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.

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)

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

Example 1 Continued 𝑃𝐷=𝑃𝐹 𝑥− 𝑥 1 2 + 𝑦− 𝑦 1 2 = 𝑥− 𝑥 2 2 + 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

Opens up or down Opens left or right

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

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.

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)

Homework Page 72 # 3, 5, 6