LC.01.6 - The Parabola
MCR3U - Santowski
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(A) Parabola as Loci

A parabola is defined as the set of points such that the distance from
a fixed point (called the focus) to any point on the parabola is the
same as the distance from this same point on the parabola to a fixed
line called the directrix  | PF | = | PD |

We will explore the parabola from this locus definition

Ex 1. Using the GSP program, we will geometrically construct a set
of points that satisfy the condition that | PF | = | PF | by following the
directions on the handout
2
(B) Parabolas as Loci
http://www.analyzemath.com/parabola/ParabolaDefinitio
n.html - Interactive applet from AnalyzeMath.com
3
(C) Parabolas as Loci - Algebra

We will now tie in our knowledge of algebra to
come up with an algebraic description of the
parabola by making use of the relationship
that | PF | = | PD|

ex 2. Find the equation of the parabola
whose foci is at (-3,0) and whose directrix is
at x = 3. Then sketch the parabola.
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(C) Parabolas as Loci - Algebra

Since we are dealing with distances, we set up our
equation using the general point P(x,y), F at (-3,0)
and the directrix at x=3 and the algebra follows on
the next slide |PF| = |PD|
5
(C) Parabolas as Loci - Algebra
PF  PD


 
y  
 x  3 2  y 2 
 x  3
2
2
2
 x  3 2  ( y  y ) 2

 x  3  ( y  y ) 2
2

2
 x  3 2  y 2   x  3 2  ( y  y ) 2
x 2  6x  9  y 2  x 2  6x  9
 12 x  y 2
1 2
1
x 
y   ( y  0) 2  0
12
12
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(D) Graph of the Parabola
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(E) Analysis of the Parabola






The actual equation is –12x = y2 and note how this is different
than our previous look at parabolas (quadratics)
Previously, we would have defined the equation as y = +(12x)
which would represent the non-function inverse of y = -1/12 x2
The domain of our parabola is {x E R | x < 0} and our range is y
ER
Our vertex is at (0,0), which happens to be both the x- and yintercept.
In general, for a parabola opening along the x-axis, the general
equation is y2 = 4cx where c would represent the x co-ordinate
of the focus
If the parabola opens along the y-axis, the general equation is
similar: x2 = 4cy
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(F) In-class Examples
 Determine
the equation of the parabola
and then sketch it, labelling the key
features, if the focus is at (5,0) and the
directrix is at x = -5
 The
equation you generate should be
the following: y2 = 20x
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(G) Internet Links

http://www.analyzemath.com/parabola/Parab
olaDefinition.html - an interactive applet fom
AnalyzeMath
 http://home.alltel.net/okrebs/page64.html Examples and explanations from OJK's
Precalculus Study Page
 http://www.webmath.com/parabolas.html Graphs of parabolas from WebMath.com
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(G) Homework
 AW,
p488, 3,4,7b,8b
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