Things to Remember: The driectrix and the axis of symmetry have to be perpendicular to each other (both are lines). The distance from the vertex to the focus determines how "skinny" or how "fat" the parabola will end up being. The distance from ANY point on the parabola to straight down to the directrix is always the SAME. The vertex and the the focus are each ordered pairs.
2. Algebraically:
(x-h)^2=(y-k)
This is just one equation of a parabola. If "x" is the one variable being squared, as this equation shows, then the parabola, once graphed, will face either up or down. "p" equals the distance between the focus point and the directrix, so if "p" is positive, then the parabola will face upwards, whereas if "p" is negative, the parabola will face downwards. (For an example of just what this means, view the image in section 3)
(y-k)^2=(x-h)
This is the other equation for the parabola where instead of "x" being squared, "y" is squared. This means that the parabola will not face up or down but instead face either left or right, depending on"p". If "p" is positive, then the parabola will face the right. If "p" is negative, then the parabola will face the left. (For an example of just what this means, view the image in section 3).
3. This image explains the above concept about where the parabola will face depending on which variable is squared and what "p" is. Instead of straight-out giving us exactly what "p" is, though, we are given the generalization of "p": if "p" is greater than 0, then it will for sure be positive and therefore face either up or right, depending on the one squared term. If "p" is less than 0, then that means that "p" will obviously be negative and therefore face either down or left, depending on the one squared term (http://jwilson.coe.uga.edu/EMAT6680Fa05/Frye/EMAT%206690%20Frye/Conics%20instructional%20unit/Lesson%204_files/image003.jpg).
The following website shows several examples of parabolas in real life, like in roller-coasters and the reflection of a light beam from a flashlight, etc. Mathematics are a basic component of parabolas in roller-coasters, as well as taking into account the force of gravity. The peak of the roller-coaster mentioned in the web-site is the vertex of the ride, which then releases force. Something else this website shows that I haven't yet referenced is how a parabola is expressed in a cone and why it's a conic section. There is even an image which shows a parabola in sliced into a cone and the parabolic trajectory is shown as well (the path of a parabola).
http://mathforum.org/mathimages/index.php/Parabola
This video shows a teacher teaching the concept of parabolas in real-life to her class and where they are found and how math actually does apply to real-life situations. She hit upon the points we have learned so far, but for a more verbal explanation of this concept, please copy/paste this link and watch.
http://www.teachertube.com/viewVideo.php?video_id=554
4.Citing URLS:
http://jwilson.coe.uga.edu/EMAT6680Fa05/Frye/EMAT%206690%20Frye/Conics%20instructional%20unit/Lesson%204_files/image003.jpg
http://mathforum.org/mathimages/index.php/Parabola
http://www.teachertube.com/viewVideo.php?video_id=554
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