1. 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:

2. 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 (𝑦−𝑘) 2 𝑏 2 =1
(𝑥−ℎ) 2 𝑏 (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

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

4. 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

5. 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.

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

7. 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).

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

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