Section 10-5
Parabolas
Definition
 A parabola is defined as the set of all points in a plane that are
the same distance from a given point, called the focus, and a
given line, called the directrix. The axis of symmetry or also
called the axis of the parabola passes through the vertex and
focus of the parabola.
Example # 1
 Consider the equation y 2+ 2x = 0. We know this is a parabola because A=0
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(General form when A or C =0 (not both), we have a parabola). Find the
coordinates of the focus and the vertex and the equations of the directrix axis of
symmetry. Graph the equation.
First we need to write the e quation in standard form by completing the square.
2
(y-0) = 4 (-.5) (x-0)
Vertex (0,0)
Focus (h+p, k) = (-1/2, 0) or (-.5,0)
focus (-.5,0)
vertex (0,0)
Directrix x= (h-p) = ½
*
1
Axis of symmetry y=k so y=0
directrix x=1/2
Gerneral form of the parabola
 When A=0 we know that the directrix is parallel to the y axis
and the parabola opens right or left
2
 (y-k) = 4p (x-h)
 When C=0 we know that the directrix is parallel to the x
axis and the parabola opens up or down
2
 (x-h) = 4p (y-k)
Example # 2
 Consider the equation x 2 -8x – y + 18 = 0. Write the equation in standard form. Find
the coordinates of the vertex and focus and the equations of the directrix and the line of
symmetry. Graph the equation of the parabola.
 First we know it is a parabola where C = 0 so we know the directrix is parallel to the x
axis and the parabola opens up or down. Since A is positive, we know it opens up.
 Standard form: Complete the square
2
(x-4) = y – 18 + 16
(x – 4 )
2
2
= y-2
( x – 4 ) = 4 ( ¼) ( y – 2 ) so h=4,
k=2, p= ¼
Vertex (4,2)
Focus (h, k+p) = (4, 9/4)
Directrix y = k – p = 7/4
Line of symmetry x = 4
directrix y=7/4
2
focus
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4
vertex
Parabolas are often used to demonstrate maximum and
minimum points in real-world situations
 N A SA ’s K C -135A aircraft flies in parabolic arcs to simulate the weightlessness experienced by
astronauts in space. The aircraft starts its ascent at 24,000 feet. During the ascent, all on board
experience 2 g’s or tw ice the pu ll of E arth’s gravity. A s the aircraft approaches its m axim um height,
the engines are stopped, and the aircraft is allowed to free fall at a precisely determined angle. Zero
gravity is achieved for 25 seconds as the plane reaches the top of the parabola and begins its descent.
After this 25 second period, the engines are throttled to bring the aircraft out of the dive. If the
height of the aircraft in feet (y) verses time in seconds (x) is modeled by the equation
x2 – 65 x + .11y – 2683.75 = 0, what is the maximum height achieved by the aircraft during its
parabolic flight?
Standard form of the equation (complete the square):
The vertex of the parabola is (32.5, 34000)
Since p < 0, we know the parabola opens
downward making the vertex a maximum point
32.5 seconds after the aircraft began its parabolic
maneuver, it reaches its maximum height of 34000 ft.
2
x - 65x = -.11y + 2683.75
2
( x – 32.5) = -.11y + 2683.75 + 1056.25
2
( x – 32.5) = -.11y + 3740
2
( x – 32.5) = -.11 (y-34000)
2
( x – 32.5) = 4 (-.0275) (y – 34000)
Locus
 All conic sections can be defined using the focus-directrix definition.
 A conic section is defined to be the locus (path) of points such that, for any
point P in the locus, the ratio ( also called the eccentricity) of the distance
between that point and a fixed point F (focus) to the distance between that
point and a fixed line  is the directrix.
Eccentricity
 We now know that the ratio of the distance of the points
(locus points) to the focus TO the distance of the points to
the directrix is the eccentricity of the conic section.
 Circle e = 0
 Parabola e = 1
 E llipse e < 1 e ≠ 0
 Hyperbola e > 1
HW # 18
Section 10-5
Pp. 659-661
#13-27 odds, 34,39,41