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.


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.

As always, feel free to share this but don't forget proper attribution.

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