We now have enough tools to draw some complicated functions from scratch. Now we know how graphing calculators do it, and why they require the energy of four triple-A's.
When drawing a rational function f(x) from scratch, we need to know a lot of information, which can be nicely grouped into three big chunks.
- We need to know where f has vertical asymptotes and/or holes.
- We need to know the horizontal/slant/curvilinear asymptotes of f, if any.
- We need to know about values of f. We found where f is undefined when we found the vertical asymptotes and holes; now we need to know where f(x) is 0, positive, and negative. We also want to know f(0), also called the y-intercept.