advertisement

Precalc 9.1 Parabolas For each parabola, find an equation for the axis of symmetry and the coordinates of the vertex. State whether the parabola opens up or down, and whether the y-coordinate of the vertex is the minimum or maximum value of the function. A parabola is defined in terms of a fixed point, called the focus, and a fixed line, called the directrix. A parabola is the set of all points P(x,y) in the plane whose distance to the focus equals its distance to the directrix. directrix Mar 30­7:52 AM focus axis of symmetry Mar 30­7:55 AM Vertical Directrix Horizontal Directrix Standard Equation of a parabola with its vertex at the origin is p > 0: opens upward p < 0: opens downward focus: (0, p) directrix: y = –p axis of symmetry: y-axis Standard Equation of a parabola with its vertex at the origin is p > 0: opens right p < 0: opens left focus: (p, 0) directrix: x = –p axis of symmetry: x-axis Mar 30­8:03 AM The standard form of the equation of a parabola with vertex at (h,k) is as follows: Mar 30­8:07 AM Graph directrix. . Label the vertex, focus, and vertical axis directrix: y = k ­ p (x ­ h)2 = 4p(y ­ k), p ≠ 0 horizontal axisÍž directrix: x = h ­ p (y ­ k)2 = 4p(x ­ h), p ≠ 0 The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0,0), the equation takes one of these forms: Vertical Axis: Horizontal Axis: x2 = 4py y2 = 4px Mar 9­7:42 AM Mar 30­8:09 AM 1 Identify the focus & the directrix Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6. Graph y = 1/8x2 Focus (­3,0) Directrix y = 2 Mar 9­8:27 AM Standard Equation of a Translated Parabola Horizontal Directrix: vertex: (h, k) focus: (h, k + p) directrix: y = k – p axis of symmetry: x = h Mar 30­8:12 AM Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. Mar 30­8:10 AM Standard Equation of a Translated Parabola Vertical Directrix: vertex: (h, k) focus: (h + p, k) directrix: x = h - p axis of symmetry: y = k Mar 30­8:14 AM Write the standard equation of the parabola with its focus at F(-6,4) and directrix x = 2. I graphed parabola from given. Find vertex & p, then write equation. Mar 30­8:16 AM Mar 30­8:20 AM 2 Graph the parabola vertex, focus, and directrix. -4( . Label the ) ( )-4 Isolate the y-terms Complete the square vertex: (h, k) = (1,-2) Find p: so, p = 1 focus: = (2,-2) directrix: x = 0 Mar 30­8:22 AM Mar 30­8:29 AM Graph the parabola x2 – 6x + 6y + 18 = 0. Label the vertex, focus, and directrix. Mar 30­8:33 AM Mar 9­7:54 AM 3