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Section 10.1 Parabolas

Section 10.1 Parabolas. Objectives: To define parabolas geometrically. Use the equation of parabolas to find relevant information. To find the equation of parabolas given certain information. Parabola—Geometric Definition.

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Section 10.1 Parabolas

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  1. Section 10.1 Parabolas Objectives: To define parabolas geometrically. Use the equation of parabolas to find relevant information. To find the equation of parabolas given certain information

  2. Parabola—Geometric Definition • A parabola is the set of points in the plane equidistant from a fixed point F (focus) and a fixed line l(directrix). • The vertex V lies halfway between the focus and the directrix. • The axis of symmetry is the line that runs through the focus perpendicular to the directrix.

  3. Parabola with Vertical Axis of Symmetry The graph of the equation y = ax2 is a parabola with these properties. • vertex: V(0,0) • focus: F(0, p) where p is the distance between the focus and vertex • directrix: y = -p • a = (recall: a is the number that determines how wide or narrow the parabola is)

  4. Parabola with Vertical Axis • The parabola opens: • Upward if p > 0. • Downward if p < 0.

  5. Ex 1. Find the equation of the parabola with vertex V(0,0) and focus F(0,2).

  6. Ex 2. Find the equation of the parabola with vertex (0,0) and directrix y = 4.

  7. Class Work 1. Find the equation of the parabola with focus (0,-5) and vertex (0,0). 2. Find the equation of the parabola with focus (0,3) and directrix y = -3

  8. Parabolas whose vertex is not at the origin: The equation of the parabola whose vertex is (h,k) is

  9. Ex 3. Find the equation of the parabola with focus (3, -1) and vertex (3, -4).

  10. Ex. 4 Find the equation of the parabola with focus (4, -1) and vertex (4, 1).

  11. Class Work 3. Find the equation of the parabola with vertex (2, 8) and focus (2, 3). 4. Find the equation of the parabola with focus (-1, -3) and vertex (-1, 1)

  12. Parabola with Horizontal Axis The graph of the equation x=ay2is a parabola with these properties: • Vertex: V(0,0) • Focus: F(p, 0) • directrix: x = -p

  13. Parabola with Horizontal Axis • The parabola opens: • To the right if p > 0. • To the left if p < 0.

  14. Ex 5. Find the equation of the parabola with focus (6, 0) and vertex (0, 0).

  15. Class Work 5. Find the equation of the parabola with focus (-3,0) and directrix x = 3

  16. Parabolas whose vertex is not at the origin: The equation of the parabola whose vertex is (h,k) is

  17. Ex 6. Find the equation of the parabola with vertex (4, -2) and focus (2, -2)

  18. Class Work • 6. Find the equation of a parabola with focus (3,2) and vertex (5, 2). • 7. Find the equation of a parabola with vertex (-4,1) and directrix x = -7.

  19. HW#1 Parabolas Worksheet

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