Chapter 8• 8-1 : parabolas

• 8-2 : ellipse

• 8-3 : hyperbolas

• 8-4 : translating and rotating conics

• 8-5 : writing conics in polar form

• 8-6 : 3-D coordinate system (plotting points, planes, vectors)

Section 8-1

• the conic sections

• definition of parabola

• standard form of the equation of a parabola

• translating a parabola

• graphing a parabola

• convert from general form to standard form

• reflective property of parabolas

Parabola: the set of all points equidistant from a particular line (the directrix) and a particular point (the focus).

focus

directrix

Parabola: the set of all points equidistant from a particular line (the directrix) and a particular point (the focus).

focus

directrix

Vertex: the vertex is midway between the focus and the directrix

focus

directrix

vertex

Vertex: the vertex is midway between the focus and the directrix

Focal Length: the distance from the focus to the vertex, denoted with the letter p

focus

directrix

vertex p {

Standard Form of a ParabolaVertex at (0 , 0)

pp

Directrix at y = – p

Focus at (0 , p)4p

2 4x pyvertex (0 , 0)

Focal length = p

Focal width = 4p

if p is negative, the graph flips over the x-axis

Standard Form of a ParabolaVertex at (0 , 0)

2 4y pxvertex (0 , 0)

Focal length = p

Focal width = 4p

pp

Focus at (p , 0)

Directrix at x = – p

4p

If the value of p is negative, the graph opens to the left (flips over the y-axis)

Translating a Parabola• if the parabola has a vertex of (h , k) the

two equations change into:

• notice that h is always with x and k is always with y

• the focus and directrix will adjust accordingly

2

2

( ) 4 ( )

( ) 4 ( )

x h p y k

y k p x h

Sketching the Graph of a Parabola• convert the equation into standard form, if

necessary• find and plot the vertex• decide which way the graph opens (based

on p and which variable is squared)• add the focus and directrix to your graph • use the focal width to find two other

points (these will give the parabola’s width)

• graph the rest of the parabola

4p

Convert Into Standard Form

• to convert from general form into standard form you must use “complete the square”

2 6 2 13 0y x y 2 2 = 6 13y y x

2 2 +1 = 6 13 1y y x 2( 1) = 6 12y x 2( 1) = 6( 2)y x

Reflective Properties of Parabolas• if a parabola is rotated to create a 3-D

version it is called a “paraboloid of revelolution”

• there are many examples of parabolic reflectors in use today involving sound, light, radio and electromagnetic waves