Assignment 3/17/15. Section 10.2(p 568) 2 12 (E) (E)

Transcription

1 Section 10.2

2 Warm Up

3 Assignment 3/17/15 Section 10.2(p 568) 2 12 (E) (E)

4 Objective We are going to find equations for parabolas identify the vertex, focus, and directrix of a parabola

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6 The parabola has many applications in situations where: Radiation often needs to be concentrated at one point (e.g. radio telescopes, pay TV dishes, solar radiation collectors) or Radiation needs to be transmitted from a single point into a wide parallel beam (e.g. headlight reflectors).

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8 Conics Circle Parabola Ellipse Hyperbola

9 Review distance

10 B(x, y ) A(0, 4) y = - 2 C(?,? )

11 B(x, y ) A(0, 2) y = - 1 C(?,? )

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14 Write an equation for a graph that is the set of all points in the plane that are equidistant from the point F(0,3) and the line y=-3

15 Write an equation for a graph that is the set of all points in the plane that are equidistant from the point F(2,0) and the line x=-2.

16 Write an equation for a graph that is the set of all points in the plane that are equidistant from the point F(0,1) and the line y= 1.

17 Write an equation for a graph that is the set of all points in the plane that are equidistant from the given point F(0,-1) and the line y=1

18 Review y=ax 2

19 Parabola Equations F (0,c) F (x,-c)

20 Consider any parabola with equation y = ax 2 and vertex at the origin F(0, c) y = c If a>0, then The parabola opens upward The focus is at (0, c) The directrix is y = c

21 Consider any parabola with equation y = ax 2 and vertex at the origin y = c If a<0, then The parabola opens downward The focus is at (0, c ) The directrix is y = c F(0, c )

22 Consider any parabola with equation x = ay 2 and vertex at the origin x = c F(c, 0 ) If a>0, then The parabola opens to the right The focus is at (c, 0 ) The directrix is x = c

23 Consider any parabola with equation x = ay 2 and vertex at the origin F( c, 0 ) x = c If a<0, then The parabola opens to the left The focus is at ( c, 0 ) The directrix is x = c

24 Write an equation of a parabola with a vertex at the origin and a focus at (-5,0)

25 Write an equation of a parabola with a vertex at the origin and a focus at 1,0 2

26 Write an equation of a parabola with a vertex at the origin and a focus at (0,-7)

27 Identify the focus and directrix of the graph of the equation x = graph. 1 8 y 2. Then The parabola is of the form x = ay 2, so the vertex is at the origin and the parabola has a horizontal axis of symmetry. Since a < 0, the parabola opens to the left.

28 Identify the focus and directrix of the graph of the equation y = graph x 2. then a = 1 4c 1 8 = 1 4c 4c = 8 c = 2 The focus is at ( 2, 0). The equation of the directrix is x = 2.

29 Then graph

30 Then graph

31 1 y x 12 Then graph 2

32 Then graph

33 F(h, k+c ) V (h, k ) y = k c

34 Vertex is not at the origin F(h, k+c ) V (h, k ) y = k c

35 Vertex is not at the origin y = k + c V (h, k ) F(h, k c )

36 Vertex is not at the origin F(h + c,k ) V (h, k ) x = h c

37 Vertex is not at the origin V (h, k ) F(h c,k ) x = h + c

38 Review completing the square x 2 6x 2 0

39 Review completing the square x 2 10x 1 0

40 Identify the vertex, the focus, and the directrix of the graph of the equation y 2 4x 4y + 16 = 0. Then graph the parabola.

41 Identify the vertex, the focus, and the directrix of the graph of the equation x 2 + 4x + 8y 4 = 0. Then graph the parabola. x 2 + 4x + 8y 4 = 0 8y = x 2 4x + 4 Solve for y, since y is the only term. 8y = (x 2 + 4x + 4) Complete the square in x. y = 1 (x + 2) vertex form The parabola is of the form y = a(x h) 2 + k, so the vertex is at ( 2, 1) and the parabola has a vertical axis of symmetry. Since a < 0, the parabola opens downward.

42 Parabolas LESSON 10-2 (continued) a = 1 8 4c = 8 Solve for c. c = 2 1 4c 1 4c = Substitute for a. The vertex is at ( 2, 1) and the focus is at ( 2, 1). The equation of the directrix is y = 3. Locate one or more points on the parabola. Select a value for x such as 6. The point on the parabola with an x-value of 6 is ( 6, 1). Use the symmetric nature of a parabola to find the corresponding point (2, 1). 1 8

43 Identify the vertex, the focus, and the directrix of the graph of the equation x 2 + 6x + 3y + 12 = 0. Then graph the parabola.

44 Identify the vertex, the focus, and the directrix of the graph of the equation y 2 4y 16x + 36 = 0. Then graph the parabola.

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