Parabolas

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Parabolas GEO HN CCSS G.GPE.2
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Standards for Mathematical Practice

• 1. Make sense of problems and persevere in
  solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
  reasoning of others.  
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning.

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Essential Question

• What is the relationship between focus and
  directrix to the equation of a parabola?

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CCSS G.GPE.2

• DERIVE the equation of a parabola given a focus
  and directrix. 

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Parabola
A parabola is a set of points in a plane that are
equidistant from a fixed line, the directrix, and
a fixed point, the focus.
For any point Q that is on the parabola, d2 d1
Q
d2
Focus
The latus rectum of a parabola is a line segment
that passes through the focus, is parallel to the
directrix and has its endpoints on the parabola.
d1
Directrix
The length of the latus rectum is 4p where p
is the distance from the vertex to the focus.
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Parabolas
Things you should already know about a parabola.
Forms of equations
y a(x h)2 k
opens up if a is positive
opens down if a is negative
vertex is (h, k)
y ax2 bx c
opens up if a is positive
V
opens down if a is negative
-b 2a
-b 2a
vertex is , f( )
Thus far in this course we have studied parabolas
that are vertical - that is, they open up or
down and the axis of symmetry is vertical
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Parabolas
In this unit we will also study parabolas that
are horizontal that is, they open right or left
and the axis of symmetry is horizontal
V
In these equations it is the y-variable that is
squared.
x a(y k)2 h
or
x ay2 by c
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Equations of a Parabola
x ay2 by c
y ax2 bx c
Horizontal Hyperbola
Vertical Hyperbola
-b 2a
-b 2a
Vertex x
y
vertex
If a gt 0, opens right
If a gt 0, opens up
If a lt 0, opens left
If a lt 0, opens down
The directrix is vertical
The directrix is horizontal
1
a
4p
Remember p is the distance from the
vertex to the focus
the directrix is the same distance from the
vertex as the focus is
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Standard Form Equation of a Parabola
(y k)2 4p(x h)
(x h)2 4p(y k)
Horizontal Parabola
Vertical Parabola
Vertex (h, k)
Vertex (h, k)
If 4p gt 0, opens right
If 4p gt 0, opens up
If 4p lt 0, opens left
If 4p lt 0, opens down
The directrix is vertical the vertex is midway
between the focus and directrix
The directrix is horizontal and the vertex is
midway between the focus and directrix
Remember p is the distance from the
vertex to the focus
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Find the standard form of the equation of the
parabola given
the focus is (2, 4) and the directrix is x - 4
The vertex is midway between the focus and
directrix, so the vertex is (-1, 4)
The directrix is vertical so the parabola must be
horizontal and since the focus is always inside
the parabola, it must open to the right
V
F
Equation (y k)2 4p(x h)
p 3
Equation (y 4)2 12(x 1)
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Find the standard form of the equation of the
parabola given
the vertex is (2, -3) and focus is (2, -5)
Because of the location of the vertex and focus
this must be a vertical parabola that opens down
Equation (x h)2 4p(y k)
p 2
V
Equation (x 2)2 -8(y 3)
F
The vertex is midway between the focus and
directrix, so the directrix for this parabola is
y -1
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Graphing a Parabola
(y 3)2 4(x 1)
Find the vertex, focus and directrix. Then graph
the parabola
Vertex (-1, -3)
The parabola is horizontal and opens to the right
4p 4
p 1
Focus (0, -3)
V
Directrix x -2
F
x ¼(y 3)2 1
x y
0
0
3
3
-1
-5
1
-7
-
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Converting an Equation
Convert the equation to standard form
Find the vertex, focus, and directrix
y2 2y 12x 35 0
y2 2y ___ -12x 35 ___
1
1
(y 1)2 -12x 36
F
(y 1)2 -12(x 3)
V
The parabola is horizontal and opens left
Vertex (3, 1)
4p -12
Focus (0, 1)
p -3
Directrix x 6
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Applications
A satellite dish is in the shape of a parabolic
surface. The dish is 12 ft in diameter and 2 ft
deep. How far from the base should the receiver
be placed?
12
2
(-6, 2)
(6, 2)
Consider a parabola cross-section of the dish and
create a coordinate system where the origin is at
the base of the dish.
Since the parabola is vertical and has its
vertex at (0, 0) its equation must be of the
form
x2 4py
The receiver should be placed 4.5 feet above the
base of the dish.
At (6, 2), 36 4p(2)
so p 4.5
thus the focus is at the point (0, 4.5)
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Application
The towers of a suspension bridge are 800 ft
apart and rise 160 ft above the road. The cable
between them has the shape of a parabola, and the
cable just touches the road midway between the
towers.
(400, 160)
(300, h)
What is the height of the cable 100 ft from a
tower?
100
300
Since the parabola is vertical and has its vertex
at (0, 0) its equation must be of the form
At (300, h), 90,000 1000h
x2 4py
h 90
At (400, 160), 160,000 4p(160)
The cable would be 90 ft long at a point 100 ft
from a tower.
1000 4p
p 250
thus the equation is x2 1000y