Understanding the Parabola: Equations, Properties, and Examples
Parabola as a Geometric Locus
The parabola is defined as the geometric locus of a point that moves in a plane such that its distance from a fixed line (the directrix) is equal to its distance from a fixed point (the focus, denoted by F).
The following figure illustrates the elements of a parabola:
Where:
- F is the focus of the parabola.
- V is the vertex of the parabola.
- The segment represented by this image
is the directrix. - The segment represented by this image
is the latus rectum (LR). - The segment represented by this image
is the focal axis of the parabola. - The segments represented by this image
are the distances from the directrix to the vertex and from the vertex to the focus, respectively. These distances are equal.
Key characteristics of the elements of a parabola:
- The focal axis is perpendicular to the directrix and the latus rectum.
- The midpoint of the latus rectum is the vertex.
- The length of the latus rectum determines the width of the parabola’s opening and is given by
where p is the distance from the vertex to the focus.
Parabola Equation with Vertex at the Origin and Axis Parallel to a Coordinate Axis
The equation of a parabola with its vertex at the origin and axis parallel to a coordinate axis can be derived from the following:
Characteristics:
- Vertex at the origin: V(0,0)
- Opens right if p is positive and left if p is negative.
- Focus at: F(p,0)
- Directrix equation: x = -p
- Latus rectum:
Example:
Find the equation of the parabola with focus at (4,0) and directrix at x = -4. Graph it.
Steps:
Procedure:
1. Graph the given information:
2. The focus is always inside the parabola’s curve, so the parabola opens right. The vertex is at the midpoint between the directrix and the focus:
3. The vertex is at (0,0). The distance between the vertex and the focus is p = 4. The latus rectum is 4p = 16, meaning it extends 8 units above and 8 units below the focus.
4. Substitute into the equation:
Answer:
The equation of the parabola is:
And its graph is:
For a parabola with a vertical axis of symmetry:
Characteristics:
- Vertex at the origin: V(0,0)
- Opens up if p is positive and down if p is negative.
- Focus at: F(0,p)
- Directrix equation: y = -p
- Latus rectum:
Example:
Determine the elements of the parabola with equation 
and graph it.
Steps:
Procedure:
1. Observe the equation:
2. Compare with the standard equation:
3. Since p is negative:
- Vertex: V(0,0)
- Opens downward
- Focus: F(0,-3)
- Directrix: y = 3
- Latus rectum: 4p = -12 (6 units to the right and left of the focus)
Answer:
The elements are:
- Vertex: V(0,0)
- Opens downward
- Focus: F(0,-3)
- Directrix: y = 3
- Latus rectum: -12 (6 units to the right and left of the focus)
Parabola Equation with Vertex at (h, k) and Axis Parallel to a Coordinate Axis
The equation of a parabola with vertex at (h,k) and axis parallel to a coordinate axis is:
Characteristics:
- Vertex: V(h,k)
- Opens right if p is positive and left if p is negative.
- Focus: F(h+p,k)
- Directrix equation: x = h-p
- Latus rectum:
Example:
Find the equation of the parabola with vertex at V(2,3) and directrix at x = -2. Graph it.
Steps:
Procedure:
1. Based on the given data, the equation is of the form:
2. Substitute the vertex:
3. Graph the data to find p (the distance between the directrix and the vertex):
4. The distance is p = 4. Substitute into the equation:
5. The parabola opens right because the directrix is on the opposite side.
6. Expand the binomial and set the equation to zero:
7. The focus and latus rectum are:
F(h+p, k) = F(2+4,3) => F(6,3)
LR = 4p = 4(4) = 16 (8 units up and 8 units down from the focus)
Answer:
The equation of the parabola is:
This is the general form of the equation. The graph is:
For a parabola with a vertical axis of symmetry and vertex at (h,k):
Characteristics:
- Vertex: V(h,k)
- Opens up if p is positive and down if p is negative.
- Focus: F(h,k+p)
- Directrix equation: y = k-p
- Latus rectum:
Example:
Determine the elements of the parabola 
.
Steps:
Procedure:
1. Compare the equation:
2. From the comparison, the vertex and p are:
V(h,k) = V(6,-2)
From the relation 
we have that 
Answer:
The elements of the parabola are:
- Vertex: V(6,-2)
- Opens upward (p is positive)
- Focus:
- Directrix: y = k – p =
- Latus rectum:
Example:
Obtain the standard form of the following equation and its elements:
Steps:
Procedure:
1. Arrange the equation:
2. Complete the square:
3. Factor out -3:
4. Compare with the standard equation:
5. From this, we get the vertex V(h,k) = V(5,7) and 
Answer:
The standard form of the equation is:
The elements of the parabola are:
- Vertex: V(5,7)
- Opens downward (p is negative)
- Focus:
- Directrix: y = k – p =
- Latus rectum:
