10.2 Parabolas
By: L. Keali’i Alicea
Parabolas
• We have seen parabolas before.
Can anyone tell me where?
• That’s right! Quadratics!
• Quadratics can take the form:
x2 = 4py or y2 = 4px
Parts of a parabola
• Focus
A point that lies on
the axis of
symmetry that is
equidistant from
all the points on
the parabola.
Parts of a parabola
• Directrix
A line perpendicular
to the axis of
symmetry used in
the definition of a
parabola.
Focus
Lies on AOS
Directrix
2 Different Kinds of Parabolas
• x2=4py
• y2=4px
Standard equation of Parabola
(vertex @ origin)
Equation
x2=4py
y2=4px
Focus
(0,p)
(p,0)
Directrix
AOS
y=-p
Vertical
(x=0)
x=-p
Horizontal
(y=0)
x2=4py, p>0
Focus (0,p)
Directrix
y=-p
x2=4py, p<0
Directrix
y=-p
Focus (0,p)
y2=4px, p>0
Directrix
x=-p
Focus (p,0)
y2=4px, p<0
Focus (p,0)
Directrix
x=-p
Identify the focus and directrix of
the parabola
x = -1/6y2
• Since y is squared, AOS is horizontal
• Isolate the y2 → y2 = -6x
• Since 4p = -6
•
p = -6/4 = -3/2
• Focus : (-3/2,0) Directrix : x=-p=3/2
• To draw: make a table of values & plot
• p<0 so opens left so only choose neg values for x
Your Turn!
• Find the focus and directrix, then graph
x=
•
•
•
•
2
3/4y
y2 so AOS is Horizontal
Isolate y2 → y2 = 4/3 x
4p = 4/3 p = 1/3
Focus (1/3,0) Directrix x=-p=-1/3
Writing the equation of a parabola.
• The
graph
shows
V=(0,0)
• Directrex
y=-p=-2
• So
substitute
2 for p
• = 4py
2
• x = 4(2)y
• x2 = 8y
2
x
• y = 1/8 and check in your
calculator
2
x
Your turn!
• Focus =
(0,-3)
• X2 = 4py
• X2 = 4(-3)y
• X2 = -12y
• y=-1/12x2
to check
Assignment
10.2 A (1-3, 5-19odd)
10.2 B (2-20 even, 21-22)