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

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 Parabolas

Focus Lies on AOS Directrix 10.2 Parabolas

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 Parabolas

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

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

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

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

Standard equation of Parabola 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

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

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/ Parabolas

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

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

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

Assignment 10.2 Parabolas