# 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:

is the focus of the parabola.*F*is the vertex of the parabola.*V*- 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
is the distance from the vertex to the focus.*p*

## 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
is positive and left if*p*is negative.*p* - 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 4*p* = 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
is positive and down if*p*is negative.*p* - 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: 4
*p*= -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
is positive and left if*p*is negative.*p* - 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* = 4*p* = 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
is positive and down if*p*is negative.*p* - 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: