1. Section 2.3---Focus of a Parabola
We will be able to: use the focus and directrix of a parabola to solve real life problems.

2. Parabola---Focus and Directrix
A parabola can be defined as the set of points that are equidistant from a fixed point (Focus) and a fixed line (Directrix)

4. Example 1
Use the Distance Formula to write an equation of the parabola with focus F (0, 2) and directrix 𝑦=−2.

5. Example 1 Continued
𝑃𝐷=𝑃𝐹 𝑥− 𝑥 𝑦− 𝑦 1 2 = 𝑥− 𝑥 𝑦− 𝑦 2 2 𝑥−𝑥 2 + 𝑦−(−2) 2 = 𝑥−0 2 + 𝑦−2 2 𝑦+2 2 = 𝑥 2 + 𝑦−2 2 𝑦+2 2 = 𝑥 2 + 𝑦−2 2 𝑦 2 +4𝑦+4= 𝑥 2 + 𝑦 2 −4𝑦+4 8𝑦= 𝑥 2 𝑦= 1 8 𝑥 2

6. Opens up or down
Opens left or right

7. Example 2: Graphing a Parabola
Identify the focus, directrix, and axis of symmetry of −4𝑥= 𝑦 2 . Graph the equation.
Step 1: Rewrite the equation in standard form:
𝑥=− 1 4 𝑦 2

8. Example 2 Continued
Step 2: Identify the focus, directrix, and axis of symmetry.
Equation is in the form 𝑥= 1 4𝑝 𝑦 2 , but since it is negative that means 𝑝=−1 . So the focus is (−1,0).
Directrix is 𝑥=−(−1) which is 𝑥=1.

9. Example 2 Continued Again
Step 2 Continued:
The vertex is at (0, 0). So the axis of symmetry is y=0 (the x-axis)
Step 3: Graph the vertex. Pick another point for y! to graph.
(−0.25, 1)

11. Homework
Page 72 # 3, 5, 6