1. Conic Sections
The Parabola

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

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

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. View Geogebra Demonstration
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

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 =
Distance = y + p

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 x2
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