8 1 Conic Sections Parabolas PreCalculus 4162007 Parabolas

8. 1 Conic Sections (Parabolas) Pre-Calculus 4/16/2007

Parabolas with vertex (h, k) Standard Equation (x – h)2 = 4 p(y – k)2 = 4 p(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) Directrix y=k–p x=h–p Axis x=h y=k Focal length p p Focal width Pre-Calculus 4/16/2007

8. 1 Conic Sections (Ellipses) Pre-Calculus 4/16/2007

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

8. 3 Conic Sections (Hyperbolas) Pre-Calculus 4/16/2007

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

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

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

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

Examples • • 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. 4/16/2007

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