The parabola is the locus of all points in a plane that are the same distance from a line in the plane, the directrix, as from a fixed point in the plane, the focus



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9.2 - parabolas 1

Parabolas

  • Section 9 - 2
  • The parabola is the locus of all points in a plane that are
  • the same distance from a line in the plane, the directrix,
  • as from a fixed point in the plane, the focus.
  • The parabola has one axis of
  • symmetry, which intersects
  • the parabola at its vertex.
  • | p |
  • The distance from the
  • directrix to the vertex is also | p |.
  • The Parabola
  • | p |
  • For a parabola with the axis of symmetry parallel to
  • the y-axis and vertex at (h, k), the standard form is …
  • The coordinates of the focus are (h, k + p).
  • When p is positive,
  • the parabola opens upward.
  • When p is negative,
  • the parabola opens downward.
  • (x - h)2 = 4p(y - k)
  • The Standard Form of the Equation with Vertex (h, k)
  • For a parabola with an axis of symmetry parallel to the
  • x-axis and a vertex at (h, k), the standard form is:
  • The equation of the axis of symmetry is y = k.
  • The coordinates of the focus
  • are (h + p, k).
  • The equation of the directrix
  • is x = h - p.
  • (y - k)2 = 4p(x - h)
  • When p is positive, the parabola
  • opens to the right.
  • The Standard Form of the Equation with Vertex (h, k)
  • Finding the Equations of Parabolas
  • The distance from the focus to the directrix is 6 units,
  • therefore, 2p = -6, p = -3. Thus, the vertex is (6, 5).
  • (6, 5)
  • The axis of symmetry is parallel to the x-axis:
  • (y - k)2 = 4p(x - h)
  • h = 6 and k = 5
  • Standard form
  • (y - 5)2 = 4(-3)(x - 6)
  • (y - 5)2 = -12(x - 6)
  • Find the equation of the parabola that has a minimum at
  • (-2, 6) and passes through the point (2, 8).
  • The axis of symmetry is parallel to the y-axis.
  • The vertex is (-2, 6), therefore, h = -2 and k = 6.
  • (x - h)2 = 4p(y - k)
  • (2 - (-2))2 = 4p(8 - 6)
  • 16 = 8p
  • 2 = p
  • x = 2 and y = 8
  • (x - h)2 = 4p(y - k)
  • (x - (-2))2 = 4(2)(y - 6)
  • (x + 2)2 = 8(y - 6)
  • Standard form
  • Finding the Equations of Parabolas
  • Find the coordinates of the vertex and focus,
  • the equation of the directrix, the axis of symmetry,
  • and the direction of opening of 2x2 + 4x - 2y + 6 = 0.
  • 2x2 + 4x - 2y + 6 = 0
  • 2(x2 + 2x + _____) = 2y - 6 + _____
  • 1
  • 2(1)
  • 2(x + 1)2 = 2(y - 2)
  • (x + 1)2 = (y - 2)
  • The parabola opens to upward.
  • The vertex is (-1, 2).
  • The focus is ( -1, 2 ¼ ).
  • The Equation of directrix is y = 1¾ .
  • The axis of symmetry is x = -1 .
  • 4p = 1
  • p = ¼
  • Graphing a Parabola
  • y2 - 10x + 4y - 16 = 0
  • 4
  • 4
  • y2 + 4y + _____ = 10x + 16 + _____
  • (y + 2)2 = 10x + 20
  • (y + 2)2 = 10(x + 2)
  • Horizontally oriented (right)
  • Vertex @ (-2, -2)
  • Line of Symmetry y = -2
  • P = 2.5
  • focus @ ( 0.5, -3)
  • Directrix X = - 4.5

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