11.4 The Parabola

Parabola: the set of all points P in a plane that are equidistant from a fixed line and a fixed point not on the line. (directrix) Basic look of a parabola Focus parabola (focus) directrix *think: food goes in a bowl on a desk *order works even if parabola is sideways or upside down c c

Parabola centered at (h, k) A of S F d a(x – h) 2 = y – k vertex: (h, k) axis of symmetry: x = h Focus: (h, k + c) directrix: y = k – c opens: up if a > 0 down if a < 0 c c V(h, k) 2c c c d A of S F V (h, k) a(y – k) 2 = x – h vertex: (h, k) axis of symmetry: y = k Focus: (h + c, k) directrix: x = h – c opens: right if a > 0 left if a < 0

vertex: (2, 3) y 2 & a is (+)  opens right F: (4, 3) d: x = 0 A of S: y = 3 count 2c up & down from focus for 2 pts on parabola Ex 1) Determine the vertex, the axis of symmetry, the focus, & the directrix of (y – 3) 2 = 8(x – 2). Graph it.  c = 2 (plot points 2 right & 2 left)

Ex 2) Find vertex, the axis of symmetry, the focus, & the directrix of 2x 2 – 4x + y + 4 = 0. *Complete the square! y + 4 = –2x 2 + 4x y –2 = –2(x 2 – 2x + 1 ) y + 2 = –2(x – 1) 2 vertex (1, –2) opens down make a sketch! (1, –2) A of S: x = 1 directrix: Focus:

Ex 3) Determine the equation of the parabola with focus (3, –4) and directrix y = 2. *Think about food-bowl-desk *Make a sketch! F(3, –4) y = 2Vertex in between! V(3, –1) halfway: c = 3  opens down so a is (–)

Homework #1104 Pg 561 #1, 5, 9, 13, 17, 22, 27, 31, 35, 39, 41, 43, 47, 49, 50