Parabolas Definition Sketch Characteristics A parabola is the curve formed Vertical (B = O) by the intersection of a plane Horizontal (A = 0) General Form: and a cone, when the plane is No x Z Ax + By + Cx + Dy + E = 0 at the same slant as the side of where A = 0 OR B = 0 the cone. ta - a OR A parabola is the set of all ta Vertex: (h,k) points in a plane equidistant - a from a fixed point (FOCUS) Focal Length = p and a fixed line Standard form: Standard form: (DIRECTRIX) in the plane. y - k = a(x - h) x - h = a(y - k) 4ap = 1 so a =- Eccentricity: e = 1 Parametric Form: Parametric Form: x=t x = a(t - h)" + k y = a(t - h) + k y= t From the equation, identify the vertex (V) of each parabola. Using the formula for focal length (p) and the coordinates of the focus (F) for each parabola. Label and find the directrix. y - 2 =1x 16 * + D y - 4 =- 0.1(x - 3)2 x + 2 =1(1 - 5) a: 76 h: 0 K: 2 a: - 0. 1 h: 3 k: 4 a: 1 h:-2 k: 5 y = 6.5 Focus vertex Vertex . Focus Vertex Focus X = -2125 y = - 2 a= 76 a= -0.I a= V = (0, 2 ) V = ( 3 , 4 ) 4 ap = 1 V= ( - 2, 5 ) 4ap= 1 p= 4R 4ap = 1 P= -2.5 4 ( - 1) p= 1 P = T Or D. 25 4 (1) p = 1 F= (0,6) 4 ( tc ) P = 1 F= ( 3 , 1.5) -. 4 p= 1 F= (-1.75, 5) 4 p = 1 4P=1 P = 4 4 away P = 4 2.5 away P =- 2.5 4 away from vertex from vertex from vertexFor each scenario below, use the given values to find the missing values and write the equation of the parabola in standard form. You will need to use the formula a = - 1) random xy paint ( 3.2, 8 ) Use X -h = a /y-k )? 2) The vertex is at (0,-2) and the focus is at (-1,-2) vertex Focus b/c parabola opens Food . Parabola opens (0/0 ) to right around the focus 3.2- 0 = 4 (8- 0 ) 3 . 2 = a (8) ? a = 40 = 3. 2= 0. 64 .05 = a a= 0. 05 4 ap= 1 p= 5 5 right of 4 (105 ) p = 1 as F: ( 5, 0 ) vertex P = 5 P= - 1 (from X= 0, went left 1) Equation: X-D = . 05 ( y-0)? Dirextric: X= 1 (if focus is one unit left, Equation: directnix is one unit right X = . 05 42 X-0 = = (y+2)2 X = = (4+2)2 3) The vertex is (-3,-1) and the directrix y =- I = a 4p parabola opens away from directrix Focus 4 ( -.25 ) = a a= =1 p= = 25 F= (-3, -1.25) Equation: 4 +1= -1(X+3)2 distance from Vertex to directnix. will move down to focus so pis neg.Standard Form of a H Parabola: Standard Form of a V Parabola: x - h = a(y - k) y - k = a(x - h) Vertex: (h,k) Parametric Form of a H Parabola: Parametric Form of a V Parabola: = a(t - h) + k x = t = y = a(t - h) + k