1. 10.2 Parabolas
Where is the focus and directrix compared to the vertex?
How do you know what direction a parabola opens?
How do you write the equation of a parabola given the focus/directrix?
What is the general equation for a parabola?

2. Parabolas
A parabola is defined in terms of a fixed point, called the focus,
A parabola is the set of all points P(x,y) in the plane whose distance to the focus
focus
and a fixed line,
called the directrix.
equals its distance to the directrix.
directrix
axis of symmetry

3. Horizontal Directrix
Standard Equation of a parabola with its vertex at the origin is x y
D(x, –p)
P(x, y)
F(0, p)
y = –p O
x2 = 4py
p > 0: opens upward
p < 0: opens downward
focus: (0, p)
directrix: y = –p
axis of symmetry: y-axis

4. Vertical Directrix
Standard Equation of a parabola with its vertex at the origin is x y
D(x, –p)
P(x, y)
F(p, 0)
x = –p O
y2= 4px
p > 0: opens right
p < 0: opens left
focus: (p, 0)
directrix: x = –p
axis of symmetry: x-axis

5. Example 1 Graph . Label the vertex, focus, and directrix. y2 = 4px
Identify p. -4 -2 2 4
y2 = 4(1)x
So, p = 1
Since p > 0, the parabola opens to the right.
Vertex: (0,0)
Focus: (1,0)
Directrix: x = -1

6. Example 1 Graph . Label the vertex, focus, and directrix. Y2 = 4x y x
Use a table to sketch a graph -4 -2 2 4 y x 2 4 -2 -4 1 4 1 4

7. Example 2
Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6.
Since the directrix is below the vertex, the parabola opens up
Since y = -p and y = -6,
p = 6
x2=4(6)y
x2 = 24y

8. Where is the focus and directrix compared to vertex?
The focus is a point on the line of symmetry and the directrix is a line below the vertex. The focus and directrix are equidistance from the vertex.
How do you know what direction a parabola opens?
x2, graph opens up or down, y2, graph opens right or left
How do you write the equation of a parabola given the focus/directrix?
Find the distance from the focus/directrix to the vertex (p value) and substitute into the equation.
What is the general equation for a parabola?
x2= 4py (opens up [p>0] or down [p<0]), y2 = 4px (opens right [p>0] or left [p<0])

9. Assignment
p. 598, 16-21, odd

10. 10.2 Parabolas, day 2
What does it mean if a parabola has a translated vertex?
What general equations can you use for a parabola when the vertex has been translated?

11. Standard Equation of a Translated Parabola
Vertical axis:
(x − h)2 = 4p(y − k)
vertex: (h, k)
focus: (h, k + p)
directrix: y = k – p
axis of symmetry: x = h

12. Standard Equation of a Translated Parabola
Horizontal axis:
(y − k)2 = 4p(x − h)
vertex: (h, k)
focus: (h + p, k)
directrix: x = h - p
axis of symmetry: y = k

13. Example 3
Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. Sketch the info.
The parabola opens downward, so the equation is of the form
(x − h)2 = 4p(y − k)
vertex: (-3,3)
h = -3, k = 3
p = -1
(x + 3)2 = 4(−1)(y − 3)

14. Example 4
Write an equation of a parabola whose vertex is at (−2,1) and whose focus is at (−3, 1).
Begin by sketching the parabola. Because the parabola opens to the left, it has the form
(y −k)2 = 4p(x − h)
Find h and k: The vertex is at (−2,1) so h = −2 and k = 1
Find p: The distance between the vertex (−2,1) and the focus (−3,1) by using the distance formula.
p = −1 (y − 1)2 = −4(x + 2)

15. What does it mean if a parabola has a translated vertex?
It means that the vertex of the parabola has been moved from (0,0) to (h,k).
What general equations can you use for a parabola when the vertex has been translated?
(y-k)2 =4p(x-h) (x-h)2 =4p(y-k)

16. Assignment
p. 598, even, even
p. 628, 15-16, 22, 28