9.1 Parabolas
Parabolas
Parabola: the set of points in a
plane that are the same distance
from a given point called the
focus and a given line called the
directrix.
Notice that the axis of
symmetry is the line
through the focus,
perpendicular to the
directrix
d2
d1Focus
d2
d3
d1
Vertex
d3
Directrix
Notice that the vertex is located at the midpoint between the focus
and the directrix...
Also, notice that the distance from the focus to any point on the
parabola is equal to the distance from that point to the directrix...
Parabola with measures.gsp
We can determine the coordinates of the focus, and the
equation of the directrix, given the equation of the parabola....
Standard Equation of a Parabola: (Vertex at the origin)
Equation
Focus
Directrix
2
x = 4py
(0, p)
y = –p
(If the x-term is squared, the parabola is up
or down)
Equation
2
y = 4px
Focus
(p, 0)
Directrix
x = –p
(If the y-term is squared, the parabola is
left or right)
Where p is the direct distance from the vertex to the focus.
* note: if p < 0, then parabola opens down or left
Standard Equation of a Parabola: (Vertex at (h, k) )
Equation
2
(x – h) = 4p(y - k)
Focus
(h, k + p)
Directrix
y=k–p
(If the x term is squared, the parabola
is up or down)
Equation
2
(y - k) = 4p(x – h)
Focus
Directrix
(h + p, k)
x=h–p
(If the y term is squared, the parabola is
left or right)
Ex 1: Determine the focus and directrix of the parabola
2
y = 4x :
Since x is squared, the parabola goes up or down…
2
2
Solve for x
x = 4py
2
y = 4x
4 4
2
x = 1/4y
Solve for p
4p = 1/4
p = 1/16
Focus: (0, p)
Directrix: y = –p
Focus: (0, 1/16) Directrix: y = –1/16
Let’s see what this parabola looks like...
Ex 2: Determine the focus and directrix of the parabola
2
–3y – 12x = 0 :
Since y is squared, the parabola goes left or right…
2
2
Solve for y
y = 4px
2
–3y = 12x
2
–3y = 12x
–3
–3
2
y = –4x
Solve for p
4p = –4
p = –1
Focus: (p, 0)
Directrix: x = –p
Focus: (–1, 0)
Directrix: x = 1
Let’s see what this parabola looks like...
Ex 3: Write the standard form of the equation of the parabola with
focus at (0, 3) and vertex at the origin.
Since the focus is on the y axis,(and vertex at the origin) the
parabola goes up or down…
x2 = 4py
Since p = 3, the standard form of the equation is
x2 = 12y
Ex 4: Write the standard form of the equation of the parabola
with directrix x = –1 and vertex at the origin.
Since the directrix is parallel to the y axis,(and vertex at the
origin) the parabola goes left or right…
y2 = 4px
Since p = 1, the standard form of the equation is
y2 = 4x
Summary…
•
•
•
•
•
•
•
•
Opens Up / Down
2
(x – h) = 4p(y - k)
p pos up
p neg down
Vertex: V (h , k)
Axis: x = h
Focus: F(h , k + p)
Directrix: y = k – p
9_1 Parabola Examples
• Opens Right / Left
• (y - k)2 = 4p(x – h)
•
•
•
•
•
•
p pos  right
p neg  left
Vertex: V (h, k)
Axis: y = k
Focus: F (h + p, k)
Directrix: x = h – p
Try It Out
• Given the equations below,
– What is the focus?
– What is the directrix?
( x  3)  ( y  2)
2
x  y  4y  9  0
2
4 x  12 x  12 y  7  0
2