Quadratics - What Do They Model?

  Have you every wondered what the a, b, and c do in the quadratic equation?  What are they other than coefficients?  Personally, I never actually gave it a thought.  I had to identify them so I could use them in the quadratic equation to find the roots.  When I studied math, that was never a question that came up.  We accepted it.

I've been exploring the idea that quadratics can represent area but during my search to find more information, I stumbled across this Geogebra applet that is awesome.   It allows students to play with the values of a, b, and c individually so they can see what happens as each coefficient is changed.  I admit, I had so much fun playing with the applet that I had to show it to others.

The thing is, I had a really hard time finding the information I was looking for.  After a nice long search, I found that Plus Maths organization has a list of 101 uses for quadratics which has lots of details to explain the formula, who came up with it, and what it represents.  The information has been spread out between two parts.

Part 1 begins back in Babylonia with their record keeping on tablets.  The first example is simply the area of a farm and what happens when you double the length of the side.  Then the author moves on to the Greeks,  the Pythagorean Theorem, and the Golden Ratio.  From here, the Greeks and conic sections are discussed and includes illustrations showing how each conic section is cut out of the original cone.

Part 2 continues the history lesson but brings it up to modern times with more modern applications. It begins with a discussion of Galileo and his accomplishments.  The connection between his discoveries and acceleration or stopping distance both of which use the quadratic.  From there the author shows the connection to ballistics.   Then as far as historical references, they moved on to Newton, his contributions and his works relation to quadratics.  They moved to Bernoulli and air travel.  The last two topics include chaos and mobile phones.

Every example includes the mathematics involved and explains why and how its a quadratic.  It would be easy to take the information out of this set of articles to create lessons in class or assign students to work on a presentation using the information provided.