Conic Sections Dr Shildneck Fall 2015 CONIC SECTIONS

Conic Sections Dr. Shildneck Fall, 2015

CONIC SECTIONS Parabolas

Parabolas DEFINITION FOCUS A parabola is the locus of points such that every point is equidistant from a fixed point (called the focus) focus and a fixed line (called the directrix). directrix DIRECTRIX

Parabolas Standard Equations (x-h)2 = 4 p(y-k)2 = 4 p(x-h) If the x-term is squared, the parabola opens vertically If the y-term is squared, the parabola opens horizontally h = x-coordinate of the vertex k = y-coordinate of the vertex p = distance from the vertex to the focus (sign indicates direction) -p = distance from the vertex to the directrix |4 p| = distance across the parabola through the focus

Quick Graphs 1. Determine the orientation 2. Find and Plot Vertex 3. Find p and plot the focus 4. Graph the directrix 5. Find 4 p and find each point (2 p) from the focus. Example 8(x+7) = (y-2)2 Direction Y is squared (left/right) 4 p is positive (up/right) Vertex (-7, 2) Find p 4 p = 8 p=2 Focus is 2 to the right Directrix is 2 left

Writing Equations 1. Know the standard form of the equation 2. Often it helps if you sketch the curve described 3. Find any critical values needed for the equation 4. Plug those into the equation

Example 1 Write the equation of the parabola Vertex: (-2, 4) Opens: down So, x is squared P: down 4; p= -4 p k (x-h)h 2 =4 p(y-k) ANSWER (x +2) 2 = – 16(y– 4 )

Examples Write the equation of the parabola with the following characteristics. 2. Vertex (4, -2) and focus (4, 7) 3. Vertex (5, -4) and directrix x = 8 4. Focus (-2, 4) and directrix y = 1 5. Vertex (3, 1) opens vertically passing through (5, 9)

Assignment Alternate Text (on website) P. 667 Vocabulary #4 -7 Exercises #37 -42, 43, 45, 51, 53, 56, 59, 65, 73, 77, 81, 93