1. M3U4D8 Warm Up
Find the vertex using completing the square: 2x2 + 8x – 12 = 0
x2 + 4x – 6 = 0
x2 + 4x = 6
x2 + 4x +__ = 6 + __
(x + )2 =
(x + 2) 2 – 10 = y
COLLECT WARM-UPS 4 4 2 10
Vertex = (-2,-10)

2. Vertex Form
y=2x2+6x+4 substitute in 0 for y then complete the square

3. Vertex Form
y=4x2+16x-28

4. Homework Check:
ON DOCUMENT CAMERA

5. Focus and Directrix
OBJ: Derive the equation of a parabola given a focus and directrix. (G-GPE.2)

6. Parabolas
We have learned that the graph of the quadratic function f (x) = ax2 + bx + c is a parabola that opens upward or downward.
The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola.

7. Parabolas Orientation Vertex (h, k) Directrix Focus Axis of symmetry
Equations:

8. Definition of Parabola
A parabola is the set of all points in the plane equidistant from a fixed line and a fixed point not on the line.
The fixed line is the directrix.
The fixed point is the focus.
axis
parabola
The axis is the line passing through the focus and perpendicular to the directrix.
focus
The vertex is the midpoint of the line segment along the axis joining the directrix to the focus.
directrix
vertex
Definition of Parabola

9. The standard form for the equation of a parabola with vertex at the origin and a vertical axis is:
x2 = 4py where p  0
focus: (0, p)
vertical axis: x = 0
directrix: y = –p, x y
x2 = 4py
(0, p) p
(0, 0)
y = –p
Note: p is the directed distance from the vertex to the focus.

10. Parabolas
Using the definition of a parabola, you can derive the following standard form of the equation of a parabola whose directrix is parallel to the x-axis or to the y-axis. WE WILL ONLY DO VERTICAL!!!

11. A ray that travels parallel to the axis of symmetry will strike the surface of the parabola or paraboloid and reflect toward the focus.
Likewise, when a ray from the focus strikes the curve, it will reflect in a ray that is parallel to the axis of symmetry.

12. Where is this property seen in our everyday life?
Satellite dishes, car headlights, flashlights, etc

13. Vertex is half the distance from the focus to the directrix. So the vertex will be at (1, ½)

14. 2) If the vertex is (-2, 2), and the focus is (-2, -4), what is the equation of the directrix?

15. The distance from the directrix to the vertex is one unit. The parabola is concave down. Therefore the focus is located at
(6, 1)

18. 5. Find all important information for the conic section with an equation:
Vertex: (-4,-4)
a: a=-1/4
Distance v to f (“p”) = 1/4a
p = ¼(-1/4) = -1
Focus: (-4,-5)
Directrix: y = -3

19. 6. A parabola has a directrix at y = 6 and a focus at (-6,0).
What is the equation of the parabola?
y = -1/12 (x + 6)2 + 3

20. Answer: Five centimeters above the vertex along the line of symmetry.
Sheila is designing a parabolic dish to use for cooking on a camping trip. She plans to make the dish 40cm wide and 20 cm deep. Where should she locate the cooking grill so that all the light that enters the parabolic dish will be reflected toward the food?
Answer: Five centimeters above the vertex along the line of symmetry.

21. Example: Graph Parabola
Example: Find the directrix, focus, and vertex, and sketch the parabola with equation
Rewrite the equation in vertex form y = -1/8 (x + 0)2 + 0.
1/4a = dist, ¼(-1/8) = dist, -2 = dist x y
x = 0
y = 2
vertex: (0, 0)
(0, 0)
vertical axis: x = 0
a: a = -1/8
(0, –2)
directrix: y = 2
focus: = (0, p)  (0, –2)
Example: Graph Parabola

22. Classwork M3U4D8 Quadratics from the Focus and Directrix Notes
Complete together with the Document Camera

23. U4D8 Using the Focus and Directrix #5-10
Homework
U4D8 Using the Focus and Directrix #5-10