Parabolas

Objective 1: Equations and Graphs of Parabolas

A parabola is one of the conic sections. A conic section

is the intersection of a plane with a right circular cone.

Definition: A parabola is a set of points in a plane equidistant from a fixed line and a fixed point. The fixed line is called the directrix and the fixed point is the focus.

Cypress College Math Department – CCMR Notes Parabolas, Page 1 of 15

The axis of symmetry is the line that goes

through the vertex and focus of the parabola.

It is perpendicular to the directrix.

The latus rectum is the line segment that is

perpendicular to the axis of symmetry that

goes through the focus. Graph the endpoints of

the latus rectum when you graph a parabola.

Distance from directrix to vertex = Distance from vertex to focus = |a|

(x,y)

focus(a,0)

directrixx= - a

vertex(0,0)

The distance from any point on the parabola to the directrix x = -a must equal the distance to the focus (a,0).

distance 1 = distance 2

( ) 2 2( )x a x a y− − = − +2 2 2( ) ( )a x x a y+ = − +

2 2 2 2 22 2a ax x x ax a y+ + = − + +

( ) ( )

2

2

4

4

y ax

y k a x h

=

− = −

Cypress College Math Department – CCMR Notes Parabolas, Page 2 of 15

( )

( )

( )

( ) ( )

( ) ( )

2

2

2

2

2

1

4

14

y c x h k

y k c x h

y kx h

c

x h y kc

x h a y k

where ac

= − +

− = −

−= −

− = −

− = −

=

Example: Match the graph to its equation.

A. 2 4y x=

B. 2 8y x= −

C. 2 8x y=

D. 2 4x y=

Example: Match the graph to its equation.

A. 2 4y x=

B. 2 8y x= −

C. 2 8x y=

D. 2 4x y=

Cypress College Math Department – CCMR Notes Parabolas, Page 3 of 15

Example: Match the graph to its equation.

A. ( ) ( )2

3 8 2x y− = − +

B. ( ) ( )2

3 4 2x y− = +

C. ( ) ( )2

2 8 3y x+ = − −

D. ( ) ( )2

3 8 2y x− = +

Example: Match the graph to its equation.

A. ( ) ( )2

5 12 1y x− = +

B. ( ) ( )2

1 4 5y x+ = −

C. ( ) ( )2

1 12 5y x+ = −

D. ( ) ( )2

5 12 1x y− = +

Pause the video to try this one on your own, then restart when you are ready to check your

answer.

Cypress College Math Department – CCMR Notes Parabolas, Page 4 of 15

Extra Practice

1. Match the graph to its equation.

2. Match the graph to its equation.

Restart when you are ready to check your answers.

Cypress College Math Department – CCMR Notes Parabolas, Page 5 of 15

Objective 2: Convert from General Form to Standard Form

General form for the equation of a conic section:

2 2 0Ax By Cx Dy F+ + + + =

Standard form for the equation of a parabola:

( ) ( )

( ) ( )

2

2

4

4

x h a y k or

y k a x h

− = −

− = −

Example: Convert the following equation to standard form. Identify the vertex, focus and directrix.

2 6 8 23 0x x y+ + − =

Example: Convert the following equation to standard form. Identify the vertex, focus and directrix.

2 12 40 0y y x− + + =

Pause the video to try this one on your own, then restart when you are ready to check your

answer.

Extra Practice

1. Convert the following equation to standard form. Identify the vertex, focus and directrix.

2 4 12 8 0y y x− − − =

Cypress College Math Department – CCMR Notes Parabolas, Page 6 of 15

2. Convert the following equation to standard form. Identify the vertex, focus and directrix.

2 10 2 19 0x x y+ + + =

Restart when you are ready to check your answers.

Objective 3: Graph Parabolas

The latus rectum is the line segment that is

perpendicular to the axis of symmetry that

goes through the focus. Graph the endpoints of

the latus rectum when you graph a parabola.

When graphing a parabola, include the vertex,

focus, directrix and endpoints of the latus

rectum. Label the coordinates of the vertex,

focus, and endpoints of the latus rectum.

Label the equation of the directrix.

Cypress College Math Department – CCMR Notes Parabolas, Page 7 of 15

Example: Graph ( ) ( )2

3 12 4y x− = − + . Make sure to label the exact coordinates of the vertex,

focus, the two points that determine the latus rectum, and the equation of the directrix.

Example: Graph ( ) ( )2

3 4x y− = − + . Make sure to label the exact coordinates of the vertex,

focus, the two points that determine the latus rectum, and the equation of the directrix.

Cypress College Math Department – CCMR Notes Parabolas, Page 8 of 15

Example: Graph 2 8 8 0y y x− − = . Make sure to label the exact coordinates of the vertex, focus,

the two points that determine the latus rectum, and the equation of the directrix.

Pause the video to try this one on your own, then restart when you are ready to check your

answer.

Extra Practice

1. Graph ( ) ( )2

2 12 3x y− = − . Make sure to label the exact coordinates of the vertex, focus,

the two points that determine the latus rectum, and the equation of the directrix.

Cypress College Math Department – CCMR Notes Parabolas, Page 9 of 15

2. Graph . Make sure to label the exact coordinates of the vertex, 2 2 2 11 0y y x+ + − =

focus, the two points that determine the latus rectum, and the equation of the directrix.

Restart when you are ready to check your answers.

Objective 4: Determine the Equation of a Parabola

Example: Determine the equation of the parabola with focus (1, 7) and vertex (4, 7).

Cypress College Math Department – CCMR Notes Parabolas, Page 10 of 15

Example: Determine the equation of the parabola with focus (3, -7) and directrix y = -3.

Example: Determine the equation of the parabola with vertex (-2,-4), axis of symmetry y = -4,

contains the point (1, -10).

Pause the video to try this one on your own, then restart when you are ready to check your

answer.

Cypress College Math Department – CCMR Notes Parabolas, Page 11 of 15

Extra Practice

1. Determine the equation of the parabola with focus (-4,1) and directrix y = 5.

2. Determine the equation of the parabola with vertex (6,-2), axis of symmetry y = -2,

contains the point (10,-10).

Restart when you are ready to check your answers.

Cypress College Math Department – CCMR Notes Parabolas, Page 12 of 15

Objective 5: Applications of Parabolas

A paraboloid of revolution is a surface

formed by rotating a parabola around the

axis of symmetry.

If a flashlight, or searchlight, is constructed where the

light source is at the focus and the mirror around the light

source is a paraboloid of revolution, then the rays of light

will emanate from the flashlight parallel to the axis of

symmetry. This gives a strong beam of light.

In telescopes, rays of light come in and strike

the mirror at the back of the telescope. The

mirror is a paraboloid of revolution. The rays

reflect toward the focus.

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Satellite dishes work similarly, all of the signals

strike the dish and are reflected back to the

focus. The receiver is placed at the focus.

Example: A reflecting telescope contains a mirror shaped like a paraboloid of revolution. If the

mirror is 18” across at its opening and it is 2 feet deep, where will its light be concentrated? Write your answer accurate to two decimal places.

Example: The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter

is 3” and its depth is 2”. How far from the vertex should the light bulb be placed so that the

rays will be reflected parallel to the axis? Write your answer accurate to two decimal places.

Example: An arch is in the shape of a parabola. It has a span of 120 feet and a maximum height

of 30 feet. Find the equation of the parabola (assuming the origin is halfway between the arch’s

feet). Determine the height of the arch 25 feet from the center. Round your answer to one

decimal place.

Cypress College Math Department – CCMR Notes Parabolas, Page 14 of 15

Pause the video to try this one on your own, then restart when you are ready to check your

answer.

Extra Practice

1. A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by

rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is

72 feet across at its opening and 6 feet deep at its center, how far above the vertex should the receiver

be placed? Write your answer accurate to two decimal places.

1. 2. The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is

4” and its depth is 1.5”. How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis? Write your answer accurate to two decimal places.

Cypress College Math Department – CCMR Notes Parabolas, Page 15 of 15