1. 9.2 Notes – Parabolas

2. Parts of a Parabola: (U-shaped)
Vertex:
Axis of Symmetry:
Focus:
Directrix:
Turning point
Through vertex, folds in half
Point inside
Line outside, perpendicular to axis of symmetry

3. Parabola: equidistant to the Focus and Directrix
vertex
directrix
Axis of symmetry
Parabola: equidistant to the Focus and Directrix

4. Remember transformations of parabolas?
(h, k)
Vertex: ____________

5. or To find the distance from the vertex to the directrix or focus:
Make sure to take the absolute value for distance!

6. Using y = ax2 where “a” determines the width and direction of the graph:
If:
width is: ________ _________ ________
normal
skinny
wide

7. vertical horizontal (h, k) up right down left y = k x = h
Where k is the ___________ shift of vertex from (0, 0) and h is the _____________ shift
Vertex: _____________
a > 0 parabola opens _______
a > 0 parabola opens to the ______
a < 0 parabola opens _______
a < 0 parabola opens to the ______
Axis of symmetry: _________
Axis of symmetry: _____________
vertical
horizontal
(h, k) up
right
down
left
y = k
x = h

8. Vertex: _____________ Horizontal or Vertical? Opens: _____________
1. Identify vertex of each parabola, whether it’s vertical or horizontal, and which way it opens: a)
(–2, –7)
Vertex: _____________
Horizontal or Vertical?
Opens: _____________
left

9. Vertex: _____________ Horizontal or Vertical? Opens: _____________
1. Identify vertex of each parabola, whether it’s vertical or horizontal, and which way it opens: b)
(2, –3)
Vertex: _____________
Horizontal or Vertical?
Opens: _____________ up

10. Vertex: _____________ Horizontal or Vertical? Opens: _____________
1. Identify vertex of each parabola, whether it’s vertical or horizontal, and which way it opens: c)
(2, –5)
Vertex: _____________
Horizontal or Vertical?
Opens: _____________
right

11. 2. Draw the directrix |d| behind the parabola
2. Draw the directrix |d| behind the parabola. Show where the focus is by going along the axis of symmetry d units toward the inside of the parabola. x y -8 -4 -2 2 4 8 8 2
1/2
1/2 2 8
Focus:
(0, 2) = 2
y = -2 2

12. Try any point on the parabola and see if the distance to the focus is about the same distance to the directrix you drew.
Notice two special points on any parabola can be drawn once you know the vertex and the distance d from the vertex to the directrix.
Additional points: 2d

13. down x = 5 1 y = 3 2. Graph the parabola. Vertical or horizontal
Opens ___________
Vertex ( , )
Axis of symmetry: _______
d =
down
x = 5 1
Focus ( , )
Directrix: __________
Additional points ( , ) ( , )
y = 3

14. left y = 2 3 x = 7 3. Graph the parabola. Vertical or horizontal
Opens ___________
Vertex ( , )
Axis of symmetry: _______
d =
left
y = 2 3 3
Focus ( , )
Directrix: __________
Additional points ( , ) ( , )
x = 7
1 –4

15. right y = –1 2 x = –4 4. Graph the parabola. Vertical or horizontal
Opens ___________
Vertex ( , )
Axis of symmetry: _______
d =
right
–2 –1
y = –1 2 2
Focus ( , )
Directrix: __________
Additional points ( , ) ( , )
0 –1
x = –4
1 –4

16. 5. Graph the parabola. 25 25

17. up x = 5 y = -7.5 5. Graph the parabola. Vertical or horizontal
Opens ___________
Vertex ( , )
Axis of symmetry: _______
d = up
5 –6
x = 5 2 3
Focus ( , )
Directrix: __________
Additional points ( , ) ( , )
y = -7.5

18. 6. Graph the parabola. 1 1

19. left y = 1 2 x = 4 3. Graph the parabola. Vertical or horizontal
Opens ___________
Vertex ( , )
Axis of symmetry: _______
d =
left
y = 1 2 2
Focus ( , )
Directrix: __________
Additional points ( , ) ( , )
x = 4
1 –4

20. 7. Write the standard equation for the parabola that has a vertex of (-1, -5) and a focus of (-1, -6). First, show a rough sketch of the layout only and determine whether a will be positive or negative, then find a.

21. 8. Write the standard equation for the parabola that has a vertex of (3, 1) and a focus of (6, 1). First, show a rough sketch of the layout only and determine whether a will be positive or negative, then find a.

22. Either x or y are squared, not both
How is the parabola equation different from circles?
Circle
Parabola
x and y are squared
Either x or y are squared, not both
Ax2 + Cy2 + Dx + Ey + F = 0
Has an A and C term
A or C = 0