1. 12.4
Conic Sections and Parabolas

2. Conic Sections
Hyperbola
Parabola
Ellipse
Circle

3. General Form Equation of Conic Sections
Where ‘A’, ‘B’, ‘C’, ‘D’, ‘E’, and ‘F’ are real numbers
AND
‘A’, ‘B’, and ‘C’ are not all zero.

4. Parabola as a function:
Parabola: A “U” shaped curve opening either up or down.
Vertex: The highest or lowest point on the parabola.
Axis of Symmetry: the vertical line that divides the parabola into
mirror images and passes through the vertex.
What you’ve learned so far are parabolas that have
vertical axes of symmetry
Now we will learn equations of parabolas that have axes of
symetry that are lines in either horizontal or vertical direction
 the equation is not necessarily a function.

5. What is the general form of an equation of a parabola with:
Vertex at the origin and
opening upward?
x-axis

6. Geometry of a Parabola Axis Directrix Point on parabola Distance to
the directrix
Distance to
the focus
Focus
Vertex
Distance to
the directrix
Distance to
the focus =
Directrix

7. Geometry of a Parabola Directrix Focus Distance to the directrix
the focus =
Directrix

8. Axis
(0, P) P
x-axis -P
Directrix: y = -P

9. Axis Distance from (x, y) to (x, -p) = distance from (x, y) to (0, p)
down: p < 0
(0, p)
up: p > 0 p
x-axis -p
(x, -p)
Directrix: y = -P

10. Your turn:
p = ?
p = ?

11. Axis Focal width (4p): the length of the chord passing thru the
vertex and perpendicular to
the axis.
Focal length (p): the distance
from the vertex to the focus.
Axis
focus p
Vertex
x-axis
Directrix: y = -P

12. What is the focal length of the following parabola?
What is the focal width of the above parabola?
Axis
Focal width (4p): the length of the chord passing thru the vertex and perpendicular to
the axis.
Focal length (p): the distance
from the vertex to the focus.
focus p
Vertex
x-axis
Directrix: y = -P

13. y-axis Parabolas opening to the left or right are inverses
of the parabolas opening up or down.
Exchange ‘x’ and ‘y’.
y-axis
(P, 0)
x-axis -P P
Directrix:
x = -P

14. Your turn: 3. Which way does the parabola open?
6. What is the focal length of the following parabola?
7. What is the focal width of the following parabola?

15. Parabolas with Vertex (0, 0)
Standard Equations:
Opening Direction:
Up/down
Right/Left
Focus:
(0, p)
(p, 0)
Directrix:
Axis:
y-axis
x-axis
Focal Length:
Focal width:

16. Example Problem: Standard Equations: Opening Direction: down Focus:
(0, -1/8)
Directrix:
Axis:
y-axis
Focal Length:
Focal width:

17. Your turn: for the following equation, find the following information about the parabola.
8. Opening Direction:
9. Focus:
10. Directrix:
12. Axis:
13. Focal Length:
14. Focal width:

18. What is the equation of the parabola?
Given the parabola with the following information:
Which way does this parabola open?
To the right
Focus:
(3, 0)
Focal Length = ?
p = 3
(0, 0)
Focal length (p): the distance
from the vertex to the focus.
Directrix:
x = -3
What is the equation of the parabola?
Focal width = ?
Focal width = 12
Focal width (4p): the length of the chord passing thru the vertex and perpendicular to
the axis.
What is the parabola’s
axis ?
x -axis

19. 15. Which way does this parabola open?
Your turn: Given the following information on a parabola:
Directrix:
y = 2
15. Which way does this parabola open?
down
Focus:
(0, -2)
16. Focal Length = ?
p = 2
(0, 0)
17. Focal width = ?
Focal width = 8
18. What is the parabola’s axis ?
y -axis
19. What is the equation of the parabola?

20. Your turn: Find the equation of the parabola that has a directrix of y = -4 and a focus of (0, 4).

21. Parent Function and transformations
How is the second parabola a transformation of the parent function?
Right 3, down 5
How is the second parabola a transformation of the parent function?
Left 2, down 7

22. Parabolas with Vertex (h, k)
Standard Equation:
Opening Direction:
Up/down
Up/down
Focus:
(0, p)
(h, k + p)
Directrix:
y-axis
(x = 0)
Axis:
x = h
Focal Length:
Focal width:

23. Vertex: (h, k) y value of vertex and y value of focus are
the same  axis is
a horizontal line.
Standard Equations:
(5, 4)
Opening Direction:
Left/right
right
Focus:
(h + p, k)
(3 + p, 4)
= (5, 4)
(3, 4)
p = 2
Directrix:
x = 3 - 2
x = 1
Axis:
y = k
y = 4
Focal Length: 2 8
Focal width:

24. Your turn: 21. Find the equation of the parabola that has a vertex of (2, 5) and a focus of (2, -2).
Vertex: (h, k)
h = 2 k = 5
Focus below vertex: opens down
Focus: (h, k + p)
= (2, -2)
 5 + p = -2
p = -7