4/16/2007 Pre-Calculus 8.1 Conic Sections (Parabolas) 8.1 Conic Sections (Parabolas)

4/16/2007 Pre-Calculus Parabolas with vertex (h, k) Standard Equation(x – h) 2 = 4p(y – k)(y – k) 2 = 4p(x – h) Opens Upward (p > 0) Downward (p < 0) To the right (p > 0) To the left (p < 0) Focus(h, k + p)(h + p, k) Directrixy = k – px = h – p Axisx = hy = k Focal lengthpp Focal width

4/16/2007 Pre-Calculus 8.1 Conic Sections (Ellipses) 8.1 Conic Sections (Ellipses)

4/16/2007 Pre-Calculus Ellipses with Center (h, k) Standard Equation Focal Axis y = kx = h Foci (h  c, k)(h, k  c) Vertices (h  a, k)(h, k  a) Semimajor Axisaa Semiminor Axisbb Pythagorean Relation a 2 = b 2 + c 2

4/16/2007 Pre-Calculus 8.3 Conic Sections (Hyperbolas) 8.3 Conic Sections (Hyperbolas)

4/16/2007 Pre-Calculus Hyperbola with Center (h, k) Standard Equation Focal Axis y = kx = h Foci (h  c, k)(h, k  c) Vertices (h  a, k)(h, k  a) Semitransverse Axis aa Semiconjugate Axis bb Pythagorean Relation c 2 = b 2 + a 2 Asymptotes

4/16/2007 Pre-Calculus Find the vertex, focus, directrix, and focal width: Examples Vertex: (3, –2) Opens: leftp: –4 Focus: (–1, –2) Directrix: x = 7 Focal width: 16

4/16/2007 Pre-Calculus Find the vertices and foci: Examples Center: (4, –2) a = √ 10 b = √ 6 c = √(10 – 6) = 2 Vertices: (4 + √ 10, –2) (4 + √ 10, –2) Foci: (6, –2) (2, –2)

4/16/2007 Pre-Calculus Find the vertices and foci: Examples Center: (4, –2) a = √ 6 b = √ 10 c = √(6 + 10) = 4 Vertices: (4 + √ 6, –2) (4 + √ 6, –2) Foci: (8, –2) (0, –2)

4/16/2007 Pre-Calculus Prove that the graph of is an ellipse. Find the center, vertices and foci. Graph the ellipse by hand first. Check the solution using your graphing calculator. Examples

4/16/2007 Pre-Calculus  (h, k) (h – a, k) (h + a, k) (h + c, k) (h – c, k)