## Saturday, February 9, 2013

### Do Parabolas Have Centers? Another PairoDucks?

Well, your chance to win a free copy of my Math Challenge book has come and gone but I thought I would post an original problem/investigation about graphs of quadratic functions.

In Alg2/Precalculus students learn about parabolas:
Vertex, axis, symmetry, intercepts then, perhaps, further into other defining properties involving focus & directrix. It's also fun to touch on other important and fascinating applications such as the reflecting properties of 3-dim parabolic surfaces. OK, enough overview.

So, is the focus of a parabola the closest analog to a "center"? It's my blog so I say no!

Consider the following sloppily drawn sketch...

Can you make sense out of this graph of y = (1/12)x^2?

Without referring to the focus-directrix form of a parabola (e.g., x^2=4py), determine the values of p and q.

Reflections...

1.  Is the "p" in this problem the same as the parameter p which defines the focus?

2.  Does the diagram suggest another way to define the focus? Explain.

3.  Of course, we do not refer to the point Q as the center. I just felt  like calling it that. Can you guess how the Circle Paradox posts led me to this? Hey, I may not be as creative as some of you but I am persistent!

4. OK, fellow (gender-free) colleagues. How might you extend this investigation? We need to share ideas, right?

5.  So what does "4p" represent geometrically? Refer to the diagram.