1. 9.2 Graph and Write Equations of Parabolas

2. Parabolas
We already know the graph of y=ax2 is a parabola w/ vertex (0,0) and AOS (axis of symmetry) x=0
Every parabola has the property that any point on it is equidistant from a point called the Focus and a line called the directrix.

3. Focus
Lies on AOS
Directrix

4. The focus and directrix each lie IpI units from the vertex
The focus and directrix each lie IpI units from the vertex. (the vertex is ½ way between the focus and directrix)

5. x2=4py, p>0
Focus (0,p)
Directrix
y=-p

6. x2=4py, p<0
Directrix
y=-p
Focus (0,p)

7. y2=4px, p>0
Directrix
x=-p
Focus (p,0)

8. y2=4px, p<0
Directrix
x=-p
Focus (p,0)

9. Standard equation of Parabola (vertex @ origin)
Focus
Directrix
AOS
x2=4py
(0,p)
y=-p
Vertical (x=0)
y2=4px
(p,0)
x=-p
Horizontal (y=0)

10. 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

12. Your Turn! y2 so AOS is Horizontal Isolate y2 → y2 = 4/3 x
Find the focus and directrix, then graph x = 3/4y2
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

14. Writing the equation of a parabola.
The graph shows V=(0,0)
Directrex y=-p=-2
So substitute 2 for p

15. x2 = 4py
x2 = 4(2)y
x2 = 8y
y = 1/8 x2 and check in your calculator

16. Your turn! Focus = (0,-3) X2 = 4py X2 = 4(-3)y X2 = -12y
y=-1/12x2 to check