In Plain Sight, but Unseen

Cross-posted from CAS Musings:

Thanks to a comment from Doug Kuhlmann on my last post, I’ve got a few new cool connections on the transformational effects of the parameters in y=a\cdot x^2+b\cdot x+c on its graph.  This is exactly why I share.  Thanks, Doug!!

THE NEW PATTERN:  Use this GeoGebraTube Web document to model the problem.  Set the value of b to any non-zero value and vary a.  The parabola’s vertex moves along a line, as shown below.

As with the changes in the b parameter, define that line and prove your claims.

[The GeoGebra link above does not produce the vertex geometry trace footprints shown in the image.  If you want to create these, download GeoGebra and create this simple document for yourself.  It is FREE.  If anyone wants explicit instructions for how to do this, email me and I’ll post instructions on the ‘blog.]

Another option to see the line is to use Geogebra’s locus tool.  It requires two inputs:  which object is the locus following, and which variable driving the variation.  After selecting the locus tool, click on the vertex and then the slider for a.  You get the next image.

SOLUTION ALERT!  Don’t read further if you want to solve the problem for yourself.

I knew the line contained the vertex and noticed that it seemed to pass through y-intercept.  Predicting the y-intercept was c, all I needed was the slope.  With my prediction of the two generic points, I could compute that, too.  I enjoy symbol manipulation for the mental exercise.  The symbols (to me) weren’t all that complicated, so I took a brief moment of fun solving that by hand.  But this is another of those situations where the symbol manipulation isn’t the point, so using my CAS is 100% legitimate.  It is also a great leveler of ability for those intimidated for any reason by algebraic manipulations.

The next image is a great use of CAS commands to find the line’s slope.  In particular, notice the use of a function definition to minimize the algebraic clutter through function notation.

Lovely and surprisingly simple.  That means the line the parabola’s vertex follows when a varies for non-zero b is y=\frac{b}{2}\cdot x+c.

Students often overlook the domain warning.  It doesn’t matter for the creation of the line, but ultimately lies at the heart of the unequal spacing of the vertex footprints in the first image and explains the unique behavior of the parabola’s movement.

If a student didn’t use the vertex and y-intercept to derive the linear equation, a CAS solve command could legitimately be used to show that those two generic points were always on the line.

MOTION ALONG THE LINE:  One of the interesting parts of this problem is how the parabola moves along y=\frac{b}{2}\cdot x+c.  After some play with the GeoGebra document, you can see that as |a|\rightarrow 0 the parabolas’ vertices move infinitely far away from the y-axis, and as |a|\rightarrow\infty the vertices approach the y-axis. This can also be seen numerically from the generic x-coordinate of the vertex, -\frac{b}{2a}.  For a fixed, non-zero value of b, the fraction representing the x-coordinate of the vertex increases in magnitude as |a|\rightarrow 0 and decreases in magnitude toward 0 as |a|\rightarrow\infty.

The vertex trace points in the first image above are separated by \Delta a=.01.  The reason for the differences in distances between the points noted above is because -\frac{b}{2a} does not change linearly when a changes linearly.  As |a|\rightarrow\infty, -\frac{b}{2a}\rightarrow 0 slower and slower, explaining the increasing density of the vertex trace points near the y-axis.

When a=0, the x-coordinate of the vertex is undefined.  At that moment, the generic quadratic, y=a\cdot x^2+b\cdot x+c, becomes the degenerate y=b\cdot x+c, a line.  Graphing that line against a trace of all possible parabolas as a varies (the red dashed line below), the degenerate parabola resulting when a=0 is precisely the tangent line to all of these parabolas at their y-intercept, (0,c), a pretty extension on a connection suggested by Dave Radcliffe on my previous post.  Nice.

A FINAL NOTE:  My memory suggests that I’ve seen this pattern before in some of the numerous times I’ve presented the b-variation of this problem in conferences and assigned it in classes.  Despite all the times I must have seen it, the pattern never rose to my active conscience.  Serendipitously, I’m currently reading Tina Seelig’s inGenius:  A Crash Course on Creativity. I offer two quotes from her Are You Paying Attention? chapter:

  • We think we understand the world and look for the patterns that we already recognize. (p. 71)
  • We focus predominantly on things that are at our eye level rather than looking around more broadly.  In addition, we pay attention to objects that we expect to find and ignore those things that don’t fit. (p. 71)

The moral:  Even after all of my attempts and success at finding unique patterns, I missed this one until Doug pointed it out to me.  I suspect my focus on what I knew about b‘s effect blinded me to the a effect.  This is a great reminder to me to always hold myself ready to see beauty and pattern in unexpected places.

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