332: Which values of x do we choose? Rational functions

For the following functions, think about "How to graph like a math teacher." Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.

• Which points are best found by inspection?
• Which points are best found by substitution?
• Which points are best found by symmetry?

$y = \dfrac{4}{(x+2)(x+11)}$

$y = \dfrac{1}{x^2+1}$

$y = \dfrac{1}{x^-1}$

$ y = \dfrac{x+2}{x^2-4}$

$y=\dfrac{x^2-1}{x^2+1}$

What are your favorite examples of this?