Definition of a parabola

1. In this section, you will be learning the definition of a parabola, what is the standard form of the
equation of a parabola is and how to determine the focus, vertex, directrix, and axis symmetry using the
standard form of the equation of a parabola. Study the illustration below and its example.
DEFINITION OF A PARABOLA
Let F be a given point, and l a given line not containing F. The set of all points P such that its distances
from F and l are the same, is called a parabola. The point F is its focus and the line l its directrix
Mathematical Key Concepts…
(1) A parabola is the graph of a quadratic function
(2) A directrix is a line used to construct and define a parabola
(3) A focus is a point about which the conic section is constructed. In other words, it is a point about
which rays reflected from the curve converge.
(4) A parabola has one focus on which the shape is constructed.
(5) A focus is the extreme point of the parabola.
(6) The axis of symmetry of a parabola is a line about which the parabola is symmetrical.
Mathematical Key Points…
The standard equation formula of parabola has two formulas depending on its axis of symmetry.
HORIZONTAL AXIS OF SYMMETRY VERTICAL AXIS OF SYMMETRY
𝑦 − 𝑘 2
= 4𝑐 𝑥 − ℎ
Vertex ℎ, 𝑘
Focus ℎ + 𝑐, 𝑘
Directrix 𝑥 = ℎ − 𝑐
Axis of Symmetry 𝑦 = 𝑘
𝑥 − ℎ 2
= 4𝑐 𝑦 − 𝑘
Vertex ℎ, 𝑘
Focus ℎ, 𝑘 + 𝑐
Directrix 𝑦 = 𝑘 − 𝑐
Axis of Symmetry 𝑥 = ℎ
NOTE: In a given equation, you can determine if it
has a horizontal axis of symmetry if the squared term
is the y-terms
NOTE: In a given equation, you can determine if it
has a vertical axis of symmetry if the squared term
is the x-terms

2. EXAMPLES
(1) Find the vertex, focus, directrix & axis of symmetry if the equation is already on the standard
form of the equation of the parabola.
a. =
SOLUTION:
Explanation
= Given
− = − *Substitute the given to the standard equation
formula of the parabola. We use the standard
equation:
− ℎ 2
= 4 −
*Since the squared term is the x-term
, *the formula for the vertex
, *substitution and the values of the vertex
= *Getting the value of c
=
*Divide both sides by 4
= *the value of c is obtained and will be used in
determining the value for the focus & directrix
, + *the formula in getting the focus
, + *substitution
, *the values of the focus
= − *the formula in getting the directrix
= − *substitution
= − *the value of the directrix
= *the formula in getting the axis of symmetry
= *the equation of the horizontal axis symmetry
b. = −
SOLUTION:
Explanation
= − Given

3. − = − − *Substitute the given to the standard equation
formula of the parabola. We use the standard
equation:
− 2
= 4 − ℎ
*Since the squared term is the y-term
, *the formula for the vertex
, *substitution and the values of the vertex
= − *Getting the value of c
=
− *Divide both sides by 4
= − *the value of c is obtained and will be used in
determining the value for the focus & directrix
+ , *the formula in getting the focus
+ − , *substitution
− , *the values of the focus
= − *the formula in getting the directrix
= − − *substitution
= *the value of the directrix
= *the formula in getting the axis of symmetry
= *the equation of the vertical axis symmetry
c. + = −
SOLUTION:
Explanation
+ = + Given
− = − *Substitute the given to the standard equation
formula of the parabola. We use the standard
equation:
− 2
= 4 − ℎ
*Since the squared term is the y-term
, *the formula for the vertex
− , − *substitution and the values of the vertex
= *Getting the value of c

4. =
*Divide both sides by 4
=
*the value of c is obtained and will be used in
determining the value for the focus & directrix
+ , *the formula in getting the focus
− + ( ) , −
*substitution
− , −
*the values of the focus
= − *the formula in getting the directrix
= − − −
*substitution
= −
*the value of the directrix
= *the formula in getting the axis of symmetry
= − *the equation of the vertical axis symmetry
d. + = − −
SOLUTION:
Explanation
a. + = − − Given
− = − − −
*Substitute the given to the standard equation
formula of the parabola. We use the standard
equation:
− ℎ 2
= 4 −
*Since the squared term is the y-term
, *the formula for the vertex
− , *substitution and the values of the vertex
= −
*Getting the value of c
= −
*Divide both sides by 4
= −
*the value of c is obtained and will be used in
determining the value for the focus & directrix
, + *the formula in getting the focus
− , + −
*substitution
− ,
*the values of the focus

5. = − *the formula in getting the directrix
= − −
*substitution
=
*the value of the directrix
= *the formula in getting the axis of symmetry
= − *the equation of the vertical axis symmetry