Solutions for The Parabola

Solutions to Try Its

1. Focus: [latex]\left(-4,0\right)[/latex]; Directrix: [latex]x=4[/latex]; Endpoints of the latus rectum: [latex]\left(-4,\pm 8\right)[/latex]

2. Focus: [latex]\left(0,2\right)[/latex]; Directrix: [latex]y=-2[/latex]; Endpoints of the latus rectum: [latex]\left(\pm 4,2\right)[/latex].

3. [latex]{x}^{2}=14y[/latex]

4. [latex]{x}^{2}=14y[/latex]

5. Vertex: [latex]\left(8,-1\right)[/latex]; Axis of symmetry: [latex]y=-1[/latex]; Focus: [latex]\left(9,-1\right)[/latex]; Directrix: [latex]x=7[/latex]; Endpoints of the latus rectum: [latex]\left(9,-3\right)[/latex] and [latex]\left(9,1\right)[/latex].

6. Vertex: [latex]\left(-2,3\right)[/latex]; Axis of symmetry: [latex]x=-2[/latex]; Focus: [latex]\left(-2,-2\right)[/latex]; Directrix: [latex]y=8[/latex]; Endpoints of the latus rectum: [latex]\left(-12,-2\right)[/latex] and [latex]\left(8,-2\right)[/latex].

7.  a. [latex]{y}^{2}=1280x[/latex]
b. The depth of the cooker is 500 mm

Solutions to Odd-Numbered Exercises

1. A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

3. The graph will open down.

5. The distance between the focus and directrix will increase.

7. yes [latex]y=4\left(1\right){x}^{2}[/latex]

9. yes [latex]{\left(y - 3\right)}^{2}=4\left(2\right)\left(x - 2\right)[/latex]

11. [latex]{y}^{2}=\frac{1}{8}x,V:\left(0,0\right);F:\left(\frac{1}{32},0\right);d:x=-\frac{1}{32}[/latex]

13. [latex]{x}^{2}=-\frac{1}{4}y,V:\left(0,0\right);F:\left(0,-\frac{1}{16}\right);d:y=\frac{1}{16}[/latex]

15. [latex]{y}^{2}=\frac{1}{36}x,V:\left(0,0\right);F:\left(\frac{1}{144},0\right);d:x=-\frac{1}{144}[/latex]

17. [latex]{\left(x - 1\right)}^{2}=4\left(y - 1\right),V:\left(1,1\right);F:\left(1,2\right);d:y=0[/latex]

19. [latex]{\left(y - 4\right)}^{2}=2\left(x+3\right),V:\left(-3,4\right);F:\left(-\frac{5}{2},4\right);d:x=-\frac{7}{2}[/latex]

21. [latex]{\left(x+4\right)}^{2}=24\left(y+1\right),V:\left(-4,-1\right);F:\left(-4,5\right);d:y=-7[/latex]

23. [latex]{\left(y - 3\right)}^{2}=-12\left(x+1\right),V:\left(-1,3\right);F:\left(-4,3\right);d:x=2[/latex]

25. [latex]{\left(x - 5\right)}^{2}=\frac{4}{5}\left(y+3\right),V:\left(5,-3\right);F:\left(5,-\frac{14}{5}\right);d:y=-\frac{16}{5}[/latex]

27. [latex]{\left(x - 2\right)}^{2}=-2\left(y - 5\right),V:\left(2,5\right);F:\left(2,\frac{9}{2}\right);d:y=\frac{11}{2}[/latex]

29. [latex]{\left(y - 1\right)}^{2}=\frac{4}{3}\left(x - 5\right),V:\left(5,1\right);F:\left(\frac{16}{3},1\right);d:x=\frac{14}{3}[/latex]

31.

33.

35.

37.

39.

41.

43.

45. [latex]{x}^{2}=-16y[/latex]

47. [latex]{\left(y - 2\right)}^{2}=4\sqrt{2}\left(x - 2\right)[/latex]

49. [latex]{\left(y+\sqrt{3}\right)}^{2}=-4\sqrt{2}\left(x-\sqrt{2}\right)[/latex]

51. [latex]{x}^{2}=y[/latex]

53. [latex]{\left(y - 2\right)}^{2}=\frac{1}{4}\left(x+2\right)[/latex]

55. [latex]{\left(y-\sqrt{3}\right)}^{2}=4\sqrt{5}\left(x+\sqrt{2}\right)[/latex]

57. [latex]{y}^{2}=-8x[/latex]

59. [latex]{\left(y+1\right)}^{2}=12\left(x+3\right)[/latex]

61. [latex]\left(0,1\right)[/latex]

63. At the point 2.25 feet above the vertex.

65. 0.5625 feet

67. [latex]{x}^{2}=-125\left(y - 20\right)[/latex], height is 7.2 feet

69. 2304 feet