1. Conic Sections
Parabola

2. Conic Sections - Parabola
The parabola has the characteristic shape shown above. A parabola is defined to be the “set of points the same distance from a point and a line”.

3. Conic Sections - Parabola
Focus
Directrix
The line is called the directrix and the point is called the focus.

4. Conic Sections - Parabola
Axis of Symmetry
Focus
Vertex
Directrix
The line perpendicular to the directrix passing through the focus is the axis of symmetry. The vertex is the point of intersection of the axis of symmetry with the parabola.

5. Conic Sections - Parabola
Focus d1
Directrix d2
The definition of the parabola is the set of points the same distance from the focus and directrix. Therefore, d1 = d2 for any point (x, y) on the parabola.

6. Finding the Focus and Directrix
Parabola

7. Conic Sections - Parabola
Therefore, the distance p from the vertex to the focus and the vertex to the directrix is given by the formula

8. Conic Sections - Parabola
Using transformations, we can shift the parabola y=ax2 horizontally and vertically. If the parabola is shifted h units right and k units up, the equation would be
The vertex is shifted from (0, 0) to (h, k). Recall that when “a” is positive, the graph opens up. When “a” is negative, the graph reflects about the x-axis and opens down.

9. Graph a parabola. Find the vertex, focus and directrix.
Example 1
Graph a parabola.
Find the vertex, focus and directrix.

10. Parabola – Example 1
Make a table of values. Graph the function. Find the vertex, focus, and directrix.

11. Parabola – Example 1
The vertex is (-2, -3). Since the parabola opens up and the axis of symmetry passes through the vertex, the axis of symmetry is x = -2.

12. Parabola – Example 1 x y -2 -1 -3 Make a table of values. 1 2 3 4 -11 2 3 4 -3
Make a table of values. -1
Plot the points on the graph! Use the line of symmetry to plot the other side of the graph.

13. Find the focus and directrix.
Parabola – Example 1
Find the focus and directrix.

14. Parabola – Example 1
The focus and directrix are “p” units from the vertex
where
The focus and directrix are 2 units from the vertex.

15. Focus: (-2, -1) Directrix: y = -5
Parabola – Example 1
2 Units
Focus: (-2, -1) Directrix: y = -5