Functions, Graphs, and Limits
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.