1. Chris Harrow, Hawken School casmusings@gmail.comcasmusings@gmail.com@chris_harrow cdharr@hawken.edu

2.  CAS & thinking of algebra and functions as objects to be manipulated  FAMILIES of functions ◦ Sliders on graphs

3.  Game-changer for me over a decade ago  USACAS-1: Speaker asked us about standard form quadratics ◦ We all know what happens when a & c change. ◦ What happens when you change b? ◦ Can you prove it?  Nspire file: Quad Explore

4.  Don’t think of as a fixed transformation. Imagine a dynamic, oscillating, bouncing curve  Ceilings and Floors  Apply this to graphs of polar functions  Nspire file: Intro Polar

5.  We had a more dynamic understanding of polar graphing and explored  Sliders gave a surprise ◦ Looks almost like  Nspire file: Polar Fractions

6.  Convert to Cartesian  Slide ½ left  Convert back to Polar.  Are they the same?

8.  Surprisingly easy with a CAS

9.  So, for  Is ?

10.  Can all four conics show with only  Assume no rotations and only horizontal.  Nspire file: Hidden Conic Behavior

12.  Complete the square to identify center and stretch  Let  Center is

14.  Either way, the distance from the center to the foci is  The foci are at

16.  You can show that ◦ When the original was a horizontal ellipse (A<C and same sign), so are these denominators making this focus graph a hyperbola. ◦ When the original was a horizontal hyperbola (A>C and opposite signs), so are these denominators making this focus graph an ellipse ◦ QED.