More Conic Sections

Objective Given a translation, I can graph an equation for a conic section.

Ellipses (x-h) 2 + (y-k) 2 = 1 a 2 b 2 where a > b Center: (h, k) Major axis: y = k (horizontal) Minor axis: x = h (vertical) Vertices: (h-a, k), (h+a, k) Co-vertices: (h, y+b), (h, y-b) Foci: (h+c, k), (h–c, k) where c 2 = a 2 – b 2

Ellipses (y-k) 2 + (x-h) 2 = 1 a 2 b 2 where a > b Center: (h, k) Major axis: x = h (vertical) Minor axis: y=k (horizontal) Vertices: (h, y+a), (h, y-a) Co-vertices: (h-b, k), (h+b, k) Foci: (h, k+c), (h, k–c) where c 2 = a 2 – b 2

Graphing Ellipses Identify Center Major axis Minor axis Vertices Co-vertices Foci

Graphing Ellipses Identify Center Major axis Minor axis Vertices Co-vertices Foci

Parabolas (x-h) 2 = 4p(y-k) Vertex: (h, k) Axis of symmetry: x = h Focus: (h, k + p) Directrix: y = k – p Length of Latus Rectum: |4p| If p > 0, opens upward If p < 0, opens downward

Parabolas (y-k) 2 = 4p(x-h) Vertex: (h, k) Axis of symmetry: y = k Focus: (h = p, k) Directrix: x = h – p Length of Latus Rectum: |4p| If p > 0, opens right If p < 0, opens left

Graphing Parabolas Identify Vertex Axis of symmetry Focus Directrix Length of Latus Rectum

Graphing Parabolas Identify Vertex Axis of symmetry Focus Directrix Length of Latus Rectum

Hyperbolas

(x-h) 2 – (y-k) 2 = r 2 a 2 b 2 Center: (h, k) Transverse axis: y = k (horizontal) Conjugate axis: x = h (vertical) Vertices: (h-a, k), (h+a, k) Foci: (h-c, k), (h+c, k) where c 2 = a 2 + b 2 Asymptotes: y = k ± (b/a)(x – h) Width of inside rectangle: 2a Height of inside rectangle: 2b Corners of rectangle: (h-a, k+b), (h+a, k+b), (h-a, k-b), (h+a, k-b)

Hyperbolas (y-k) 2 – (x-h) 2 = r 2 a 2 b 2 Center: (h, k) Transverse axis: x = h (horizontal) Conjugate axis: y = k (vertical) Vertices: (h, k-a), (h, k+a) Foci: (h, k-c), (h, k+c) where c 2 = a 2 + b 2 Asymptotes: y = k ± (a/b)(x – h) Width of inside rectangle: 2b Height of inside rectangle: 2a Corners of rectangle: (h-b, k+a), (h+b, k+a), (h-b, k-a), (h+b, k-a)

Graphing Hyperbolas Identify Center Transverse axis Conjugate axis Vertices Foci Asymptotes Corners of rectangle

Graphing Hyperbolas Identify Center Transverse axis Conjugate axis Vertices Foci Asymptotes Corners of rectangle