Recently I read a post by Bob Lochel (low-shell).  His stuff is great and he is one of the desmasters I’d turn to with any burning questions or wonderings.  He brings to light some simple ideas that have been playing in my head recently.  Let’s ask the bigger question:

What do these even look like?

I feel like I’ve often looked at all of these relationships as completely different, nearly unrelated.  The bigger picture gives us two expressions, each on the side of a comparison symbol I remember from undergrad referred to as the trichotomy.  Each of these expressions can be input into @desmos, and then the greater, equal, lesser relationship can be visualized.  I’ve put together a handful here:

Absolute Value vs Constant a|x-h|+k [?] c
Linear Vs Constant ax+b [?] c
Linear Vs Linear ax+b [?] cx+d
Quadratic Vs Constant a(x-h)²+k [?] c
Cubic Vs Constant (x-h₁)(x-h₂)(x-h₃)+k [?] d

Some Previews:

I don’t want to shy away from the algebraic process in articulating where these relationships are precisely defined.  I’m just a fan of see what you’re doing.  I’ve come to understand that students Believe it when they See it, and Own it when they Control it.  Supporting an answer with visual information only strengthens the understanding.  Desmos and Geogebra have make this possible.