Unit 8.1

1. 8.1
Conic Sections
and Parabolas
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Conic Sections
 Geometry of a Parabola
 Translations of Parabolas
 Reflective Property of a Parabola
… and why
Conic sections are the paths of nature: Any free-moving
object in a gravitational field follows the path of a conic
section.
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3. A Right Circular Cone (of two nappes)
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4. Conic Sections and
Degenerate Conic Sections
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5. Conic Sections and
Degenerate Conic Sections (cont’d)
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6. Second-Degree (Quadratic) Equations
in Two Variables
The conic sections can defined algebraically as the
graphs of second - degree (quadratic) equations
in two variables, that is, equations of the form
Ax2  Bxy  Cy2  Dx  Ey  F  0,
where A, B, and C, are not all zero.
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7. Parabola
A parabola is the
set of all points in
a plane equidistant
from a particular
line (the directrix)
and a particular
point (the focus)
in the plane.
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8. Graphs of x2 = 4py
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9. Parabolas with Vertex (0,0)
 Standard equation x2 = 4py y2 = 4px
 Opens Upward or To the right or to the
downward left
 Focus (0, p) (p, 0)
 Directrix y = –p x = –p
 Axis y-axis x-axis
 Focal length p p
 Focal width |4p| |4p|
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10. Graphs of y2 = 4px
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11. Example Finding an Equation of a
Parabola
Find an equation in standard form for the parabola
whose directrix is the line x  3 and whose focus is
the point (  3,0).
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12. Example Finding an Equation of a
Parabola
Find an equation in standard form for the parabola
whose directrix is the line x  3 and whose focus is
the point (  3,0).
Because the directrix is x  3 and the focus is (  3,0),
the focal length is  3 and the parabola opens to the left.
The equation of the parabola in standard from is:
y2  4 px
y2  12x
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13. Parabolas with Vertex (h,k)
 Standard equation (x– h)2 = 4p(y – k) (y – k)2 = 4p(x – h)
 Opens Upward or To the right or to the left
downward
 Focus (h, k + p) (h + p, k)
 Directrix y = k-p x = h-p
 Axis x = h y = k
 Focal length p p
 Focal width |4p| |4p|
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14. Example Finding an Equation of a
Parabola
Find the standard form of the equation for the parabola
with vertex at (1,2) and focus at (1,  2).
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15. Example Finding an Equation of a
Parabola
Find the standard form of the equation for the parabola
with vertex at (1,2) and focus at (1,  2).
The parabola is opening downward so the equation
has the form
(x  h)2  4 p( y  k).
(h,k)  (1,2) and the distance between the vertex and
the focus is p  4.
Thus, the equation is (x 1)2  16( y  2).
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16. Quick Review
1. Find the distance between ( 1,2) and (3,  4).
2. Solve for y in terms of x. 2y2  6x
3. Complete the square to rewrite the equation in vertex form.
y  x2  2x  5
4. Find the vertex and axis of the graph of f (x)  2(x 1)2  3.
Describe how the graph of f can be obtained from the graph
of g(x)  x2 .
5. Write an equation for the quadratic function whose graph
contains the vertex (2,  3) and the point (0,3).
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17. Quick Review Solutions
1. Find the distance between (  1, 2) and (3, 
4).
2. Solve for y in terms of x . 2 y 2

6
x
3. Complete the square to rewrite the equation in vertex form.
y x x
y x
  2  2  5
 
(  1
)
2

4. Find
52
  3
x
the ver
4
tex
y
2
f x x
f
 

Describe how the graph of can be obtained from the graph
g x x x
of ( ) 2
.
and axis of the graph of ( ) 2( 1) 3.
vertex:(  1,3); axis:  
1; translation left 1 unit,

vertical stretch by a factor of
2,
translation up 3 u
nits.
5. Write an equation for the quadratic function whose graph
 2
 y  x  
contains the vertex (2, 3) and
the point (0,3).
3
2 3
2
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