1. 1 LC.01.6 - The Parabola MCR3U - Santowski

2. 2 (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

3. 3 (B) Parabolas as Loci http://www.analyzemath.com/parabola/ParabolaDefinitio n.htmlhttp://www.analyzemath.com/parabola/ParabolaDefinitio n.html - Interactive applet from AnalyzeMath.com

4. 4 (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.

5. 5 (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|

6. 6 (C) Parabolas as Loci - Algebra

7. 7 (D) Graph of the Parabola

8. 8 (E) Analysis of the Parabola  The actual equation is –12x = y 2 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 x 2  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 y 2 = 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: x 2 = 4cy

9. 9 (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: y 2 = 20x

10. 10 (G) Internet Links  http://www.analyzemath.com/parabola/Parab olaDefinition.html - an interactive applet fom AnalyzeMath http://www.analyzemath.com/parabola/Parab olaDefinition.html  http://home.alltel.net/okrebs/page64.html - Examples and explanations from OJK's Precalculus Study Page http://home.alltel.net/okrebs/page64.html  http://www.webmath.com/parabolas.html - Graphs of parabolas from WebMath.com http://www.webmath.com/parabolas.html

11. 11 (G) Homework  AW, p488, 3,4,7b,8b