a

1. 1π
Jeff Bivin -- LZHSLast Updated: March 11, 2008
By: Jeffrey Bivin
Lake Zurich High School
jeff.bivin@lz95.org

2. 1π
Jeff Bivin -- LZHS
Parabola
 The set of all points that are equidistant
from a given point (focus) and a given line
(directrix).

3. 1π
Jeff Bivin -- LZHS
Parabola
Distance between focus and vertex = p
Distance between vertex and directrix = p
a
pnote
4
1
: = cbxaxy ++= 2

4. 1π
Jeff Bivin -- LZHS
Parabola
The line segment through the focus
perpendicular to the axis of symmetry with
endpoints on the parabola is called the
Latus Rectum (LR)
Length of the LR = 4p

5. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = 3x2
+ 24x + 53
y = 3(x2
+ 8x ) + 53
y + 48 = 3(x2
+ 8x + (4)2
) + 53
y = 3(x + 4)2
+ 5
Axis of symmetry: x = -4
Vertex: (-4, 5)
y = 3(x2
+ 8x + (4)2
) + 53 - 48
3●(4)2
= 48
x + 4 = 0

6. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = 3(x + 4)2
+ 5
Axis of symmetry: x = -4
Vertex: (-4, 5)

7. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = 3(x + 4)2
+ 5
Axis of symmetry: x = -4
Vertex: (-4, 5)
Focus:
( )12
1
5,4−
Directrix: 12
11
4=y
Length of LR:
12
1
)3(4
1
4
1
=== ap
( ) ( )12
1
12
1
5,45,4 −→+−
12
1
5−=y
( ) 3
1
12
1
44 ==p

8. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = -2x2
+ 12x + 11
y = -2(x2
- 6x ) + 11
y - 18 = -2(x2
- 6x + (-3)2
) + 11
y = -2(x - 3)2
+ 29
Axis of symmetry: x = 3
Vertex: (3, 29)
y = -2(x2
- 6x + (-3)2
) + 11 + 18
-2●(-3)2
= -18
x - 3 = 0

9. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = -2(x - 3)2
+ 29
Axis of symmetry: x = 3
Vertex: (3, 29)
Focus:
( )8
7
28,3
Directrix:
8
1
29=y
Length of LR:
8
1
)2(4
1
4
1 −
− === ap
( )8
1
29,3 −
8
1
29 +=y
( ) 2
1
8
1
44 ==p

10. 1π
Jeff Bivin -- LZHS
Graph the following parabola
x = y2
+ 10y + 8
x = (y2
+ 10y ) + 8
x + 25 = (y2
+ 10y + (5)2
) + 8
x = (y + 5)2
- 17
Axis of symmetry: y = -5
Vertex: (-17, -5)
x = (y2
+ 10y + (5)2
) + 8 - 25
(5)2
= 25
y + 5 = 0

11. 1π
Jeff Bivin -- LZHS
Graph the following parabola
x = (y + 5)2
- 17
Axis of symmetry: y = -5
Vertex: (-17, -5)
Focus:
( )5,16 4
3
−−
Directrix:
4
1
17−=x
Length of LR:
4
1
)1(4
1
4
1
=== ap
( )5,17 4
1
−+−
4
1
17 −−=x
( ) 144 4
1
==p

12. 1π
Jeff Bivin -- LZHS
Graph the following parabola
x = -2y2
- 8y - 1
x = -2(y2
+ 4y ) - 1
x - 8 = -2(y2
+ 4y + (2)2
) - 1
x = -2(y + 2)2
+ 7
Axis of symmetry: y = -2
Vertex: (7, -2)
x = -2(y2
+ 4y + (2)2
) - 1 + 8
-2(2)2
= -8
y + 2 = 0

13. 1π
Jeff Bivin -- LZHS
8
1
)2(4
1
4
1 −
− === ap
Graph the following parabola
x = -2(y + 2)2
+ 7
Axis of symmetry: y = -2
Vertex: (7, -2)
Focus:
( )2,6 8
7
−
Directrix:
8
1
7=x
Length of LR:
( )2,7 8
1
−−
8
1
7 +=x
( ) 2
1
8
1
44 ==p

14. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = 5x2
- 30x + 46
y = 5(x2
- 6x ) + 46
y + 45 = 5(x2
- 6x + (-3)2
) + 46
y = 5(x - 3)2
+ 1
Axis of symmetry: x = 3
Vertex: (3, 1)
y = 5(x2
- 6x + (-3)2
) + 46 - 45
5●(-3)2
= 45
x - 3 = 0

15. 1π
Jeff Bivin -- LZHS
Graph the following parabola
y = 5(x - 3)2
+ 1
Axis of symmetry: x = 3
Vertex: (3, 1)
Focus:
( )20
1
1,3
Directrix: 20
19
=y
Length of LR:
20
1
)5(4
1
4
1
=== ap
( )20
1
1,3 +
20
1
1−=y
( ) 5
1
20
1
44 ==p

16. 1π
Jeff Bivin -- LZHS
Graph the following parabola
x = y2
- 4y + 11
x = (y2
- 8y ) + 11
x + 8 = (y2
- 8y + (-4)2
) + 11
x = (y - 4)2
+ 3
Axis of symmetry: y = 4
Vertex: (3, 4)
x = (y2
- 8y + (-4)2
) + 11 - 82
1
2
1
2
1
2
1
2
1
8)16(
2
1
=
y - 4 = 0

17. 1π
Jeff Bivin -- LZHS
Graph the following parabola
x = (y - 4)2
+ 3
Axis of symmetry: y = 4
Vertex: (3, 4)
Focus:
Directrix:
Length of LR:
2
1
( )4,3 2
1
2
1
2=x
2
1
)(4
1
4
1
2
1 === ap
( )4,3 2
1
+
2
1
3−=x
( ) 244 2
1
==p

18. 1π
Jeff Bivin -- LZHS
A Web Site & Sketchpad demo
• http://www.xahlee.org/SpecialPlaneCurves_dir/Parabola_dir/parabolaReflect.mov
• A sketchpad demo:

19. 1π
Jeff Bivin -- LZHS