Question

Write an equation of the parabola with the given characteristics. directrix: \(y=\frac{8}{3}\) vertex: \((0,0)\)

Short answer

The equation of the parabola is \(x^2 = -\frac{32}{3}y\).

Step-by-step solution

Determine the value of p

The value of p can be determined from the equation of the directrix, \(y = -p\). So, \(-p = \frac{8}{3}\): therefore, \(p = -\frac{8}{3}\).

Write the equation of the parabola

The standard form of the equation for a parabola that opens downwards from the vertex at the origin is \(x^2 = 4py\). Plug in the value of \(p\) from Step 1 to find the equation, so it becomes: \(x^2 = 4(-\frac{8}{3})y\). Simplifying this equation gives: \(x^2 = -\frac{32}{3}y\).

Key concepts

Directrix

The directrix is a crucial component when studying the properties of a parabola. It is a fixed straight line that, in conjunction with the focus, helps define the set of points that constitute the parabola. In the context of our example, the directrix is given by the equation \(y = \frac{8}{3}\). This tells us it is a horizontal line located at \(y = \frac{8}{3}\). When you have a parabola, every point on it is equidistant from the focus and the directrix. This balanced relationship between the directrix and the parabola is what gives the parabola its symmetrical, curved shape. To identify the directrix's role:

• It serves as a reference line used to dynamically calculate and describe the layout and orientation of the parabola.
• In this problem, the directrix helps us deduce important parameters like \(p\), which is the distance from the vertex to the directrix along the axis of symmetry.

Vertex

The vertex of a parabola is a point from which the parabola originates, and it represents the parabola's peak or lowest point, depending on its orientation. In our problem, the vertex is given as \((0,0)\). This is one of the simplest cases where the vertex is at the origin, which simplifies the parabola's equation significantly. The vertex is important because it:

• Defines the central axis or line of symmetry of the parabola, which runs vertically through the vertex.
• Acts as a pivot point around which the parabola opens either up, down, or sideways, as determined by other parameters like \(p\).

Understanding the vertex allows for determining other characteristics of the parabola with respect to both its geometric and algebraic properties. In this exercise, the vertex location influences how parameters like the directrix and \(p\) relate in the equation.

Standard Form of Parabola

The standard form of a parabola is an algebraic representation that makes it easier to analyze and understand its geometric properties. For a parabola that opens vertically, the standard form is generally \(x^2 = 4py\). In this equation:

• \(x\) and \(y\) represent the coordinates of any point on the parabola.
• \(p\) is a constant that indicates the distance from the vertex to the focus, and also from the vertex to the directrix.

In this lesson, since our parabola opens downwards, we found the equation to be \(x^2 = -\frac{32}{3}y\). The negative sign signifies that the parabola opens in the downward direction, reflecting a key piece of information about its orientation. The value \(-\frac{32}{3}\) comes from substituting \(p = -\frac{8}{3}\) into the standard equation, allowing you to express the entire curve.

Conic Sections

Parabolas are classified under conic sections, which are the curves obtained by intersecting a plane with a double-napped cone. Understanding this background can provide more depth about why the shape behaves the way it does. Conic sections also include ellipses, circles, and hyperbolas, but the parabola is distinct for its symmetry and singular directrix. Key aspects of parabola under conic sections include:

• It is the set of all points that are equidistant from a given point (focus) and a line (directrix).
• The geometric nature of parabolas makes them useful in real-world applications, like designing satellite dishes and headlights.

This problem illustrates a specific case of a parabola opening downward, which is just one instance of how versatile and broad "parabola shapes" can be in both theory and practical applications!