertical parabolas, it's \(y = a(x-h)^2 + k\).

The coefficient \(a\) plays a significant role in determining the shape and direction of the parabola. A positive \(a\) indicates the parabola opens in the positive direction – to the right for horizontal and upwards for vertical. Conversely, a negative \(a\) would open the parabola in the opposite direction.

• In the example \(x = \frac{1}{8} y^2\), since \(a = \frac{1}{8}\) is positive, the parabola opens to the right.
• The vertex is the starting base point and we graph from there.

Understanding these equation properties helps in graphing parabolas, predicting their behavior, and solving related mathematical problems with ease.