Transcript CONIC SECTIONS: PARABOLAS

Worksheet Key
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9.2: Parabolas
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Worksheet Key
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9.2: Parabolas
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Section 11.3
CONIC SECTIONS: PARABOLAS
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9.2: Parabolas
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Shape of a parabola from a cone
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Real-Life Examples
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Definitions
A. Parabola: Graph of a quadratic equation to which a set of
points in a plane that are the same distance from a given point
B. Vertex: Midpoint of the graph; the turning point
C. Focus: Distance from the vertex; located inside the parabola
D. Directrix: A fixed line used to define its shape; located outside
of the parabola
E. Axis of Symmetry: A line that divides a plane figure or a graph
into congruent reflected halves.
F. Latus Rectum: A line segment through the foci of the shape in
which it is perpendicular through the major axis and endpoints
of the ellipse
G. Eccentricity: Ratio to describe the shape of the conic, e = 1
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Formulas to know:
Horizontal Axis Standard Form:
If the ‘x’ is not being squared:
P is placement point; distance from the
focus to the vertex and vertex to the
directrix
4 p  x  h   y  k 
2
If p is positive, the parabola opens right
If p is negative, the parabola opens left
 h  p, k 
Focus Point:
x h p
yk
Directrix:
Axis of Symmetry:
h  p, k  2 p
Points of Latus Rectum
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Formulas to know:
Vertical Axis Standard Form:
If the ‘y’ is not being squared:
P is placement point; distance from the
focus to the vertex and vertex to the
directrix
4 p  y  k    x  h
2
If p is positive, the parabola opens up
If p is negative, the parabola opens down
Focus Point:
 h, k  p 
Directrix:
yk p
xh
Axis of Symmetry:
h  2 p, k  p
Points of Latus Rectum
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Formulas to know:
All Standard Form Equations
h, k 
Center
Length of Latus Rectum
Eccentricity e = 1
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4p
c
a
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Review of Parent Function Parabolas
y = x2 or x2 = y
x = y2 or y2 = x
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The Parabola
Brief Clip
Where do we see the focus point used in reallife?
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Horizontal Parabola
Horizontal parabola
due to its ‘Axis of
Symmetry’
y=k
(h, k)
Vertex: (h, k)
x = –p
Focus point: (p, 0)
Directrix: x = –p
Axis of Symmetry: y = k
Length of Latus Rectum: |4p|
Latus Rectum: (h + p, k + 2p)
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F
(p, 0)
4 p  x  h   y  k 
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2
Vertical Parabola
(p, 0)
F
(h, k)
y = –p
Vertex: (h, k)
Focus point: (0, p)
Directrix: y = –p
Axis of Symmetry: x = k
Length of Latus Rectum: |4p|
Latus Rectum: (h + 2p, k + p)
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x=k
Vertical parabola due
to its ‘Axis of
Symmetry’
4 p  y  k    x  h
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2
Review of Parent Function Parabolas
y = x2 or x2 = y
x = y2 or y2 = x
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Steps in Writing Equations of Parabolas
A. Identify whether the equation opens Up/Down or
Left/Right
B. Divide the coefficient (if necessary) to keep the
variable by itself
C. On the side without the squared into the equation
(which usually is a fraction), drop off all the variables
D. Multiply the coefficient (not involved with squared)
with ¼ to solve for p
E. Put it in standard form and graph
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Example 1
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
y = 4x2.
First, figure out what variable is squared? Put into the
suitable equation.
4 p  y  k    x  h
2
What is the vertex? (0, 0)
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Example 1
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
y = 4x2.
To find p:
4 p  y  k    x  h
y  4x
2
2
Isolate the equation to where the variable squared has a
2
coefficient of 1.
y 4x

4
4
1
2
yx
4
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Example 1
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
y = 4x2.
To find p:
1
2
yx
4
Take the coefficient in front of the isolated un-squared variable
and multiply it by ¼
11
  p
44
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p
16
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Example 1
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
y = 4x2.
To determine the Latus Rectum:
L  2p
1
L  2 
 16 
1
LR 
8
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Example 1
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
1
y = 4x2.
p
16
Vertex: (0, 0)
P:1/16 Vertical
Focus Point: (0, 1/16)
F
Directrix: y = –1/16
Axis of Symmetry: x = 0
Latus Rectum: (+1/8, 1/16)
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Your Turn
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
x = (1/20)y2.
Vertex: (0, 0)
P: 5
Focus Point: (5, 0)
F
Directrix: x = –5
Axis of Symmetry: y = 0
Latus Rectum: (5, +10)
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Example 2
Write in standard form equation of a parabola with the
vertex is at the origin and the focus is at (2, 0).
4 p  x  h   y  k 
2
4  2  x  0    y  0 
F
8x  y
P=2
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2
2
y  8x
2
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Your Turn
Write a standard form equation of a parabola where the
directrix is y = 6 and focus point (0, –6).
x  24 y
2
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Example 3
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
(y + 1)2 = 8(x + 1).
Vertex: (–1, –1) P: 2
Focus Point: (1, –1)
F
Directrix: x = –3
Axis of Symmetry: y = -1
Latus Rectum: (1, 3), (1, –5)
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Example 4
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
(x – 7)2 = –8(y – 2).
Vertex: (7, 2)
P: –2
Focus Point: (7, –2)
F
Directrix: y = 4
Axis of Symmetry: x = 7
Latus Rectum: (7, 5), (7, 9)
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Your Turn
Determine the vertex, focus point, directrix, axis of
symmetry, latus rectum, and graph the parabola for
(y – 1)2 = –4(x – 1).
Vertex: (1, 1)
P: –1
Focus Point: (0, 1)
F
Directrix: x = 2
Axis of Symmetry: y = 1
Latus Rectum: (0, 3), (0, –1)
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Example 5
Write a standard form equation of a parabola where the
vertex is (–7, –3) and focus point (2, –3).
F
 y  3
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 36  x  7 
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Example 6
Write a standard form equation of a parabola where the
vertex is (–2, 1) and the directrix is at x = 1.
 y  1
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 12  x  2 
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Your Turn
Write a standard form equation of a parabola where the
axis of symmetry is at y = –1, directrix is at x = 2 and
the focus point (4, –1).
 y  1
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 4  x  3
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Assignment
Worksheet
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