So, yesterday…yes, Christmas…I proved (or perhaps, better said “derived”) the Quadratic Formula. The fun didn’t stop there. 🙂

My oldest son (visiting for the holiday), seemed to enjoy what I was doing…so I figured I’d blow his mind with how polar equations act in graphs, first showing him a simple \(r = 2\sin \theta\) one. After a bit discussing that, I thought I’d blow his mind more and create the equivalent of \(x = 1\), which is \(r = \frac{1}{{\cos \theta }}\).

Having a brain cramp, I thought I’d shift it one by adding 1 to it (\(r = \frac{1}{{\cos \theta }} + 1\)), but that results in something quite weird. You see, in a \(g(x) = f(x) + 1\) world, \(g(x)\) is just \(f(x)\) shifted up 1. Not so in the polar equation world. 🙂

The green line is \(r = \frac{1}{{\cos \theta }}\). The red lines are what happened when I added one to it (\(r = \frac{1}{{\cos \theta }} + 1\)): [Read more…]