PC 11.4 Translations & Rotations of Conics Let h & k be constant. Replacing x with (x – h) and y with (y – k) in an equation shifts the graph of the equation:   +h shifts the graph right h units -h shifts the graph left h units +k shifts the graph up k units -k shifts the graph down k units Center = (h, k) The following are characteristics of Standard Equations of Conics:

ELLIPSES: (𝑥−ℎ) 2 𝑎 2 + (𝑦−𝑘) 2 𝑏 2 =1 (𝑥−ℎ) 2 𝑏 2 + (𝑦−𝑘) 2 𝑎 2 =1 (𝑥−ℎ) 2 𝑎 2 + (𝑦−𝑘) 2 𝑏 2 =1 (𝑥−ℎ) 2 𝑏 2 + (𝑦−𝑘) 2 𝑎 2 =1 Major axis: y = k Major axis: x = h Minor axis: x= h Minor axis: y = k Vertices: (h ± a, k) Vertices: (h, k ± a) Foci: (h ± c, k) given that c = 𝑎 2 − 𝑏 2 Foci: (h, k ± c) given that c = 𝑎 2 − 𝑏 2

Example #1: Given: (𝑥−5) 2 9 + (𝑦+4) 2 36 =1 Find: Center - __________ Major Axis - ________ Minor Axis - ________ Vertices - _______________ Foci - _____________  

HYBERBOLA: (𝑥−ℎ) 2 𝑎 2 − (𝑦−𝑘) 2 𝑏 2 =1 (𝑦−𝑘) 2 𝑎 2 − (𝑥−ℎ) 2 𝑏 2 =1 (𝑥−ℎ) 2 𝑎 2 − (𝑦−𝑘) 2 𝑏 2 =1 (𝑦−𝑘) 2 𝑎 2 − (𝑥−ℎ) 2 𝑏 2 =1 Focal Axis: y = k Focal Axis: x = h Vertices: (h ± a, k) Vertices: (h, k ± a) Foci: (h ± c, k) given that c = 𝑎 2 + 𝑏 2 Foci: (h, k ± c) given that c = 𝑎 2 + 𝑏 2 Asymptotes: y = ± 𝑏 𝑎 (x – h) + k Asymptotes: y = ± 𝑎 𝑏 (x – h) + k

PARABOLA: (x – h)2 = 4p(y – k) (y – k)2 = 4p(x – h) Focus: (h, k + p) Focus: (h + p, k) Directrix: y = k – p Directrix: x = h – p Axis: x = h Axis: y = k Opens upward if p > 0. Opens right if p > 0. Opens downward if p < 0. Opens left if p < 0.

Example #2: Given (y – 7)2 = 12(x – 8) Find: Center - ____________ Focus - _____________ Directrix - ___________ Axis - _______________ Opens - _____________

Example #3: Write the equation given the following information: a) Ellipse with center (-5, 2) and endpoints of major & minor axis at: (0, 2), (-5, 17), (-10, 2), (-5, -13).  

b) Hyperbola with center (-5, 1), vertex (-3, 1), & passing through (-1, 1 - 4 3 ).

c) Parabola with vertex (-3, 0), axis y = 0, & passing through (-1, 1).