Skinny or Tall Parabola

Post date: Jun 16, 2012 4:22:50 AM

For y=a*x^2, what happens when 'a' is greater than one and increases? Prompted by a discussion on the AP Calculus electronic discussion group I started the following story. Also enjoy the TI-Nspire file to make the story more 'hands-on' interactive.  Parent Parabola is now available with the recently updated TI-Nspire Player.  

Chapter 1 (In a world defined from -2≤x≤2) Parent Parabola had a daughter. When she was d(x)=1/2 x^2, she was shorter than her parent, but as soon as she was older than 1 she was taller. In fact, when she was d(x)=2x^2, she was twice as tall as her parent.  One day the Daughter felt like a zero. She would not get up, but she stayed horizontal in bed all day long. Then for a while, this daughter had a really negative attitude; this caused her to always wear a frown. But oh, the smiles and rejoicing that both Parent and Daughter had when she became positive again!

Chapter 2 (See page 2.2 of the TI-Nspire file and/or the image below)- Eventually the daughter moved away from home, or where she originated.  Some thought it confusing, but when her address was d(x)=(x-3)^2 she had moved to the right 3 "blocks." This makes sense if you consider ...

 (Okay, I'm ready for help. I can explain it, but it isn't funny nor does it fit in with a nice story. If you ask yourself what is x if d(x)=0, then x-3=0 gives us that the x-value must be 3.)

There was a great activity I did recently in a Laying the Foundation module using graph paper and beads where we had a function defined on a limited domain of say, -2,-1,0,1,2. The beads were placed on the x-axis and then each was physically moved up or down to the appropriate location based on the rule given by the function. There are many cool things that can be done with this that students will physically see. The one that pertains to y=a*x^2 is instead of getting skinnier, a point moves higher as 'a' increases and is greater than 1. This 'a' gives a vertical scale factor.