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Conics

Parabolas. Parabolas. Circles. Conics . Ellipses. Hyperbolas. c = b. V( ) c. F( ) d. x 2 or y 2 e. directrix _________ f. axis _____________ . focus. vertex. directrix. directrix. General form for x 2 parabola: y = (x – h) 2 + k. 1 4c. axis.

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Conics

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  1. Parabolas Parabolas Circles Conics Ellipses Hyperbolas

  2. c = b. V( ) • c. F( ) d. x2 or y2 • e. directrix _________ • f. axis _____________ focus vertex directrix directrix General form for x2 parabola: y = (x – h)2 + k 1 4c axis axis g. equation:___________________ Parabolas (h, k) 2, 3 b. V( ) 1 “c” is the distance from the vertexto the focus. 2, 4 F( ) x2 opens up or down y2 opens right or left directrix y = 2 There are “c” units from the directrixto the vertex. axis x = 2 The axis is the line that goes through the vertex and focus. y = ¼(x – 2)2 + 3

  3. V( ) • c = b. V( ) • c. F( ) d. x2 or y2 • e. directrix _________ • f. axis _____________ F( ) directrix axis g. equation:___________________ Parabolas -3 -1, 2 directrix -1, -1 y = 5 x = -1 axis y = -1/12(x + 1)2 + 2

  4. V( ) • c = b. V( ) • c. F( ) d. x2 or y2 • e. directrix _________ • f. axis _____________ F( ) General form for y2 parabola: x = (y – k)2 + h directrix 1 4c axis g. equation:___________________ Parabolas (h, k) 2 -1, 4 axis 1, 4 x = -3 directrix y = 4 x = 1/8(y - 4)2 - 1

  5. That's it for Parabolas Day 1

  6. Parabolas Day #2

  7. c =___ b. V( ) • c. F(-2, 0) d. x2 or y2 • e. directrix x = -6 • f. axis _____________ directrix F( ) axis directrix axis g. equation:___________________ Parabolas Day #2 2 V( ) -4, 0 y = 0 x = 1/8(y - 0)2 - 4

  8. V( ) • c =___ b. V(1, 4) • c. F(1, 7) d. x2 or y2 • e. directrix _________ • f. axis _____________ axis F( ) directrix axis directrix g. equation:___________________ Parabolas Day #2 3 y = 1 x = 1 y = 1/12(x - 1)2 + 4

  9. c =___ b. V( ) • c. F(-3, -2) d. x2 or y2 • e. directrix x = 3 • f. axis _____________ directrix F( ) directrix axis axis g. equation:___________________ Parabolas Day #2 -3 0, -2 V( ) y = -2 x = -1/12(y + 2)2 + 0

  10. That's it for Parabolas Day #2

  11. Parabolas Circles Conics Circles Ellipses Hyperbolas

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

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

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

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

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

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

  18. k h C =(-1, 7) and (2, 8) -1 7 2 8 (x - )² + (y - )² = ² -1 7 ( + 1)² + ( - 7)² = r² 2 8 (3)² + (1)² = r² 10 = r² (x + 1)² + (y - 7)² = 10

  19. More Circle Fun Next Time!!

  20. Circles Day #2 Putting into standard form and graphing.

  21. Remember how to Complete the Square ?!? If a quadratic equation isn’t in Standard form for a Circle (x - h)² + (y - k)² = r² you will need to Complete the Square to get it in the correct form.

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

  23. 112 121 Center (-8, 11) radius = x2 + 16x +() + y2 – 22y +() = 20 +() +() 82 64 (x + 8)2 + (y – 11)2 = 205 Completethe square • Half the linear term and square it. • Add to both sides. • Do this for both x and y. • Factor and simplify.

  24. 62 36 42 16 Center (6, -4) radius = Now you try it: x2 + y2 - 12x + 8y + 32 = 0 x2 - 12x +() + y2 + 8y +() = -32 +() +() (x - 6)2 + (y + 4)2 = 20

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

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

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

  28. That's all for circles!!

  29. Parabolas Circles Conics Ellipses Ellipses Hyperbolas

  30. x2 + y2 = 1 16 25 x2 + y2 = 1 16 25 name of ellipse: center: a: b: major axis: minor axis: vertices: foci: Square root of the smaller denominator. Square root of the larger denominator. 2a 2b c2 = a2 – b2 Ellipses (0, 0) name of ellipse: center: vertical b: 4 a: 5 b was under the x2, so you move b units from the center in a x direction. major axis: 10 focus (0, 3) minor axis: 8 a was under the y2, so you move a units from the center in a y direction. (0, 5), (0, -5), (4, 0), (-4, 0) center (0, 0) vertices: focus (0, -3) (0, 3), (0, -3) foci:

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

  32. x2 x2 y2 y2 __ + __ = 1 25 9 __ + __ = 1 25 9 5 3 How many units from the center to the curve in an “x” direction? How many units from the center to the curve in an “y” direction? Where is the center of this ellipse?

  33. x2 x2 y2 y2 __ + __ = 1 16 36 __ + __ = 1 36 16 4 6 How many units from the center to the curve in an “x” direction? How many units from the center to the curve in an “y” direction? Where is the center of this ellipse?

  34. 10 10 10 Divide to make the constant 1. x2 + y2 = 1 10 1 x2 + 10y2 = 10 SF: center: vertices: foci: SF: (0, 0) center: (±√10, 0) (0, ±1) vertices: foci: (±3, 0)

  35. 72 72 72 Divide to make the constant 1. x2 + y2 = 1 3 24 24x2 + 3y2 = 72 SF: center: vertices: foci: SF: (0, 0) center: (0, ±2√6) (±√3, 0) vertices: foci: (0, ±√21)

  36. bye-bye ellipses ellipses ellipses ellipses ellipses ellipses ellipses

  37. Parabolas Circles Conics Hyperbolas Ellipses Hyperbolas

  38. “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. c2 = a2 + b2 x2 - y2 = 1 9 16 x2 - y2 = 1 9 16 Hyperbolas (0, 0) center: a: b: vertices: foci: center: 4 a: 3 b: vertices: (3, 0) (-3, 0) foci: (5, 0) (-5, 0)

  39. 4(y + 1)2 – 25(x – 3)2 = 100 4(y + 1)2 – 25(x – 3)2 = 100 100 100 100 (y + 1)2 – (x – 3)2 = 1 25 4 (y + 1)2 – (x – 3)2 = 1 25 4 Divide each term by 100 to get into form. (3, -1) center: a: b: vertices: foci: center: 2 a: 5 b: vertices: (3, -6) (3, 4) (3, -1±√29) foci:

  40. Note: The+54ybecomes-6y 9 32 -144 –144 -144 Getting it into Standard Form 16x2 - 9y2 + 54y + 63 = 0 16x2 + (-9y2 + 54y + ( ))= -63 + ( ) Factor the –9 out of the “y” terms. 16x2 + -9(y2 - 6y + ( ))= -63 + -9( ) Remember: Put the –9 on the right too. 16x2 + -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?

  41. (y – 3)2 - x2 = 1 16 9 (y – 3)2 - x2 = 1 16 9 (0, 3) center: a: b: vertices: foci: center: 3 a: 4 b: vertices: (0, 7) (0, -1) foci: (0, 8) (0, -2)

  42. 1 12 9 32 36 36 36 9x2 - 4y2 + 54x + 8y + 41 = 0 (9x2+54x+( ))+(-4y2+8y+( ))= -41+ ( ) + ( ) 9(x2+6x+( )) + -4(y2-2y+( )) = -41+ 9( ) + -4( ) 9(x + 3)2– 4(y - 1)2 = 36 (x + 3)2 – (y – 1)2 = 1 4 9

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

  44. Hit the road!! Hyperbolas

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