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

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.

• Point Focus = Point Directrix
• PF = PD

• The parabola has one axis of
• symmetry, which intersects
• the parabola at its vertex.

• | p |

• The distance from the
• vertex to the focus is | 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 equation of the axis of symmetry is x = h.

• The coordinates of the focus are (h, k + p).

• The equation of the directrix
• is y = 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 negative, the parabola
• opens to the left.

• 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

• Write the equation of the parabola with a focus at (3, 5) and
• the directrix at x = 9, in standard form and general form

• 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.

• Substitute into the standard form of the equation
• and solve for p:

• (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

• Analyzing a Parabola

• 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