10.110.1 conic sections and parabolas. quick review
TRANSCRIPT
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10.110.110.110.1
Conic Sections and ParabolasConic Sections and Parabolas
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Quick Review 2
2
1. Find the distance between ( 1,2) and (3, 4).
2. Solve for in terms of . 2 6
3. Complete the square to rewrite the equation in vertex form.
2 5
4. Find the vertex and axis of the graph of
y x y x
y x x
f
2
2
( ) 2( 1) 3.
Describe how the graph of can be obtained from the graph
of ( ) .
5. Write an equation for the quadratic function whose graph
contains the vertex (2, 3) and the point (0,3).
x x
f
g x x
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Quick Review Solutions 2
2 2
1. Find the distance between ( 1,2) and (3, 4).
2. Solve for in terms of . 2 6
3. Complete the square to rewrite the equation in vertex form.
2 5
4. Find
52
3
( 1
the ver
4
tex
)
y x y x
y xy
y
x x
x
2
2
and axis of the graph of ( ) 2( 1) 3.
Describe how the graph of can be obtained from the graph
vertex:( 1,3); axis: 1; translation left 1 unit,
vertical stretch by a factor of
of ( ) .
2,
f x x
f
g xx x
2
5. Write an equation for the quadratic function whose graph
contains the vertex (2, 3) and
translation up 3 u
the point (0,3).
nits.
32 3
2y x
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What you’ll learn about• Conic Sections• Geometry of a Parabola• Translations of Parabolas• Reflective Property of a Parabola
… and whyConic 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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ParabolaA 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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A Right Circular Cone (of two nappes)
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Conic Sections and Degenerate Conic
Sections
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Conic Sections and Degenerate Conic Sections (cont’d)
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Second-Degree (Quadratic) Equations
in Two Variables
2 2 0, where , , and , are not all zero.Ax Bxy Cy Dx Ey F A B C
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Structure of a Parabola
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Graphs of x2=4py
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Parabolas with Vertex (0,0)
• Standard equation x2 = 4py y2 = 4px• Opens Upward or To the Right
Downword or to the 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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Graphs of y2 = 4px
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Example Finding an Equation of a
Parabola
Find an equation in standard form for the parabola whose directrix
is the line 3 and whose focus is the point ( 3,0).x
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Example Finding an Equation of a
Parabola Find an equation in standard form for the parabola whose directrix
is the line 3 and whose focus is the point ( 3,0).x
2
2
Because the directrix is 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:
4
12
x
y px
y x
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Parabolas with Vertex (h,k)
• Standard equation (x-h)2 = 4p(y-k) (y-k)2 = 4p(x-h)
• Opens Upward or downward To the right or to the left
• 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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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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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).
2
2
The parabola is opening downward so the equation has the form
( ) 4 ( ).
( , ) (1,2) and the distance between the vertex and the focus is
4. Thus, the equation is ( 1) 16( 2).
x h p y k
h k
p x y