RWA #1: Unit M Concept 4: Graphing parabolas given equation

1. Definition:     "The set of all points the same distance from a point and a line"

  ("http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet")

 2. Properties:  

Algebraically:        

                       Equations/formulas-

                                                  Vertical graph:

Horizontal graph:

Graphically:

     A parabola is a graph that can go up, down, left or right. It has a center at it's vertex and focuses around the focus. The directrix is a line outside the parabola that is perpendicular to the axis of symmetry. The axis of symmetry is a line in the middle of the parabola where the focus, vertex and part of the directrix lie.

How to find it's key features algebraically and graphically:

     

     In order to put the equation into standard form you need to complete the square (note that only one term is squared). If the x term of the equation is squared and the value of p is negative it will go down and if the value of p is positive the graph will go up. If the y term is squared and the value of p is positive the graph will go right, if p is negative the graph will go left. The vertex is an ordered pair that is the center of the parabola. The values of h,k is the vertex. Remember that h always goes with x and y always goes with k.

Graphing a parabola!

("http://www.youtube.com/watch?v=AQngdAoPIgE")

     

Key features of a parabola!

 ("http://www.mathsisfun.com/geometry/parabola.html")

      The focus is an ordered pair that is a point inside the parabola that is algebraically found by subtracting the value p with the term of the vertex that is changing. The distance that the focus is from the vertex determines how skinny or how fat the parabola is. In addition, the distance from the focus to any point on the parabola to the directrix is always equal and that is called the eccentricity. A parabola's eccentricity is equal to 1 which is why the two distances are equal. The directrix is a line outside of the parabola that is x=# (vertical line) or y=# (horizontal line). The directrix is determined by subtracting  the value of p from the value that is not changing withing the vertex. "p" is a point that is determined by setting the term outside of the non-squared portion of the formula equal to 4p and solving. "p" is the value that determines how far away the focus and the directrix are from the vertex. The axis of symmetry is a line that lies in the middle of the parabola that is across the x or y axis and is x=#  or y=#. Notice that the value of the axis of symmetry is the x or y value between the vertex and focus that is not changing.  The focus, vertex, and a part of the directrix all lie on the axis of symmetry. 

3. Real World Application: Satellite Dishes!

("http://www.ips-intelligence.com/ips/wp-content/uploads/2013/08/2-satellite1.jpg")

 

("http://www.youtube.com/watch?v=fV9YuF__fM4")

     A satellite dish is an example of where parabolas are applied in the real world. Radio waves that are parallel to the axis of symmetry hit any curve on the surface and gets reflected off and directly to the focus.  The radio waves then create a signal when the wave concentrates off the focus. Note that there will not be a signal if the focus is not correctly built within the shape of the dish.

4. References

http://www.mathsisfun.com/geometry/parabola.html

http://www.sophia.org/tutorials/unit-m-concept-4a?cid=embedplaylist

http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/parabola-drawn-from-definition-geogebra-dynamic-worksheet

http://www.ips-intelligence.com/ips/wp-content/uploads/2013/08/2-satellite1.jpg

http://www.youtube.com/watch?v=fV9YuF__fM4

http://www.youtube.com/watch?v=AQngdAoPIgE

http://www.mathsisfun.com/geometry/parabola.html