81 : parabolas

1
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)

2
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

3

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

focus
directrix
4

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

focus
directrix
5

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

focus
vertex
directrix
6

• 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
p
vertex
directrix
7
Standard Form of a ParabolaVertex at (0 , 0)
Focus at (0 , p)
4p
p
p
Focal length p Focal width
Directrix at y p
if p is negative, the graph flips over the x-axis
8
Standard Form of a ParabolaVertex at (0 , 0)
p
p
4p
Focus at (p , 0)
Focal length p Focal width
Directrix at x p
If the value of p is negative, the graph opens to
the left (flips over the y-axis)
9
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

10
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 parabolas width)
• graph the rest of the parabola

11
Convert Into Standard Form

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

12
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