Homework Log Wed 4/27 Lesson Rev Learning Objective: To remember everything on conics Hw: STUDY! Stamp log due Thurs.

4/27/16 Chapter 10 Conics Algebra II

Learning Objective To graph conics To write equations of conics

Write an equation of a parabola with the given information 1. Vertex (1, 4), Focus (1, -2) V(h, k) V c = -6 𝑦= 1 4𝑐 (𝑥−ℎ) 2 +𝑘 F 𝑦= 1 4(−6) (𝑥−1) 2 +4 𝑦=− 1 24 (𝑥−1) 2 + 4

Write an equation of a parabola with the given information 2. Focus (2, 3), Directrix y = -1 F 𝑦= 1 4𝑐 (𝑥−ℎ) 2 +𝑘 c = 2 D 𝑦= 1 4(2) (𝑥−2) 2 +1 V(2, 1) 𝑦= 1 8 𝑥−2 2 +1 V(h, k)

Write an equation of a parabola with the given information 3. V(0, -3), Directrix x = 1.25 c = -1.25 V(h, k) 𝑥= 1 4𝑐 (𝑦−𝑘) 2 +ℎ V 𝑥= 1 4(−1.25) (𝑦+3) 2 +0 D 𝑥=− 1 5 (𝑦+3) 2

Get into vertex form. Find vertex, focus, and directrix. Sketch a graph. 4. y=2 𝑥 2 +8𝑥+11 V(-2, 3) 𝑦=2 𝑥 2 +4𝑥+_____ +11+____ 4 −8 D: y = 2 7 8 𝑏 2 2 = 4 2 2 = 2 2 =4 𝑦=2 (𝑥+2) 2 +3 F D V 𝑦= 1 4𝑐 (𝑥−ℎ) 2 +𝑘 2 1 = 1 4𝑐 8𝑐=1 𝑐= 1 8 Opens “y” up

D Get into vertex form if it isn’t already. Find vertex, focus, and directrix. Sketch a graph. 5. 𝑥=− 1 8 𝑦+2 2 +5 𝑥= 1 4𝑐 (𝑦−𝑘) 2 +ℎ V(5, -2) − 1 8 = 1 4𝑐 F(3, -2) V F 4𝑐=−8 𝑐=−2 D: x = 7 Opens “x” left

Write an equation of a circle with the given information 6. Center (-4, 0), radius 9 (𝑥−ℎ) 2 + (𝑦−𝑘) 2 = 𝑟 2 (𝑥−(−4)) 2 + (𝑦−0) 2 = 9 2 (𝑥+4) 2 + 𝑦 2 =81

Write an equation of a circle with the given information (h, k) 7. Center (-3, 2), point on the circle (1, 5) 𝑟= (1−(−3)) 2 + (5−2) 2 𝑟= (4) 2 + (3) 2 𝑟=5 (𝑥−ℎ) 2 + (𝑦−𝑘) 2 = 𝑟 2 (𝑥−(−3)) 2 + (𝑦−2) 2 = 5 2 (𝑥+3) 2 + (𝑦−2) 2 =25

Write an equation of a circle with the given information 8. Endpoints of a diameter (2, 1) & (-2, 3) 𝑟= 1 2 𝑑 𝑟= 1 2 (3−1) 2 + (−2−2) 2 𝑟= 1 2 (2) 2 + (−4) 2 = 1 2 20 = 1 2 2 5 = 5 𝐶= 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 = −2+2 2 , 3+1 2 center=(0, 2) (h, k) (𝑥−ℎ) 2 + (𝑦−𝑘) 2 = 𝑟 2 𝑥−0 2 + 𝑦−2 2 = 5 2 𝑥 2 +( 𝑦−2) 2 =5

Find the center & radius of the circle & graph 9. 𝑥 2 + 𝑦 2 −2𝑥+8𝑦=9 ( 𝑥 2 −2𝑥+____)+( 𝑦 2 +8𝑦+____)=9+____ +____ 1 16 1 16 𝑏 2 2 = −2 2 2 = −1 2 =1 𝑏 2 2 = 8 2 2 = 4 2 = 16 (𝑥−1) 2 +( 𝑦+4) 2 =26 Center: (1, -4) Radius: 26 ≈5.1

Find the center, vertices, co-vertices & foci Find parts Sketch Graph Look at Graph 10. 𝑥 2 4 + 𝑦 2 25 =1 Major Axis: y a = 5 b = 2 Center (0, 0) V(0,±5) CV(±2,0) F(0,± 21 ) 𝑐 2 = 𝑎 2 − 𝑏 2 𝑐 2 = 5 2 − 2 2 𝑐 2 =21 c= 21

Write an equation of an ellipse with the given information 11. Co-vertices (0, ±3), major axis length 8 Major Axis: x Sketch Graph Find parts (look at graph) a = 4 b = 3 b = 3 (𝑥−ℎ) 2 𝑎 2 + (𝑦−𝑘) 2 𝑏 2 =1 Plug in (𝑥−0) 2 4 2 + (𝑦−0) 2 3 2 =1 a = 4 𝑥 2 16 + 𝑦 2 9 =1

Find the center, vertices, co-vertices & foci 12. 𝑥 2 9 − 𝑦 2 25 =1 Opens: x a = 3 b = 5 Center (0, 0) V(±3, 0) F(± 34 ,0) Asym: y=± 5 3 𝑥 𝑐 2 = 𝑎 2 + 𝑏 2 𝑐 2 = 3 2 + 5 2 𝑐 2 =34 c= 34

Write an equation of a hyperbola with the given information 13. Foci (0, ±4), Vertices (0, ±3) Center is between vertices Center (0, 0) Opens: y c = 4 a = 3 c = 4 𝑐 2 = 𝑎 2 + 𝑏 2 a = 3 (𝑦−𝑘) 2 𝑎 2 − (𝑥−ℎ) 2 𝑏 2 =1 4 2 = 3 2 + 𝑏 2 (𝑦−0) 2 (3) 2 − (𝑥−0) 2 7 2 =1 b= 7 𝑦 2 9 − 𝑥 2 7 =1

Identify the conic section, get into standard form, & graph 14. 4 𝑥 2 − 𝑦 2 −16𝑥−2𝑦−1=0 Hyperbola (4 𝑥 2 −16𝑥+____)+ − 𝑦 2 −2𝑦+____ =1+____+____ 4 𝑥 2 −4𝑥+____ − 𝑦 2 +2𝑦+____ =1+____+____ 4 1 16 -1 𝑏 2 2 = −4 2 2 𝑏 2 2 = 2 2 2 = −2 2 =4 = 1 2 =1 4 (𝑥−2) 2 − (𝑦+1) 2 =16 (𝑥−2) 2 4 − (𝑦+1) 2 16 =1 Center (2, -1) Opens: x

#14 Cont’d (𝑥−2) 2 4 − (𝑦+1) 2 16 =1 𝑐 2 = 𝑎 2 + 𝑏 2 𝑐 2 = 2 2 + 4 2 Opens: x a = 2 b =4 (look at graph) Center (2, -1) V(0, -1) (4, -1) F(2±2 5 ,−1) 𝑐 2 = 𝑎 2 + 𝑏 2 𝑐 2 = 2 2 + 4 2 c=2 5 𝑐 2 =20 Add along x

Identify the conic section, get into standard form, & graph 15. 𝑥 2 +4 𝑦 2 +6𝑥−32𝑦+57=0 Ellipse 𝑥 2 +6𝑥+____ + 4𝑦 2 −32𝑦+____ =−57+____+____ 𝑥 2 +6𝑥+____ +4 𝑦 2 −8𝑦+____ =−57+____+____ 9 16 9 64 𝑏 2 2 = 6 2 2 𝑏 2 2 = −8 2 2 = 3 2 =9 = −4 2 =16 (𝑥+3) 2 +4( 𝑦−4) 2 =16 (𝑥+3) 2 16 + ( 𝑦−4) 2 4 =1 Center (-3, 4) Major Axis: x

#15 Cont’d (𝑥+3) 2 16 + ( 𝑦−4) 2 4 =1 Major Axis: x a = 4 b = 2 (look at graph) Center (-3, 4) V(1, 4) (-7, 4) CV (-3, 2) (-3, 6) F(-3±2 3 , 4) 𝑐 2 = 𝑎 2 − 𝑏 2 𝑐 2 = 4 2 − 2 2 c=2 3 𝑐 2 =12 Add along x