Question

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation. \(x=\frac{1}{8} y^2\)

Short answer

The focus of the parabola is at (2, 0), the directrix is at \(x = -2\) and the axis of symmetry is the line \(y = 0\).

Step-by-step solution

Identify the Parameters of the Parabola

The given equation \(x=\frac{1}{8} y^2\) is of the form \(x=ay^2\), where \(a=\frac{1}{8}\). Thus the vertex of the parabola is at the origin \(0\).

Identify the Axis of Symmetry

Since this parabola is expressed in terms of \(x\), the axis of symmetry is a horizontal line through the vertex. Hence, the axis of symmetry is the line \(y=0\).

Find the Focus and Directrix

For a parabola of the form \(x=ay^2\), the focus is at \( \left(\frac{1}{4a}, 0 \right) \) and the directrix is the line \(x=-\frac{1}{4a}\). So, here the focus will be at \( \left(2, 0 \right)\) and the directrix will be the line \(x=-2\).

Graph the Equation

The vertex is at the origin, the focus is to the right of it and the directrix is a vertical line to the left of the origin. The graph opens to the right because the coefficient of \(y^2\) is positive.

Key concepts

Axis of Symmetry

The axis of symmetry of a parabola is a line that passes through the vertex and divides the parabola into two mirror-image halves. For parabolas, the axis of symmetry can either be vertical or horizontal depending on the orientation of the parabola.

In the given equation, \(x = \frac{1}{8}y^2\), the parabola opens horizontally because it involves \(y^2\). Therefore, the axis of symmetry is a horizontal line. Specifically, it is the line \(y = 0\), which passes through the vertex at the origin. This line acts as a reflection line for the parabola, meaning that if you fold the parabola over this line, both sides would match perfectly.

Finding the axis of symmetry helps in understanding the orientation and direction of the parabola, and is crucial in graphing the parabola.

Vertex of Parabola

The vertex is one of the key characteristics of a parabola. It is the point where the parabola changes direction. For most parabolas, this point acts as the minimum or maximum point on the curve.

In our equation, \(x = \frac{1}{8}y^2\), the vertex is located at the origin \((0,0)\). This specific location is derived from the general form of a parabola equation \(x = ay^2\), where there are no \(x\) or \(y\) shifts involved, indicating the vertex remains at the origin.

The vertex forms the centerpiece for plotting the rest of the parabola, as it offers a starting point and balances the symmetry of the graph. Always ensure to identify the vertex first when graphing or analyzing parabolas as it dictates the general shape and position of the parabola.

Equations of Parabolas

Understanding the different forms of parabola equations is essential for analyzing their behavior and features. A parabola can be expressed in different formats, typically based on whether it opens up and down or left and right.

For parabolas that open horizontally, like \(x = ay^2\), you can notice distinct characteristics:

• The axis of symmetry is horizontal.
• The focus determines the direction of the opening.
• The vertex remains the center of symmetry.

In our example, \(x = \frac{1}{8}y^2\), the equation signifies a horizontal orientation, opening to the right. The parameter \(a = \frac{1}{8}\) influences the parabola's width and overall shape. A small \(|a|\) value results in a wider parabola, while a larger \(|a|\) produces a steeper one.By knowing the equation and identifying \(a\), you can effectively graph and predict the features of the parabola, including the direction, focus, and vertex. Equations such as \(x = ay^2\) thus provide a comprehensive structure and set of tools to interpret the parabola's properties and visual representation.