1. Conic Sections The Parabola

2. Introduction Consider a cone being intersected with a plane Note the different shaped curves that result

3. Introduction We will consider various conic sections and how they are described analytically  Parabolas  Hyperbolas  Ellipses  Circles They can be described or defined as a set of points which satisfy certain conditions

4. Parabola Definition  A set of points on the plane that are equidistant from  A fixed line (the directrix) and  A fixed point (the focus) not on the directrix

5. Parabola Note the line through the focus, perpendicular to the directrix  Axis of symmetry Note the point midway between the directrix and the focus  Vertex View Geogebra Demonstration View Geogebra Demonstration

6. Equation of Parabola Let the vertex be at (0, 0)  Axis of symmetry be y-axis  Directrix be the line y = -p (where p > 0)  Focus is then at (0, p) For any point (x, y) on the parabola Distance = y + p Distance =

7. Equation of Parabola Setting the two distances equal to each other What happens if p < 0? What happens if we have... simplifying...

8. Working with the Formula Given the equation of a parabola  y = ½ x 2 Determine  The directrix  The focus Given the focus at (-3,0) and the fact that the vertex is at the origin Determine the equation

9. When the Vertex Is (h, k) Standard form of equation for vertical axis of symmetry Consider  What are the coordinates of the focus?  What is the equation of the directrix? (h, k)

10. When the Vertex Is (h, k) Standard form of equation for horizontal axis of symmetry Consider  What are the coordinates of the focus?  What is the equation of the directrix? (h, k)

11. Try It Out Given the equations below,  What is the focus?  What is the directrix?

12. Another Concept Given the directrix at x = -1 and focus at (3,2) Determine the standard form of the parabola

13. Applications Reflections of light rays  Parallel rays strike surface of parabola  Reflected back to the focus View Animated Demo Build a working parabolic cooker Build a working parabolic cooker How to Find the Focus Proof of the Reflection Property Proof of the Reflection Property Spreadsheet Demo MIT & Myth Busters

14. Applications Light rays leaving the focus reflect out in parallel rays Used for Searchlights Used for Searchlights Military Searchlights Military Searchlights

15. Assignment See Handout Part A 1 – 33 odd Part B 35 – 43 all