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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

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(B) Parabolas as Loci

http://www.analyzemath.com/parabola/ParabolaDefinition.html - Interactive applet from AnalyzeMath.com

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(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|

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(C) Parabolas as Loci - Algebra

PF PD

x y x y y

x y x y y

x y x y y

x x y x x

x y

x y y

3 3

3 3

3 3

6 9 6 9

12

1

12

1

120 0

2 2 2 2

2 22

2 22

2 2 2 2

2 2 2

2

2 2

( )

( )

( )

( )

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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

E R Our vertex is at (0,0), which happens to be both the x- and y-

intercept. 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