Transcript Ch. 10.2 Parabolas

Parabolas
Date: ____________
Parabolas
Parabola—Set of all points in a plane that
are the same distance from a given point
called the focus and a given line called the
directrix.
Focus
Directrix
Vertex
Parabolas with Vertex (0,0) that Open Up/Down
y = ax2
+a → opens up, -a → opens down
1
vertex: (0, 0) focus: (0, )
4a
1
directrix: y = –
4a
1
c=
4a
c is the distance from the vertex
Parabolas with Vertex (0,0) that Open Left/Right
x = ay2
+a → opens right, -a → opens left
1
focus: ( , 0)
4a
1
directrix: x = –
4a
1
c=
4a
vertex: (0, 0)
c is the distance from the vertex
Parabola Opens Up/Down
y – k = a(x – h)2
+a → opens up, -a → opens down
vertex: (h, k)
1
focus: (h, k +
)
4a
1
directrix: y = k –
4a
Parabola Opens Left/Right
x – h = a(y – k)2
+a → opens right, -a → opens left
1
vertex: (h, k) focus: (h + , k)
4a
1
directrix: x = h –
4a
Write an equation of a parabola with a
vertex at the origin and the given focus.
focus at (6,0)
1
6=
· 4a
4a
opens to the right
Solve for a
24a = 1
24 24
1
a=
24
x = ay2
x=
1
24
y²
Write an equation of a parabola with a
vertex at the origin and the given directrix.
directrix x = 9
x = ay2
opens to the left
1
9=· -4a Solve for a
4a
-36a = 1
-36 -36
1
a=
-36
x =-
1
36
y²
Write an equation of a parabola with a
vertex at the origin and the given focus.
focus at (0,7)
1
7=
· 4a
4a
opens up
y = ax2
Solve for a
28a = 1
28 28
1
a=
28
y=
1
28
x²
2
x
=
⅛y
Graph
opens right
vertex: (0, 0)
1
= 2
4(⅛)
y
focus: (2,0)
directrix: x = –2
x
1
2
y
=
x
Graph
12
opens down
vertex: (0, 0)
1
= -3
1
4(- 12 )
y
focus: (0,-3)
directrix: y = 3
x
2
y
=
¼x
Graph
opens up
vertex: (0, 0)
1
=1
1
4( 4 )
y
focus: (0,1)
directrix: y = -1
x
1
2
x
=
y
Graph
16
opens left
vertex: (0, 0)
1
= -4
4(-1/16)
y
focus: (-4,0)
directrix: x = 4
x
1
2
y
–
3
=
−
(x
+
3)
Graph
16
opens down
vertex: (-3, 3)
1
= -4
1
4(- 16 )
y
focus: (-3,-1)
directrix: y = 7
x
Write the equation in standard form
and graph the parabola.
y2 – 8y + 8x + 8 = 0
y2 – 8y = -8x – 8
16 = -8x – 8 +16
y2 – 8y + ___
(y – 4)2 = -8x + 8
(y – 4)2 = -8(x – 1)
-8
-⅛(y – 4)2 = x – 1
x – 1 = -⅛(y – 4)2
x – 1 = -⅛(y – 4)2
vertex: (1, 4)
y
opens left
1
= -2
1
4(- 8 )
focus: (-1,4)
directrix: x = 3
x
Write the equation in standard form
and graph the parabola.
x2 – 6x + 6y + 18 = 0
x2 – 6x = -6y – 18
9 = -6y − 18+9
x2 – 6x + ___
(x – 3)2 = -6y – 9
(x – 3)2 = -6(y + 1.5 )
-6
- 6 (x – 3)2 = y +1.5
1
y + 1.5 = - 6 (x − 3)2
1
y + 1.5 = - 6 (x − 3)2
1
vertex: (3,-1.5)
y
opens down
1
= -1.5
1
4(- 6 )
focus: (3,-3)
directrix: y = 0
x