1. parabola
Last Updated: October 11, 2005

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

3. Distance between focus and vertex = p
Parabola
Distance between focus and vertex = p p p
Distance between vertex and directrix = p
Jeff Bivin -- LZHS

4. Parabola
The line segment through the focus perpendicular to the axis of symmetry with endpoints on the parabola is called the Latus Rectum (LR) 4p 2p 2p p 2p p
Length of the LR = 4p
Jeff Bivin -- LZHS

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

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

7. Graph the following parabola
y = 3(x + 4)2 + 5
x = - 4
Axis of symmetry: x = -4
Vertex: (-4, 5)
p = 1/4a = 1/(4●3) = 1/12
Focus: (-4 , 5 + 1/12)
(- 4, 5)
Directrix: y = 5 – 1/12
Length of LR: 4p = 4(1/12) = 1/3
Jeff Bivin -- LZHS

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

9. Graph the following parabola
y = -2(x - 3)2 + 29
Axis of symmetry: x = 3
Vertex: (3, 29)
Jeff Bivin -- LZHS

10. Graph the following parabola
y = -2(x - 3)2 + 29
Axis of symmetry: x = 3
(3, 29)
Vertex: (3, 29)
p = 1/4a = 1/(4●(-2)) = -1/8
Focus: (3 , /8)
Directrix: y = /8
x = 3
Length of LR: 4p = 4(1/8) = 1/2
Jeff Bivin -- LZHS

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

12. Graph the following parabola
x = (y + 5)2 - 17
Axis of symmetry: y = -5
Vertex: (-17, -5)
Jeff Bivin -- LZHS

13. Graph the following parabola
x = (y + 5)2 - 17
Axis of symmetry: y = -5
Vertex: (-17, -5)
(-17, -5)
y = -5
p = 1/4a = 1/(4●1) = 1/4
Focus: (-17+1/4 , -5)
Directrix: x = /4
Length of LR: 4p = 4(1/4) = 1
Jeff Bivin -- LZHS

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

15. Graph the following parabola
x = -2(y + 2)2 + 7
Axis of symmetry: y = -2
Vertex: (7, -2)
Jeff Bivin -- LZHS

16. Graph the following parabola
x = -2(y + 2)2 + 7
Axis of symmetry: y = -2
Vertex: (7, -2)
focus
(7, -2)
y = -2
p = 1/4a = 1/(4●(-2)) = -1/8
Focus: (7-1/8 , -2)
Directrix: x = 7 + 1/8
Length of LR: 4p = 4(1/8) = 1/2
Jeff Bivin -- LZHS

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

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

19. Graph the following parabola
y = 5(x - 3)2 + 1
x = 3
Axis of symmetry: x = 3
Vertex: (3, 1)
p = 1/4a = 1/(4●5) = 1/20
Focus: (3 , 1 + 1/20)
(3, 1)
Directrix: y = 1 – 1/20
Length of LR: 4p = 4(1/20) = 1/5
Jeff Bivin -- LZHS

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

21. Graph the following parabola
x = (y - 4)2 + 3
Axis of symmetry: y = 4
Vertex: (3, 4)
(3, 4)
y = 4
p =
Focus: (3+1/2 , 4)
Directrix: x = 3 – 1/2
Length of LR: 4p = 4(1/2) = 2
Jeff Bivin -- LZHS

22. Some Web Sites
Jeff Bivin -- LZHS

23. That's
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