Introduction
- Parabola is the set of all points in a plane equidistant from a fixed point (focus) and a line (directrix).
- Parabola is formed when the intersecting plane to a right circular cone is parallel to the slant height of the cone.
- The eccentricity of a parabola is equal to 1.
- The axis of symmetry is given by
when the parabola opens to the right or opens to the left. - The axis of symmetry is given by
when the parabola opens upward or downward. - The parabola
opens upward when and downward when - The parabola
opens to the right when and opens to the left when - The general equations of the parabola are as follows:
- In other references, it is important to note the following general equations of the parabola:
- No matter where we put point C in the parabola, as shown above, the distances between Focus A and point C and between point C and point B are always equal. The line BD is the directrix of the parabola.
Basic Concepts: Standard Equations and Graphs of the Parabola with vertex at (0,0)
The standard equations of the parabola with center at
1.
This is the standard equation of the parabola that opens to the right.
The graph of
Directrix is
Focus:
Endpoints of the Latus Rectum:
2.
This is the standard equation of the parabola that opens to the left.
The graph of
Directrix is
Focus:
Endpoints of the Latus Rectum:
3.
This is the standard equation of the parabola that opens upward.
The graph of
Directrix is
Focus:
Endpoints of the Latus Rectum:
4.
This is the standard equation of the parabola that opens downward.
The graph of
Directrix is
Focus:
Endpoints of the Latus Rectum:
Basic Concepts: Standard Equations and Graphs of the Parabola with vertex at (h,k)
The standard equations of the parabola with vertex at
1.
This standard equation represents a parabola that opens to the right.
The graph of
Vertex
Equation of the Directrix:
Focus:
Endpoints of the Latus Rectum:
2.
This standard equation represents a parabola that opens to the left.
The graph of
Vertex
Equation of the DIrectrix:
Focus:
Endpoints of the Latus Rectum:
3.
This standard equation represents a parabola that opens upward
The graph of
Vertex
Equation of the Directrix:
Focus:
Endpoints of the Latus Rectum:
4.
This standard equation represents a parabola that opens downward.
The graph of
Vertex
Equation of the Directrix:
Focus:
Endpoints of the Latus Rectum:
Solved Examples
Example 1. Determine the opening, coordinates of the vertex, coordinates of the focus, endpoints, and length of the latus rectum, and equation of the directrix of the parabola
Solution:
The parabola
The graph of this parabola opens upward since
The vertex is at the origin
The coordinates of the focus follow the form
Hence, the focus is at
To solve for the endpoints of the latus rectum, we follow the form
Hence, the endpoints of the latus rectum are
The length of the latus rectum is given by
In this case, we have:
Since this parabola is in the form of
Equation of the Directrix:
Example 2. Determine the opening, coordinates of the vertex, coordinates of the focus, endpoints, and length of the latus rectum, and the equation of the directrix of the parabola
Solution:
The parabola
This parabola opens to the left since
The coordinates of the focus of this parabola follows
To solve for
In solving for the coordinates of the focus and the endpoints of the latus rectum, disregard the sign of
The focus is
The endpoints of the latus rectum are solved using the form
In this example, the endpoints of the latus rectum are
The length of the latus rectum is
The equation of the directrix is
Cheat Sheet
Standard Forms of a Parabola | Coordinates | Axis of Symmetry | Equation of the Directrix | ||
Vertex | Focus | Latus Rectum | |||
Blunder Areas
- Always inspect the sign of
to know the opening of the parabola, the correct coordinates of the vertex, and other properties. - In solving for the coordinates of the latus rectum and focus, neglect the sign of
. - In solving for the equation of the directrix, always consider the sign of
. - Parabolas in the form of
are not representations of quadratic functions. - Use the algebraic method in transforming the equation of the parabola in
in vertex form (standard form). This is done using completing the square. - The algebraic method and graphical method can both be used to determine, study, and interpret the characteristics/properties of a parabola.
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Keith Madrilejos
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