Conic Sections Parabola (Part 1)

Irish Anne Ubalde

January 29, 2015

APOLLONIUS OF PARGA

What is a Conic Section?

Conic Section

Sections formed when planes cut a right circular cone of two nappes.

Conic Sections

General Parts of Conics

1. Focus is the fixed point F

2. Directrix

is the fixed line D

3. Eccentricity is the positive constant ratio (ratio from the segment connecting F to the conic section and from D to cs)

If e=1, parabola

e1, hyperbola

Parabola Definition:

Is the locus of a point that moves in a plane so that its distance from a fixed point is equal to its distance from a fixed line. Its eccentricity is 1.

Geogebra

Parts of Parabola

A. Focus

B. Directrix

C. Vertex

D. Axis of symmetry

E. Focal Chord

F. Latus Rectum

Geogebra

Important Measures of Parabola

1. a = distance from F to V

= distance from V to D

2. 2a = distance from F to D

= distance from F to an end of LR

3. 4a = distance from one end of LR to the other

4. = eccentricity

Geogebra

Standard Forms of Parabola

When V (0,0)

1. Axis of Symmetry on y-axis a. Upward opening b. Downward opening = =

Geogebra

2. Axis of Symmetry on x-axis

a. Right opening b. Left opening = =

Example 1: Determine the opening of each parabola

1. 2 = 4

2. 2 = 4

3

3. 2 = 6y

4. 2 = 8

right

downward

upward

left

5. 2 16 = 0 right

6. 22 + 30 = 0 downward

Example 3: Reduce 2 + 12 = 0 to standard form and

determine the following:

a. Opening of the parabola

b. Vertex

c. Focus

d. Equation of directrix

e. Ends of Latus Rectum

downward

(0,0)

= =

F(0,-3)

=

, &(, )

Example 2:

Sketch the graph of the parabola 2 16 = 0

a. 2 16 = 0

Solution: 2 = 16 Transform in s.f 2 = 4

Opening is right Vertex at (0,0)

4 = 16 length of latus rectum

2 = 8 length of (a) F to one end of LR; (b) F to D

= 4 length of (a) F to V; (b) V to D

Answer: Opening is right V: (0,0) F: (4,0) LR: (4,8) , (4, 8) D: = 4

Seatwork:

P. 100 # 2

Sketch the graph of the parabola 2 = 20.

Determine the opening, locate the vertex, focus, ends of latus rectum and the equation of directrix

Standard Forms of Parabola When V (h,k)

3. Axis of Symmetry vertical

a. Upward opening b. Downward opening ( )2= 4 ( )2= 4( )

Geogebra

4. Axis of Symmetry horizontal a. Right opening b. Left opening ( )2= 4 ( )2= 4( )

Proof:

(, )

(, )

=

Example 3:

Reduce 2 + 16 32 = 0 to s.f, find the direction of opening, vertex, focus, endpoints of latus rectum, determine the equation of the directrix and draw the parabola.

Exercise:

Reduce 2 4 + 8 20 = 0 to s.f, find the direction of opening, vertex, focus, endpoints of latus rectum, determine the equation of the directrix and draw the parabola.