328: Which values of x do we choose? Conics

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?

$x^2+y^2=25$ 

$\dfrac{(x-1)^2}{9}+\dfrac{(y+2)^2}{16} = 1$

$\dfrac{(x-4)^2}{9}-\dfrac{(y-4)^2}{16} = 1$

What are your favorite examples of this?