1. 10.2 Parabolas What you should learn: Goal1 Goal2 Graph and write equations of parabolas. Identify the FOCUS and DIRECTRIX of the parabola. 10.2 Parabolas Goal3 Write the Standard form of the equation of the parabola.

2. Parabolas  We already know the graph of y = ax 2 is a parabola with vertex (0,0) and AOS x = 0  Every parabola has the property that any point on it is equidistant from a point called the Focus and a line called the directrix. 10.2 Parabolas

3. Focus Lies on AOS Directrix 10.2 Parabolas

4. The Focus is a point. The Directrix is a line (the vertex is ½ way between the focus and directrix) y = ax 2 standard equation before a =4p x 2 = 4py new equation. 10.2 Parabolas

5. x 2 =4py, p>0 Focus (0,p) Directrix y=-p 10.2 Parabolas

6. x 2 =4py, p<0 Focus (0,p) Directrix y=-p 10.2 Parabolas

7. y 2 =4px, p>0 Directrix x=-p Focus (p,0) 10.2 Parabolas

8. y 2 =4px, p<0 Focus (p,0) Directrix x=-p 10.2 Parabolas

9. Standard equation of Parabola (vertex @ origin) EquationFocusDirectrixAOS x 2 =4py(0,p)y = -p Vertical (x=0) y 2 =4px(p,0)x = -p Horizontal (y=0) 10.2 Parabolas

10. Identify the Focus and Directrix of the parabola x = -1/6y 2 Since y is squared, AOS is horizontal Isolate the y 2 → y 2 = -6x Since 4p = -6 p = -6/4 = -3/2 Focus : (-3/2,0) Directrix : x=-p=3/2 To draw: make a table of values & plot p<0 so opens left so only choose neg values for x 10.2 Parabolas Goal2

12. Your Turn! Find the focus and directrix, then graph x = 3/4y 2 y 2 so AOS is Horizontal Isolate y 2 → y 2 = 4/3 x 4p = 4/3 p = 1/3 Focus (1/3,0) Directrix x=-p=-1/3 10.2 Parabolas

14. Writing the equation of a parabola. The graph shows V=(0,0) Directrex y=-p=-2 So substitute 2 for p 10.2 Parabolas Goal3

15. x 2 = 4py x 2 = 4(2)y x 2 = 8y y = 1/8 x 2 and check in your calculator 10.2 Parabolas

16. Your turn! Focus = (0,-3) X 2 = 4py X 2 = 4(-3)y X 2 = -12y y=-1/12x 2 to check 10.2 Parabolas

17. Assignment 10.2 Parabolas