I find graphical solutions to be the quickest method to approach many arduous tasks when a graphing utility is involved. Solve a system? Graph and locate the intersection. Find real roots? Graph and look for intercepts. Identify asymptotes? Graph and see where that weird squiggle is.
My honors Pre-Calculus students on the other hand, are not convinced. They thought I was torturing them when I asked them to graph to find real roots. I knew they studied all the other methods of solving polynomials last year and this year I wanted to focus on the more interesting task of complex roots. Several students announced they would rather do synthetic division than pull roots off a graph. I consider that torture! But even with graphing calculators they had such a hard time estimating the values of intercepts, and I promised to only give integer solutions so they knew to round.
We have since moved on to rational functions, yet the same issues are still cropping up. On the test I’m grading, one question is: “Describe the graph of Include domain, range, holes and asymptotes (vertical and horizontal).” It is certainly a question that involves synthesizing a number of ideas, but they have graphing calculators, are good at factoring and we did plenty of problems just like this. I shouldn’t be getting multiple vertical asymptotes, domains like -6<x<6 or ranges of y<1. I don't know what to do.
How do you help students pull important information out of a graph?