Post on 14-Jan-2016
Lets study Parabolas by exploring the focus and directrix
There is a chart on page 171 of the Pearson textbook where you can copy formulas and memorize. I plan to show you how to do these problems with just memorizing the vertex form of parabolas. Copy chart below if you wish to see both.
A parabola is the set of all points in a plane that are the same distance from a fixed line and a fixed point not on the line.
The fixed point is called the FOCUS.
The fixed line is called the DIRECTRIX.
Notes: Parabola, Focus, DIrectrix
Note that the distance from the vertex to the focus point is the same as the distance as the vertex to the directrix…
This distance is the “c” value.
To be a parabola……The two red line segments must be the same length!!
Vertex form : with a slight new look since we are studying the focus and directrix
Previous Form using “a” (vertical stretch)
New Form using “c” (focus point)
2axy 2
4
1xc
y
khxay 2 khxc
y 2
4
1
ca
4
1
Opens UP if Opens DOWN if
04
1
0
c
ora
04
1
0
c
ora
Vertex form : How about parabolas that open sideways vs. up and down
Previous Form using “a” (vertical stretch)
New Form using “c” (focus point)
2ayx 2
4
1yc
x
hkyax 2 hkyc
x 2
4
1
ca
4
1
Opens Right if Opens LEFT if
04
1
0
c
ora
04
1
0
c
ora
Write an equation for this parabola!
Vertex is: or
Focus point is at :
Directrix is at:
We could write the equation for this if we knew the value of “c”
Opens up so
kh,
X goes with h and y goes with k
c = distance from vertex to Focus and directrix!!!
2,1 3,1
F
1y
1c
2)1(4
1 2 xy
Now you will graph parabolasExtra hint to help get 2 easy points
on the graph of parabolas
Latus Rectum: move 2c out from the focus to get 2 more points
on the parabola
This will make more sense when we apply it to the next example.
Graph the PARABOLA
vertex :Axis of Symmetry:
382 2 xy
(-3,2)y = 2
Parabola opens: right
Find the focus and directrix :
8
1
4
1c
2c
21X = -5
Latus Rectum = 2c = 4
Vertex Form: 328
1 2 yx
Convert the equation to a parabola in vertex form
025432 yxy
25342 xyy
222 225324 xyy
2132 2 xy
xy 723
1 2
x= a(y-k)2 + h
Find the:
vertex:
focus:
directrix:
(7,2)
3
1
4
1c 4
3c
2,4
37
4
16x
Complete the square
xy 3212 2
Find the vertex, focus and directrix:
242 xxy
Find the:
vertex:
focus:
directrix:
(2,-2)
14
1c
a4
1c
4
31,2
4
12y
Complete the square orTry this:
1 tcoefficienleadinga
22
4
2
a
b2)2(4)2( 2
2
Find the equation of a parabola with a focus (2,-3) and a directrix of x = -4 (the book shows this a different way p.174)
Find the:
direction:
vertex:
c value:
Opens right
)3,1(
3c
),( kh
Sketch to find the direction the parabola opens, the c value, and the vertex. Find a, h, and k to get vertex form
ca
4
1
12
1a 1)3(
12
1 2 yx
HW: Parabolas Due next class!