1. Goal: Find the equation, focus, and directrix of a parabola.
8.1
Parabolas

2. What you’ll learn about
Geometry of a Parabola
Translations of Parabolas
Reflective Property of a Parabola
… and why
Conic sections are the paths of nature: Any free-moving
object in a gravitational field follows the path of a conic
section.

3. Golden Arches

4. Incoming Waves are concentrated to the focus.
Satellite Dishes
Incoming Waves are concentrated to the focus.

5. Heaters
Heaters are sold which make use of the reflective property of the parabola. The heat source is at the focus and heat is concentrated in parallel rays. Have you walked by the parabolic reflector heater at COSTCO?

6. Path of a Ball
Gallileo was the first to show that the path of an object thrown in space is a parabola.

7. Suspension Cables on the Golden Gate Bridge

8. Parabola
A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.

9. Graphs of x2 = 4py

10. Graphs of y2 = 4px

11. 𝒚= 𝟏 𝟒𝒑 𝒙 𝟐 𝒙= 𝟏 𝟒𝒑 𝒚 𝟐 (0, 0) (0, p) (p, 0) 𝑦=−𝑝 𝑥=−𝑝 4𝑝 Vertex
Form of Equation
𝒚= 𝟏 𝟒𝒑 𝒙 𝟐
𝒙= 𝟏 𝟒𝒑 𝒚 𝟐
Vertex
(0, 0)
Direction of Opening
Focus
(0, p)
(p, 0)
Directrix
𝑦=−𝑝
𝑥=−𝑝
Length of Focal Chord 4𝑝

12. (h, k) 𝒚−𝒌= 𝟏 𝟒𝒑 𝒙−𝒉 𝟐 𝒙−𝒉 𝟐 =𝟒𝒑 𝒚−𝒌 𝒙−𝒉= 𝟏 𝟒𝒑 𝒚−𝒌 𝟐 𝒚−𝒌 𝟐 =𝟒𝒑 𝒙−𝒉
Form of Equation
𝒚−𝒌= 𝟏 𝟒𝒑 𝒙−𝒉 𝟐
𝒙−𝒉 𝟐 =𝟒𝒑 𝒚−𝒌
𝒙−𝒉= 𝟏 𝟒𝒑 𝒚−𝒌 𝟐
𝒚−𝒌 𝟐 =𝟒𝒑 𝒙−𝒉
Vertex
(h, k)

13. For each parabola, find the vertex, focus, directrix, and focal chord length then sketch.
vertex: ____________ focus: _____________ directrix: _________ focal chord: _________
𝑦= 𝑥 2

14. vertex: ____________ focus: _____________ directrix: _________ focal chord: _________

15. vertex: ____________ focus: _____________ directrix: _________ focal chord: __________

16. vertex: ____________ focus: _____________ directrix: _________ focal chord: __________

17. vertex: ____________ focus: _____________ directrix: ________ focal chord: ________

18. Find the equation of the parabola and sketch its graph.
Vertex at (0, 0) and directrix of x = 5.

19. Find the equation of the parabola and sketch its graph.
Focus at (3, -2) and directrix of y = 4.

20. Find the vertex, focus, and length of the focal chord for the parabola below.
vertex: _____________ focus: ______________ focal chord: __________

21. Example Finding an Equation of a Parabola

22. Example Finding an Equation of a Parabola