Chapter 10. Exploring Conic Sections

Transcription

1 Chapter 10 Exploring Conic Sections

2 Conics A conic section is a curve formed by the intersection of a plane and a hollow cone. Each of these shapes are made by slicing the cone and observing the shape of the slice.

3 Lesson 10- Parabolas

4 Uses of Parabolas

5 Uses of Parabolas

6 Uses of Parabolas

7 Parabola A parabola is the set of all points in a plane that are the same distance from a fixed line (directrix) and a fixed point (focus) not on the line

8 Standard Form of a Parabola Standard form x 4py ( x h) 4 p( y k) Vertex: ( hk, ) Focus: Directrix: ( h, k p) y k p Axis of symmetry is vertical (y-axis) p p F V D p 0 opens upward p 0 opens downward

9 Standard Form of a Parabola Standard form y 4px ( y k) 4 p( x h) Vertex: ( hk, ) Focus: ( h p, k) Directrix: x h p Axis of symmetry is horizontal (x-axis) p 0 opens to the right D p V p F p 0 opens to the left

10 Example 5 Page 547, #4 Identify the vertex, the focus, and the directrix and sketch the graph. x 1 4 y Step 1 Locate which variable has the square. y -term; the parabola has horizontal symmetry (x-axis)

11 Example 5 Page 547, #4 Step Rewrite the equation in standard form. x 1 4 y y 4px x y 1 4 y 4x

12 Example 5 Page 547, #4 Step 3 Identify the vertex, focus and directrix. y 4x Vertex (0,0) Focus (0 6,0) (6,0) Directrix: x 06 x 6 Solve for p y y 4x 4p 4 p 4px 6 x (0,0) 6 (6,0)

13 Example 5 Page 547, #30 Identify the vertex, the focus, and the directrix and sketch the graph. ( x) 4y Step 1 Locate which variable has the square. x -term the parabola has vertical symmetry (y-axis)

14 Example 5 Page 547, #30 Step Rewrite the equation in standard form. ( x) 4y ( x h) 4 p( y k) Already in standard form.

15 Example 5 Page 547, #30 Step 3 Identify the vertex, focus and directrix. ( x) 4y Vertex (,0) Focus (,0 1) (,1) Directrix y 01 y 1 Solve for p ( x) 4y 4p 4 p 1 ( x h) 4p( y k) y 1

16 Example 5 Page 547, #34 Identify the vertex, the focus, and the directrix and sketch the graph. x 4y 8x 16 Step 1 Locate which variable has the square. x -term the parabola has vertical symmetry (y-axis)

17 Example 5 Page 547, #34 Step Rewrite the equation in standard form. x 4y 8x 16 ( x h) 4 p( y k) Complete the square on the x-terms x 8x 4y 16 x 8x 4y x 8x 16 4y ( x 4) 4y

18 Example 5 Page 547, #34 Step 3 Identify the vertex, focus and directrix. ( x 4) 4y ( x h) 4p( y k) Vertex Solve for p (4,0) ( x4) 4y y 6 Focus (4,0 ( 6)) (4, 6) 4p 4 p 6 Directrix y 0 ( 6) y 6

19 Lesson 10-3 Circles

20 Circle A circle is the set of all points in a plane that are a distance r (radius) from a given point (center) Center r

21 Standard Form of a Circle Standard form x y r ( x h) y k r Vertex: ( hk, ) r ( hk, )

22 Example 5, Page 553, #34 Use the center and the radius to graph the circle x ( y 4) 144 ( x h) y k r Center: ( hk, ) (0, 4) Radius: r 144 1

23 Example 5, Page 553, #30 Use the center and the radius to graph the circle x y 1 ( 3) 16 ( x h) y k r Center: ( hk, ) (1, 3) Radius: r 16 4

24 Example 5, Page 553, #6 Find the center and the radius of the circle. x x 1 y 4 Step 1 Rewrite the equation in standard form. x x y 3 ( x h) y k r x x y 3 x x 1 y

25 Example 5, Page 553, #6 x x 1 y 3 1 ( x h) y k r x y 1 4 Center: Radius: ( 1,0) 4

26 Lesson 10-4 Ellipse

27 Uses of Ellipses

28 Uses of Ellipses

29 Uses of Ellipses

30 Ellipses An ellipse can be defined as the set of all points (P) in plane the sum of whose distances from two fixed points (F 1 and F ) is constant. These two fixed points are called the foci. P F1 F

31 Standard Form of a Ellipse Standard form Center: ( hk, ) Vertices: Major Axis Endpoints: Minor Axis Focus: ( h a, k) ( h a, k) ( h, k b) ( h, k b) ( h c, k) ( h c, k) x a c a b y 1 b V F x h y k a 1 b E F V c c E

32 Standard Form of a Ellipse Standard form Center: ( hk, ) Vertices: Major Axis Endpoints: Minor Axis Focus: ( h, k a) ( h, k a) ( h b, k) ( h b, k) ( hk, c) ( h, k c) x b c a b y 1 a E x h y k b 1 V F c E c F V a

33 Example 3, Page 559, #18 Find the foci for each equation of an ellipse. Then graph the ellipse. x y x b y 1 a Step 1 Rewrite the equation in standard form in standard form

34 Example 3, Page 559, #18 x y x y b a Step Identify a² and b² to determine which axis is the major a b 9 4 major axis is the y

35 Example 3, Page 559, #18 x y a b 9 Step 3 Find the center, vertices, endpoints and focus. Center: ( hk, ) (0,0) Vertices: ( h, k a) (0,0 3) Endpoints: ( h b, k) (0,0) Focus: c a b c 94 c ( h, k c) (0,0 5) major axis is the y

36 Example 3, Page 559, #4 Find the foci and graph the ellipse x 4y 16 x a y 1 b Step 1 Rewrite the equation in standard form x 4y x y

37 Example 3, Page 559, #4 x y x a y 1 b Step Identify a² and b² to determine which axis is the major a b 16 4 major axis is the x

38 Example 3, Page 559, #4 x y a b 16 4 major axis is the x Step 3 Find the center, vertices, endpoints and focus. Center: ( hk, ) (0,0) Vertices: ( h a, k) (0 4,0) Endpoints: ( h, k b) (0,0 ) Focus: ( h c, k) (0 3,0) c a b c c

39 Example 3, Page 559, #40 Find the foci of the ellipse x 8x y 4 0 Step 1 Rewrite the equation in standard form (x 8 x) y 4 ( x 4 x) y 4 x 4x y x 4x 4 y 4 4

40 Example 3, Page 559, #40 x 4x 4 y 4 4 x y 4 8 x y 4 x y x y 4 1

41 Example 3, Page 559, #40 x y 4 1 Step Identify a² and b² to determine which axis is the major Focus: ( h, k c) (, ) a 4 b center: ( hk, ) (,0) c a b c c 4

42 Lesson 10-5 Hyperbolas

43 Uses of Hyperbolas

44 Definition A hyperbola is the set of points in a plane the difference of whose distances from two fixed point (foci) is constant. The point midway between the two foci is the center. F d C F d 1

45 Standard Form of a Hyperbola Standard Form Center: ( hk, ) x a y x h y k b a b 1 1 Vertices: h h Rectangle: h, k Focus: h a, k a, k b c, k c a b F Transverse Axis is the x-axis F

46 Standard Form of a Hyperbola Standard Form Center: ( hk, ) y a x y k x h b a b 1 1 Vertices: h, k a h, k a Rectangle: h b, k Focus: h, k c c a b Transverse Axis is the y-axis F F

47 Example 1, Page 566, # Graph the equation. y x y a x 1 b Step 1 - Rewrite the equation in standard form. Equation already in standard form

48 Example 1, Page 566, # Graph the equation. y x y a x 1 Step Identify the a² and b² to determine the transverse axis. b a b Transverse axis is the y-axis. The graph is opening up and down.

49 Example 1, Page 566, # x 1 a 169 b 16 y y x 1 a b a 13 b 4 Step 3 Find the center, vertices, rectangle, and focus. Center: hk, 0,0 Vertices: h, k a 0,0 13 Rectangle: h, k a 0,0 13 h b, k 0 4,0 Focus: c a b h, k c 0,0 185 c c

50 Example 1, Page 566, #8 Graph the equation. 5x 35y 875 x a y 1 b Step 1 - Rewrite the equation in standard form. 5x 35y 875 5x 35y x y

51 Example 1, Page 566, #8 x y x a y 1 b Step Identify the a² and b² to determine the transverse axis. a b 35 5 Transverse axis is the x-axis. The graph is opening left and right.

52 Example 1, Page 566, #8 a 35 x y b 5 x a b 5 a b Step 3 Find the center, vertices, rectangle, and focus. y 1 Center: hk, 0,0 Vertices: h a, k 0 35,0 Rectangle: h a, k 0 35,0 Focus: c a b h, k b 0,0 5 h c, k 0 60,0 c c

53 Lesson 10-6 Translating Conic Sections

54 Example 4, Page 574, #14 Identify the conic section by writing the equation in standard form and graph. 3x 6x y 6y 3 Circle or Ellipse Step 1 - Rewrite the equation in standard form. x x y y x x y y 3 6 3

55 Example 4, Page 574, #14 x x y y x x y y x y x y

56 Example 4, Page 574, #14 x y x1 y Ellipse 1

57 Example 4, Page 574, #14 x1 y 3 a 15 a b 5 b 5.4 Step Find the center, vertices, endpoints and focus. Center:( hk, ) ( 1,3) Vertices: ( h, k a) ( 1,3 3.87) Endpoints: ( h b, k) ( 1.4,3) Focus: ( h, k c) ( 1,3 3.16) c a b c c

58 Example 4, Page 574, #18 Identify the conic section by writing the equation in standard form and graph. x y 14y 13 Circle or Ellipse Step 1 - Rewrite the equation in standard form. x y y x y 14y 13

59 Example 4, Page 574, #18 x y 14y x y 14y x y 7 36

60 Example 4, Page 574, #18 x y 7 36 ( x h) y k r Center: ( hk, ) (0, 7) Radius: r r

61 Example 4, Page 574, #16 Identify the conic section by writing the equation in standard form and graph. y x 6x 4y 6 Hyperbola Step 1 - Rewrite the equation in standard form. y y x 6x 4 6 y y 1 x x 4 6 6

62 Example 4, Page 574, #16 y y 1 x x y y 1 x x y x 3 1

63 Example 4, Page 574, #16 y x 3 1 a 1 b 1 Step Find the center, vertices, rectangle, and focus. h, k a 3, 1 Center:, 3, hk Vertices: Rectangle: h, k a 3, 1 h b, k 3 1, Focus: c a b h, k c 3, 1.41 c c y k x h a 1 b

64 Example 4, Page 574, #1 Identify the conic section by writing the equation in standard form and graph. x 8x y 19 0 Parabola Step 1 - Rewrite the equation in standard form. x 8x y 19 x 8x y x 8x 16 y 19 16

65 Example 4 Page 574, #1 x 4 ( y 3) ( x h) 4 p( y k) Step Identify the vertex, focus and directrix. hk, 4,3 Solve for p Vertex: Focus: h, k p 4,3 0.5 (4,3.5) Directrix: y k p y y p 1 p 1 4 y.75