1. Conics: The Parabola Objectives:
Use the standard and general forms of the equation of a parabola
Graph parabolas
©2002 Roy L. Gover (

2. Applications
Trajectory
The Parabola:
Parabolic Reflectors

3. Definition
A parabola is the locus (set) of all points that are equidistant from a point called the focus and a line called the directrix.

4. Important Idea They may point up and down or left and right
The graphs of parabolas come in different shapes and sizes

5. Definition
The vertex is the lowest, highest, leftmost or rightmost point on the graph

6. Definition
The vertex is half-way between a point called the focus and a line called the directrix.
Vertex
Focus
Directrix
Focus
Directrix
Vertex

7. Important Idea The focus is a point and is always inside the parabola
Directrix
Vertex
Vertex
Directrix

8. Important Idea
The directrix is a line and is always outside the parabola
Focus
Focus
Directrix
Vertex
Vertex
Directrix

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

10. Important Idea
The distance between the focus and vertex is p units where p is a real number. p

11. Important Idea
The distance between the vertex and directrix is also p units p
These distances are always the same.

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

13. Try This Match the letter to the name of the parts of a parabola: A C

14. Try This Match the letter to the name of the parts of a parabola:
What is true about the distance from C to A and the distance from C to B? C B A C A

15. Try This
What appears to be true about the distance from the focus to the points on the parabola opposite the focus? 2p

16. Definition Standard Equation for the parabola-2 forms: 1.
Opens left if p is negative
Opens right if p is positive
Vertex is at (h,k)
P is distance from vertex to focus

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

18. Definition
p>0
p<0
p>0
p<0

19. Example
Sketch the parabola, label the vertex, focus, directrix & axis of symmetry for the parabola with vertex at (2,3), p=2, & directrix parallel to the y axis. Write the equation.

20. Try This
Sketch the parabola, label the vertex, focus, directrix & axis of symmetry for the parabola with vertex at (3,2), p=2, & directrix parallel to the x axis. Write the equation.

21. Solution h k 4p Vertex: (3,2) Focus:(3,4) Directrix:y=0
Axis of Sym:x=3

22. Example
Graph using a graphing calculator:

23. Example
Graph using a graphing calculator:

24. Example Sketch the graph of the parabola:
Hint: write in standard form by completing the square.

25. Try This
Write the standard form of the following parabola:

26. Try This
Find the coordinates of the vertex and the value of p for the parabola. Which way does the parabola open?
Vertex: (0,4);p=4;open up

27. Lesson Close
Which of the following are equations of circles and which are equations of parabolas?