3. directrix axis a.c = b. V( ) c. F( ) d. x 2 or y 2 e. directrix _________ f. axis _____________ g. equation:___________________ 2, 3 2, 4 b. V( ) focus vertex F( ) “c” is the distance from the vertex to the focus. x 2 opens up or down y 2 opens right or left y = 2 x = 2 There are “c” units from the directrix to the vertex. directrix 1 The axis is the line that goes through the vertex and focus. axis General form for x 2 parabola: y = (x – h) 2 + k 1 4c y = ¼(x – 2) 2 + 3 (h, k)

4. directrix axis a.c = b. V( ) c. F( ) d. x 2 or y 2 e. directrix _________ f. axis _____________ g. equation:___________________ -3-1, 2 -1, -1 y = 5 x = -1 y = -1/12(x + 1) 2 + 2 directrix axis F( ) V( )

5. directrix axis a.c = b. V( ) c. F( ) d. x 2 or y 2 e. directrix _________ f. axis _____________ g. equation:___________________ 2-1, 4 1, 4 x = -3 y = 4 x = 1/8(y - 4) 2 - 1 directrixaxis F( ) V( ) General form for y 2 parabola: x = (y – k) 2 + h 1 4c (h, k)

8. a.c =___ b. V( ) c. F(-2, 0) d. x 2 or y 2 e. directrix x = -6 f. axis _____________ g. equation:___________________ 2-4, 0 y = 0 x = 1/8(y - 0) 2 - 4 directrix axis F( ) V( )

9. a.c =___ b. V(1, 4) c. F(1, 7) d. x 2 or y 2 e. directrix _________ f. axis _____________ g. equation:___________________ 3 y = 1 y = 1/12(x - 1) 2 + 4 directrix axis F( ) V( ) x = 1

10. a.c =___ b. V( ) c. F(-3, -2) d. x 2 or y 2 e. directrix x = 3 f. axis _____________ g. equation:___________________ -30, -2 y = -2 x = -1/12(y + 2) 2 + 0 directrix axis F( ) V( )

13. Circles General form: (x - h)² + (y - k)² = r² hkr Center (h, k) radius = r

14. (x - h)² + (y - k)² = r² Using the form: Given: Center and radius (x - )² + (y - )² = ² Ex. 1:C(5, 2) r = 7 5 27 5 27 (x - 5)² + (y - 2)² = 49 h k

15. Ex. 2:C(-3, 4) r = (x - h)² + (y - k)² = r² (x - )² + (y - )² = ²-3 4 4 (x + 3)² + (y - 4)² = 20 h k

16. (x - h)² + (y - k)² = r² Given: Center & Another Point Ex. 3:C(4, -7) & (5, 3) (x - )² + (y - )² = ² 4 -7 4 h k To find r 2, you can plug in the point or use the distance formula ( - 4)² + ( + 7)² = r² (1)² + (10)² = r² 101 = r² (x - 4)² + (y + 7)² = 101 53 53

17. To find r 2, you can plug in the point or use the distance formula (x - h)² + (y - k)² = r² Ex. 4:C origin & (-5, 2) (x - )² + (y - )² = ² 0 0 x² + y² = 29 (x - 0)² + (y - 0)² = ²

18. 1 st Find the center using the midpoint formula: Given: Endpoints of diameter Ex. 5:(2, 8) & (-4, 6) are endpoints of the diameter. = (-1, 7) C = Then…choose either endpoint and finish like before. Let’s use C =(-1, 7) and (2, 8)

19. (x - )² + (y - )² = ²7 7 h k ( + 1)² + ( - 7)² = r² 28 28 (3)² + (1)² = r² 10 = r² (x + 1)² + (y - 7)² = 10

22. Remember how to If a quadratic equation isn’t in ?!? you will need to get it in the correct form.

23. x2 + y2 + 16x – 22y – 20 = 0 Rewrite the problem: Here’s how to do it: x2 x2 + 16x + ( ) +y2 y2 – 22y + ( ) = 20 +( ) + ( ) Group your x’s and leave a space. Group your y’s and leave a space. Move the constant and leave 2 spaces.

24. (x + 8) 2 + (y – 11) 2 = 205 x 2 + 16x +( ) + y 2 – 22y +( ) = 20 +( ) +( ) Complete the square Half the linear term and square it. Add to both sides. Do this for both x and y. Factor and simplify. 8 28 2 6411 2 121 Center (-8, 11) radius =

25. x 2 - 12x +( ) + y 2 + 8y +( ) = -32 +( ) +( ) x2 + y2 - 12x + 8y + 32 = 0 (x - 6) 2 + (y + 4) 2 = 20 6 26 2 364 24 2 16 Center (6, -4) radius = Now you try it:

26. Ex. 1: (x)² + (y)² = 36 Center (0, 0) radius = 6 Center (0, 0) left 6 down 6 up 6 right 6

27. Ex. 2: (x - 3)² + (y - 4)² = 25 Center (3, 4) radius = 5 Center (3, 4) right 5left 5 up 5 down 5

28. Ex. 3:(x - 5)² + (y +4)² = 41 Center (5, -4) radius = = 6.4 Center (5, -4) right 6.4left 6.4 down 6.4 up 6.4

31. name of ellipse: center: a: b: major axis: minor axis: vertices: foci: name of ellipse: (0, 0) 5 4 10 8 (0, 5), (0, -5), (4, 0), (-4, 0) (0, 3), (0, -3) vertical center: foci: vertices: a: center (0, 0) focus (0, 3) focus (0, -3) x 2 + y 2 = 1 16 25 x 2 + y 2 = 1 16 25 2b c 2 = a 2 – b 2 2aSquare root of the larger denominator. a was under the y 2, so you move a units from the center in a y direction. b: major axis: minor axis: b was under the x 2, so you move b units from the center in a x direction. Square root of the smaller denominator.

32. name of ellipse: center: a: b: major axis: minor axis: vertices: foci: x2 + y2 = 1 9 20 center: (0, 0) center (0, 0) 2√5 a: 3 4√5 6 foci: vertices: b: minor axis: (0, ±2√5) (±3, 0) (0, ±√11) name of ellipse: vertical major axis:

33. Where is the center of this ellipse? __ + __ = 1 x2x2x2x2 y2y2y2y2 How many units from the center to the curve in an “x” direction? 3 9 How many units from the center to the curve in an “y” direction? 25 5 __ + __ = 1 x2x2x2x2 y2y2y2y29 25

34. Where is the center of this ellipse? __ + __ = 1 x2x2x2x2 y2y2y2y2 How many units from the center to the curve in an “x” direction? 4 36 How many units from the center to the curve in an “y” direction? 16 6 __ + __ = 1 x2x2x2x2 y2y2y2y236 16

35. x2 + 10y2 = 10101010 Divide to make the constant 1. x2 + + + + y2 = 1 10 1 SF: center: vertices: foci: SF: center: (0, 0) foci: vertices: (±√10, 0) (0, ±1) (±3, 0)

36. 24x2 + 3y2 = 72727272 Divide to make the constant 1. x2 + y2 = = = = 1 3 24 SF: center: vertices: foci: SF: center: (0, 0) foci: vertices: (0, ±2√6) (±√3, 0) (0, ±√21)

39. x 2 - y 2 = 1 x 2 - y 2 = 1 9 16 9 16 x2 - y2 = 1 9 16 center: a: b: vertices: foci: center: (0, 0) foci: vertices: (3, 0) (-3, 0) (5, 0) (-5, 0) a:b:3 4 “a” is the square root of the positive variable. “b” is the square root of the negative variable. Will go in the direction of the positive variable. c 2 = a 2 + b 2

40. 4(y + 1) 2 – 25(x – 3) 2 = 100 center: a: b: vertices: foci: center: (3, -1) foci: vertices: (3, -6) (3, 4) (3, -1±√29) a:b:5 2 Divide each term by 100 to get into form. 100100100 (y + 1) 2 – (x – 3) 2 = 1 (y + 1) 2 – (x – 3) 2 = 1 25 4 25 4 (y + 1) 2 – (x – 3) 2 = 1 (y + 1) 2 – (x – 3) 2 = 1 25 4 25 4

41. 16x2 - 9y2 + 54y + 63 = 0 Getting it into Standard Form Factor the –9 out of the “y” terms. Remember: Put the –9 on the right too. 16x 2 + (-9y 2 + 54y + ( )) = == = -63 + ( ) 16x 2 + -9(y 2 - 6y + ( )) = == = -63 + -9( ) 16x 2 + -9(y - 3) 2 = -144 Divide each term by -144. (y -3)2 – x2 = 1 16 9 Why did the x and y terms trade places? -144 –144 -144 3232 9 Note: The +54y becomes -6y

42. (y – 3) 2 - x 2 = 1 (y – 3) 2 - x 2 = 1 16 9 16 9 (y – 3) 2 - x 2 = 1 (y – 3) 2 - x 2 = 1 16 9 16 9 center: a: b: vertices: foci: center: (0, 3) foci: vertices: (0, 7) (0, -1) (0, 8) (0, -2) a:b:4 3

43. 9x2 - 4y2 + 54x + 8y + 41 = 0 (9x 2 +54x+( ))+(-4y 2 +8y+( )) = == = -41+ ( ) + ++ + ( ) 9(x 2 +6x+( )) + ++ + -4(y 2 -2y+( )) = == = -41+ 9( ) + ++ + -4( ) 9(x + 3) 2 – 4(y - 1) 2 = 36 (x + 3)2 – (y – 1)2 = 1 4 9 36 36 36 3232 9 1212 1

44. (x + 3) 2 – (y – 1) 2 = 1 4 9 4 9 (x + 3) 2 – (y – 1) 2 = 1 4 9 4 9 center: a: b: vertices: foci: center: (-3, 1) foci: vertices: (-5, 1) (-1, 1) (-3±√13, 1) a:b:2 3