1. Important Idea
Every point on the parabola is the same distance from the focus and the directrix

2. The distance between the focus and vertex is units where p is a real number.

3. p The distance between the vertex and directrix is also |p| units
These distances are always the same.

4. Definition
The line connecting the focus and vertex and perpendicular to the directrix is the axis of symmetry

5. Question
What appears to be true about the distance from the focus to the points on the parabola opposite the focus? 2p

6. What is the Equation? distance from P(x, y) to A(x,-p) is
distance from P(x, y) to F(0, p) is
Therefore:

7. squaring both sides:
(y + p)2 = (x2 + (y-p)2
y2 + 2py + p2 = x2 + y2 -2py + p2
4py = x2

8. Equation of a Parabola
p>0
p<0
p>0
p<0

9. Try This
Sketch the parabola, label the directrix & axis of symmetry for the parabola with vertex at (3,2) & focus at (3,4). Write the equation.

10. Solution
Vertex: (3,2)
Focus:(3,4)
Directrix:y=0
Axis of Sym:x=3

11. Standard Equation for the parabola-2 forms:1.
Opens down if p is negative
Opens up if p is positive
Vertex is at (h,k)
|p| is distance from vertex to focus

12. Standard Equation for the parabola-2 forms:2.
Opens left if p is negative
Opens right if p is positive
Vertex is at (h,k)
|P| is distance from vertex to focus

13. Equation of a Parabola
p>0
p<0
p>0
p<0