Understanding the Elements of a Parabola: Vertex, Focus, Latus Rectum, and Directrix

When working with parabolas, it's essential to understand the various elements that define the shape and behavior of the curve. This article delves into the key elements: the vertex, the focus, the latus rectum, the directrix, and the axis of symmetry. We will use a given parabola equation to illustrate these concepts:

The equation of the parabola provided is:

y - 3^2 4x^4

This equation is similar to the standard form of a parabola:

y - k^2 4p(x - h)

Identifying Key Parameters

From the given equation, we can extract the following values:

4p 4, thus, p 1

The Vertex

The vertex of the parabola is given by the point (h, k). In the standard form, these values are:

(h, k) (-4, 3)

The Axis of Symmetry

The axis of symmetry of a parabola is a line that passes through the vertex and divides the parabola into two symmetrical halves. For a parabola in the form above:

Axis of symmetry: y k

Substituting the value of k, we get:

Axis of symmetry: y 3

The Focus

The focus of the parabola is a point that lies on the axis of symmetry and is equidistant from the vertex. The coordinates of the focus are:

F(h p, k) (-4 1, 3) (-3, 3)

The focus is located at (-3, 3).

The Directrix

The directrix is a line perpendicular to the axis of symmetry and equidistant from the vertex as the focus. The equation of the directrix is:

x h - p

Substituting the values of h and p, we get:

x -4 - 1 -5

The directrix is the vertical line x -5.

The Latus Rectum

The latus rectum of a parabola is the line segment that passes through the focus and is parallel to the directrix. The ends of the latus rectum are given by:

(h p, k 2p) (-4 1, 3 2) (-3, 5)

and

(h p, k - 2p) (-4 1, 3 - 2) (-3, 1)

The length of the latus rectum can be calculated using the distance formula:

AB sqrt{(-3 - (-3)^2 (5 - 1)^2) sqrt{0 4^2} 4}

The length of the latus rectum is 4 units.

Understanding these elements helps in visualizing and manipulating parabolic shapes. This knowledge is crucial in various fields such as physics, engineering, and computer graphics.

Keywords: parabola, focus, vertex, directrix, latus rectum