Parabola PowerPoint

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Chapter 3 Conics
3.6
The Parabola
MATHPOWERTM 12, WESTERN EDITION
3.6.1
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The Parabola
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 .
3.6.2
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The Standard Form of the Equation of a Parabola
with Vertex (0, 0)
The equation of a parabola with vertex (0, 0) and
focus on the x-axis is y2 4px.
The coordinates of the focus are (p, 0). The
equation of the directrix is x -p.
If p gt 0, the parabola opens right. If p lt 0, the
parabola opens left.
3.6.3
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The Standard Form of the Equation of a Parabola
with Vertex (0, 0)
The equation of a parabola with vertex (0, 0) and
focus on the y-axis is x2 4py.
The coordinates of the focus are (0, p). The
equation of the directrix is y -p.
If p gt 0, the parabola opens up. If p lt 0, the
parabola opens down.
3.6.4
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Sketching a Parabola
A parabola has the equation y2 -8x. Sketch the
parabola showing the coordinates of the focus
and the equation of the directrix.
The vertex of the parabola is (0, 0). The focus
is on the x-axis. Therefore, the standard
equation is y2 4px.
Hence, 4p -8 p -2.
The coordinates of the focus are (-2, 0).
F(-2, 0)
The equation of the directrix is x
-p, therefore, x 2.
x 2
3.6.5
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Finding the Equation of a Parabola with Vertex
(0, 0)
A parabola has vertex (0, 0) and the focus on an
axis. Write the equation of each parabola.
a) The focus is (-6, 0).
Since the focus is (-6, 0), the equation of the
parabola is y2 4px.
p is equal to the distance from the vertex to the
focus, therefore p -6.
The equation of the parabola is y2 -24x.
b) The directrix is defined by x 5.
Since the focus is on the x-axis, the equation of
the parabola is y2 4px.
The equation of the directrix is x -p,
therefore -p 5 or p -5.
The equation of the parabola is y2 -20x.
c) The focus is (0, 3).
Since the focus is (0, 3), the equation of the
parabola is x2 4py.
p is equal to the distance from the vertex to the
focus, therefore p 3.
The equation of the parabola is x2 12y.
3.6.6
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The Standard Form of the Equation with Vertex (h,
k)
For a parabola with the axis of symmetry parallel
to the y-axis and vertex at (h, k)

• 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.

• The standard form for parabolas
• parallel to the y-axis is

(x - h)2 4p(y - k)
The general form of the parabola is Ax2 Cy2
Dx Ey F 0 where A 0 or C 0.
3.6.7
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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 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.

• When p is positive, the parabola
• opens to the right.

• When p is negative, the parabola
• opens to the left.

• The standard form for parabolas
• parallel to the x-axis is

(y - k)2 4p(x - h)
3.6.8
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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).
The axis of symmetry is parallel to the x-axis
(y - k)2 4p(x - h)
h 6 and k 5
(y - 5)2 4(-3)(x - 6) (y - 5)2 -12(x - 6)
(6, 5)
Standard form
y2 - 10y 25 -12x 72 y2 12x -
10y - 47 0
General form
3.6.9
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Finding the Equations of Parabolas
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)
x 2 and y 8
(2 - (-2))2 4p(8 - 6) 16 8p
2 p
(x - h)2 4p(y - k) (x - (-2))2 4(2)(y -
6) (x 2)2 8(y - 6)
Standard form
x2 4x 4 8y - 48 x2 4x 8y 52
0
General form
3.6.10
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Analyzing a Parabola
Find the coordinates of the vertex and focus,
the equation of the directrix, the axis of
symmetry, and the direction of opening of y2 -
8x - 2y - 15 0.
4p 8 p 2
y2 - 8x - 2y - 15 0 y2 - 2y _____ 8x
15 _____
1
1
(y - 1)2 8x 16 (y - 1)2 8(x 2)
Standard form
The vertex is (-2, 1). The focus is (0, 1). The
equation of the directrix is x 4 0. The axis
of symmetry is y - 1 0. The parabola opens to
the right.
3.6.11
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Graphing a Parabola
y2 - 10x 6y - 11 0
9
9
y2 6y _____ 10x 11 _____
(y 3)2 10x 20 (y 3)2 10(x 2)
3.6.12
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General Effects of the Parameters A and C
When A x C 0, the resulting conic is an
parabola.
When A is zero If C is positive, the parabola
opens to the left. If C is negative, the parabola
opens to the right.
When C is zero If A is positive, the parabola
opens up. If A is negative, the parabola opens
down.
When A D 0, or when C E 0, a degenerate
occurs.
E.g., x2 5x 6 0
x2 5x 6 0 (x 3)(x 2) 0 x 3
0 or x 2 0 x -3 x
-2
The result is two vertical, parallel lines.
3.6.13
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Assignment
Suggested Questions Pages 167-169 A 1, 3, 5, 7,
8 (general form)
B 9, 12, 13, 14, 16, 18, 27, 28, 29, 46, 47
3.6.14